# – 2008 Sample Paper – X Class

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– 2008 Sample Paper – X Class

Sample Paper – 2008 Class – X Subject - Mathematics Roll. No. Code. No.- Time: 3 hrs Maximum Marks: 80 General Instructions: 1. All questions are compulsory. 2. The question paper consists of 30 questions divided into four sections – A, B, C and D. Section A contains 10 questions of 1 mark each, Section B is of 5 questions of 2 marks each, Section C is of 10 questions of 3 marks each and section D is of 5 questions of 6 marks each. 3. There is no overall choice. However, an internal choice has been provided in one question of two marks each, three questions of three marks each and two questions of six marks each. 4. In question on construction, the drawing should be neat and exactly as per the given measurements 5. Use of calculator is not permitted. SECTION –A [1 mark each] 1. Explain 7 1113 13 and 7 6 5 4 3 2 1 5 are composite numbers. 2 2. Sketch a rough graph for a quadratic polynomial ax +bx+c, which has no zeroes. 3. What is the solution of the system of equation a1x+b1y+c1 = 0 and a2x+ b2y+c2 = 0, if the graph of the equation intersect each other at point P (a , b). 2 4. If one root of the quadratic equation 5 x +px – 4 =0 is 4, find the value of p. 5. Is the sequence –1, –1, –1 …in an A.P, why? 6. Ratio of the areas of two similar triangles is 4 : 5, if one of the side of first triangle is 10 cm. Find the length of the corresponding side of second triangle. 7. If the centroid of triangle ABC with A(3x+5, –y–2),B(2x–1,4y+3), C(2–x,6y+5) coincides with origin. Determine the coordinates of centroid. 8. If shadow of a pillar is 3 times of its height, find the altitude of sun. 0 9. If tanA= cotB, where A and B are acute angles and (A – B) is 30 . Calculate the value of A and B. 3 2 10. What is the relationship between sum of zeros of a cubic polynomial ax +bx +cx+d (where a 0) and its coefficients? SECTION – B [2 marks each] 11. State the fundamental theorem of arithmetic and factorize the 9240 by using it. (or) State Euclid‟s division algorithm, find the HCF of 18864 and 6075. sin 2 630 sin 2 27 0 sin25 0. cos65 0 cos25 0. sin65 0 sin 30 0 2 0 2 cos 17 cos 73 12. Evaluate: 2 2 2 2 13. In triangle ABC, if AD ┴ BC, prove that AB + CD = BD +AC 4 3 3 y 14 4 y 23 14. Solve for x and y: x ; x . Where x 0 (or) For which value(s) of „a‟ and „b‟, the following pair of linear equations have infinite number of solutions 2x + 3y = 7; (a – b)x + (a+b)y = 3a + b – 2. 3 2 15. If α, β and γ are roots of the equation x –3 x – x + 3 = 0, then find (i) α + β + γ (ii) αβγ SECTION – C [3marks each] 16. If SinΦ + CosΦ = CosΦ, then show that CosΦ – SinΦ = SinΦ. 17. Prove analytically that area of the triangle formed by joining mid points of the sides of the triangle is one-fourth of the area of the given triangle, where vertices of the triangle are (0, -1), (2, 1) and (0, 3). (or) Analytically prove that mid point of hypotenuse is equidistance from all the three vertices of a right triangle. 18. ABC and DBC are two triangles on the same BC such that A and D are on the opposite side of the BC. If AD intersects BC at O. show that (or) 2 In an obtuse triangle, obtuse angled at B, if AD is perpendicular to CB produced, prove that AC = 2 2 AB +BC +2BC.BD. 2 th 19. Sum of first „n‟ terms of an AP is given by n + 8n. Find the nth term of the AP. Also find its 100 term. 1 Chandan’s Sample Papers : Sure Success and Excellent Marks (or) The angles of a triangle are in an AP. The least angle being half the greatest. Find the angles. 2 20. Determine value(s) of „p‟ for which the quadratic equation 4x -3px + 9 = 0 has real roots. ; bx –ay + 2ab = 0 21. Solve the system of equations: 4 3 2 22. Find all zeros of the polynomial x +x -9x –3x+18, if it is given that two of its zero as are – and 23. Show that one and only one out of n, n+2 or n+4 is divisible by 3. Where „n‟ is any positive integer. . 24. Prove that 5 – is an irrational number. 25. In what ratio does the x-axis divide the line segment joining the points (2,-3) and (5, 6) SECTION – D [6marks each] 2 2 2 2 26. a) if x = r sinAcosC , y = r sinAsin C and z = r cosA. Then prove that r = x +y +z . b) In the given figure ABCD is a rectangle in which segments AP and AQ are drawn as shown. If AB 0 0 = 60m, BC = 30m, AQD = 30 and PAB = 60 .Find the length of (AP + AQ). Q D C P A 27. 28. 29. 30. B (or) If the angle of elevation of a could from a point „h‟ meters above a lake is „‟ and the angle of h tan tan depression of its reflection in the lake is „β‟, prove that the height of the cloud is tan - tan . a) The two opposite vertices of a square are (–1, 2) and (3, 2), find the co-ordinates of the other two vertices. b) Find the value(s) of y for which the distance between the points P(2,-3) and Q (10,y) is 10 units. In a potato race, a bucket is placed at the starting point, which is 5m from the first potato and other potatoes are placed 3m apart in a straight line. There are ten potatoes in a line. A competitor starts from the bucket, picks up the nearest potato, runs back to pick up the next potato, runs to the buckets to drop it in and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run? A motorboat takes 6 hours to cover 100km down stream and 30km upstream. If the motorboat goes 75km down stream and returns back to its starting point in 8hours, find the speed of the motorboat in still water and the rate of the stream. Prove that the ratio of the areas of two similar triangles is equal to the ratio of the square of any corresponding sides. Using this theorem prove that ∆ ABC = area ∆ DEF. If ∆ ABC ∆ DEF and area . (or) Sate and prove converse of Pythagoras theorem. In the following figure, AB||CD|| EF, if AB = 6cm, CD = x cm, EF = 12cm, BD = 4cm and DE = y cm. Find x and y. F B 4 6 D x A C 1 2 y E “A great teacher never strives to explain his vision. He simply invites you to stand beside him and see for yourself.” ___________________________________________________________________________________ 2 Chandan’s Sample Papers : Sure Success and Excellent Marks Sample Paper – 2 SECTION –A [1mark each] 13 1. Without actually performing the long division, state whether will have a terminating decimal 3125 expression or a non terminating repeating decimal expression. 2. How many zeroes can a polynomial of degree „n‟ have? 3. Without drawing graphs, of the system of equations:x+2y =4 and 2x+4y =12determine the natures of the graph. 2 4. At what condition, a quadratic equation ax +bx+c = 0 has real roots. 5. What is common difference of the AP: a+2b, a, a – 2b … … … 6. If ratio of areas of two equilateral triangles is 25:49, then what is ratio of their corresponding sides? 7. If AM is the median of triangle ABC with A(x , y), B(x , x–3), C(y–9, –y) and M coincides with origin then find the coordinates of A. 8. If sin 63 - 1 , then find the value of K. k sec 2 2 2 9. There are two points „A‟ and „B‟ on the ground, which are at distances x units and y units from the 0 0 foot of the tower. The angle of elevation of the top of the tower at points „A‟ and „B‟ are 57 and 33 respectively. If the line AB passes though the foot of the tower, then find the height of the tower . 10. What is the zero of a linear polynomial ax + b where a 0? SECTION – B [2marks each] n 11. Prove that unit place digit of 6 cannot be zero. 12. Find the value of „k‟ for which the points (8,1), (k, 4), (2, –5) are collinear. th 13. Two AP‟s have the same common difference. The difference between their 100 terms is 100, what th is difference between 100000 terms? 14. Find the value(s) of „k‟ for which ky(y–2) + 6 = 0 has two equal roots. (or) 2 Find the real roots of the equation 4x +3x+5 = 0 by the method of completing the square. 15. If A, B, and are interior angles of a triangle ABC, then show that sin SECTION – C 16. If secθ = x + ; prove that secθ + tanθ = 2x or [3marks each] . (or) If a cosθ – b sinθ = c, Prove that a sinθ – b cosθ = a b c 17. The line segment joining the points (3, – 4) and (1, 2) is trisected at the points P and Q. If the co2 ordinates of P and Q are (p, –2) and 2 2 respectively, find the values of „p‟ and „q‟. 18. In ∆ABC, D and E are two points lying on the side AB such that AD = BE. If DP || BC and EQ || AC, then prove that PQ || AB. (or) In a right-angled triangle ABC, right angled at C, a point D is taken on AB. Prove that 1 1 1 2 2 AC BC CD 2 19. How many terms of the AP; 72, 69, 66 … … … … make the sum 897? Explain the double answer. 20. Sum of two numbers „a‟ and „b‟ is 15 and sum of their reciprocals is 3 . Find the numbers „a‟ and „b‟. 10 (or) Find the values of „‟ and „β‟ for which the pair of liner equations 2x+3y = 7, 2x + (+β)y = 28 has infinite number of solutions. 21. Solve the following system of equation. 2(ax-by) + (a+4b) = 0; 2(bx+ay) + (b - 4a) = 0 22. Find a cubic polynomial with the sum of zeros, sum of its zeros taken two at a time and product of its zeros are 4, 1 and –6 respectively. 23. Show that one and only one out of n, n+3, n+6 or n+9 is divisible by 4. 2 24. If a prime number „p‟ divides a , then „p‟ divides „a‟, where „a‟ is positive Integer. 3 Chandan’s Sample Papers : Sure Success and Excellent Marks 25.Prove analytically that a median of a triangle divides it into two triangles of equal areas. If AM is the median of the given triangle ABC with vertices A (4,-6), B (3,-2) and C (5, 2). SECTION – D [6marks each] 26. A round balloon of radius „r‟ subtends an angle α at the eye of the observer while the angle of elevation of its center is β. Prove that the height of the center of the balloon is r sinβ cosec . (or) The angle of elevation of a cliff from a fixed point is θ. After going up a distance of „k‟ meters towards the top of the cliff at angle of . It is found that angle of elevation is show that the height of the cliff is k(cos - sin cot ) by using above result, answer the following. (cot - cot ) 0 At the foot of a mountain, the elevation of summit is 45 . After ascending 1 km to wards the elevation 0 0 up an incline of 30 , the elevation changes to 60 find the height of the mountain. 27. Two water taps together fill a tank in 9 hours. The tap of larger diameter takes 10hours less than the smaller one to fill the tanks separately. Find the time in which each tap can separately fill the tank. (or) (a) Solve: (x+2) (x-5) (x-6)(x+1) =144. (b) Solve: = x+2 28. (a)State and prove Pythagoras theorem. Use this theorem to answer the Following: 2 2 In an equilateral triangle ABC, D is a point on side BC such that BD= BC, Prove that 9AD =7AB . 29.State and prove basic proportionality theorem. Use this theorem to find „x‟, from the given figure, where ABCD is a trapezium and AB|| DC, AO = 3x–1, OC = 5x–3, BO= 2x+1 and OD = 6x–5 A B O D C 30. (a) Solve graphically the pair of equation 2x – y = 2; 4x – 4 = 8. (b) Find „a‟ if y = ax + 15 Use graph to answer the following: (c) Write the co-ordinate of point where the lines meet the x-axis. (d) Find the co-ordinates of the vertices of the triangle formed by these two lines and x –axis. (e) Shade the above triangle and find its area. “Success or achievement is not the final goal. It is 'spirit' in which you act that puts the seal of beauty upon your life.” : Swami Chinmayananda ---------------------------------------------- Sample Paper- 3 SECTION - A [1 marks each] 2 1. Find the discriminant of the quadratic equation by factorization: 2x + x – 6= 0 2. Given H.C.F ( 96, 404 ) = 4, find L.C.M (96, 404 ) 3. Write the condition for which the given pair of equations has unique solutions. ax+by+c=0; Ax+By+d=0 th 4. Find the 10 term of the AP: 2,7,12… 5. Evaluate: cos 700 cos550 cos ec350 Sin200 tan 500 tan 250 tan 450 tan 650 tan 850 4 Chandan’s Sample Papers : Sure Success and Excellent Marks 6. In fig. DE // BC. Find EC A 3cm 6c m D B 2c m E C 7. The length of a tangent from a point A at a distance of 5 cm from the center of the circle is 4 cm. What will be the radius of the circle? 8. A card is drawn at random from a well-shuffled deck of 52 cards. Find the probability that it is neither an ace nor a queen. 9. The minute hand of a clock is 10 cm. long. Find the area on the face of the Clock described by the minute hand in 10 minutes. 10. Find the value of „f‟, if the mean of the following distribution is 18. Class 11-13 13-15 15-17 19-21 21-23 23-25 f Frequency 7 6 9 5 4 SECTION B 3 3 3 3 3 11. Factorize p (q – r) + q (r – p) + r (p – q) 12. If [2 marks each] 3 3 tan = 3 Sin , Find the value of sin2 - Cos2. (or) Prove that sec A.(1 – sin A).( sec A + tan A) = 1. 13. Prove that (a, 0), (0, b) and (1, 10) are collinear if 1/a + 1/b =1 2 14. D is any point on the side BC of ABC such that ADC= BAC. Prove that CA = BC.CD 15. Two dice are thrown together. What is the probability that the sum of the numbers on the two faces is neither 9 nor 11. SECTION C [3 marks each] 16. Prove that 5 + 2 3 is irrational. 17 The sum of two numbers is 15. If the sum of their reciprocals is 3/10, find the two numbers. 18. Prove quadratic formula. 19 Solve 4x + 6y = 3xy, 8x + 9y = 5xy given (x y, x 0) 20. Prove that: cos A SinA CosA SinA 1 tan A 1 tan A (or) 2 Sin + cos = p and sec + cosec = q, then prove that q. (p –1) = 2.p 21. If the point (x, y) is equidistant from the points (a+b, b-a) and (a-b, a+b,),prove that bx=ay. (or) Determine the ratio in which the point (-6, a) divides the join of A (-3,-1) and B (-8, 9). Also find the value of a. 22. Find the area of the triangle ABC formed by joining the mid-points of the sides of the triangle whose vertices are A( 4, -6 ), B( 3, -2 ) and C ( 5, 2 ). 23. Construct a triangle similar to a given triangle ABC with its sides 3/5 th of the corresponding side of 0 0. triangle ABC .It is given that AB=5cm, angle B=60 and angle C=55 Write the steps of contraction also. 24. If triangle ABC is isosceles with AB= AC and C (o,r) is the incircle of triangle ABC touching BC at F, Prove that the point F bisects BC. 25. A well with 10m inside diameter is dug 14m deep Earth taken out of it is spread all around to a width of 5m to form an embankment. Find the height of embankment. 5 Chandan’s Sample Papers : Sure Success and Excellent Marks SECTION D [6 marks each] 26. The mean of the following frequency distribution is 57.6 and the sum of the observation is 50. Find the missing frequencies f1 and f2. Class 0-20 20-40 40-60 60-80 80-100 100-120 Frequency 7 f1 12 f2 8 5 27. Prove that the ratio of the area of two similar triangles is equal to the ratio of squares of their corresponding sides. Use the above in the following: Similar triangle ACD and ABE are constructed on sides AC and AB. Find the ratio between the areas of ABE and ACD. 28. A vertical tower is surmounted by a flagstaff of height h meters. At a point on the ground, the angles of elevation of the bottom and top of the flagstaff are and respectively. Prove that height of the tower is h tan tan tan (or) From a window (60 meters high above the ground) of a house in a street, the angles of elevation and 0 0 depression of the top and the foot of another house on opposite side of street are 60 and 45 respectively. Show that the height of the opposite house is 60(1+3) meters. 29. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder. If the height of the cylinder is 10 cm, and its base of radius 3.5cm, find the total surface area and volume of the article. (or) The diameters of the ends of a bucket 45 cm high are 56 cm and 14 cm. Find its volume, the curved surface area and the total surface area. (Use = 22/7) 30. A boat goes 35 km upstream and 55 Km downstream in 12 hrs. It can go 30Km upstream and 44 Km downstream in 10 hrs. Find the speed of the stream and that of the boat in still water. “To love and to be loved is the greatest happiness.” ___________________________________________________________________________________ Sample Paper – 4 SECTION A [1 marks each] 1. For what value of k the system of linear equations 2x + 5y = k, k x + 15y =18 has infinitely many solutions? 2 2 2. Areas of two similar triangles ABC and DEF are 64 cm and 121 cm . If EF=13.2 cm. what will be the length of BC? 3. A die is tossed once. What will be the probability of getting a number divisible by 3? 4. For what value of k the roots of the equation 2 x kx 6 0, are equal? 5. Which of the following is not a quadratic equation? 2 2 (b) x 2 x (2)(3 x) , ( x 1)2 2( x 3) , 3 2 3 (c) ( x 2)( x 1) ( x 1)( x 3) , (d) x 4 x x 1 ( x 1) (a) 6. How many terms between 12 to 99 are divisible by 3? 3 7. If the volume of a cube is 1728 cm . What will be the length of its edge? 8. The mean of 20 numbers is 15. If each number is multiplied by 7. Determine the new mean 3 . Find P (not E). 7 2 2 10. If x 2Sin and y 2Cos 1 . Calculate the value of x + y. 9. If E be an event such that P (E) = SECTION B [2 marks each] 11. Divide 3x x 3x 5 by x 1 x , and verify the division algorithm. 12. A ladder is placed against a wall such that its foot is at a distance of 4.5 m from the wall and its top reaches a window 6 m above the ground. Find the length of the ladder. 2 13. Evaluate: 3 2 5 sin2 30 cos 2 45 4 tan2 60 2 sin 30 cos 60 tan 45 (or) Without using trigonometric table Evaluate: tan 5 tan10 tan15 tan 45 tan80 tan 75 tan85 4 2 14. Solve the following equation: x - 25x + 144 = 0 6 Chandan’s Sample Papers : Sure Success and Excellent Marks (or) Find the sum and product of the roots of the following equation without actually solving it: 3 ax2 = –10ax – 7 3 15. How many balls, each of radius 0.5 cm, can be made from a solid sphere of metal of radius 10 cm by melting the sphere? SECTION C th [3 marks each] th th 16. If m times the m term of an A.P. is equal to n times its n term, prove that the (m +n) term of the A.P. is zero. (or) Find the sum of the following A.P. 109 + 104 + 99 + ……………………+ (-6) 17. Determine graphically whether the given pair of equations is consistent or not.3x – 5y = 1; 2x – y = –3 If consistent, find the solution of the above equations. 18. Prove that: 1 sin A (sec A tan A) 2 1 sin A (or) Prove that sec tan 1 1 sin tan sec 1 cos 19. In the adjacent figure, four circles of radius 3.5 cm each touch other externally. Find the area of the shaded region. each 20. Prove that any line parallel to parallel sides of a trapezium divides the non-parallel sides proportionally (i.e. in the same ratio) 21. A bag contains 7 red balls, 8 white balls, 3 green balls and 4 blue balls. One ball is drawn at random. Find the probability that the ball is: (a) white (b) red or green (c) not green (d) red or white 22. Solve for u and v: 4 u – v = 14 u v; 3 u + 2 v = 16 u v. 23. In triangle ABC, BC = 5.5 cm, AB = 4.6 cm, B = 60 . Construct a triangle A' BC ' similar to ABC, th who‟s each side is 4 of the corresponding sides of the triangle ABC. 5 24. Calculate the median and mode of the following distribution? C.I. f 20-30 14 30-40 34 40-50 15 50-60 37 60-70 18 70-80 5 80-90 7 25. In the adjacent figure, radius of The circle is 21 cm, if AOB 135 , Find the area of shaded of region. O A B 7 Chandan’s Sample Papers : Sure Success and Excellent Marks SECTION D [6 marks each] 26. Prove the ratio of the areas of two similar triangles is equal to the ratio of the square of their corresponding sides. Using this theorem, prove that the ratio of the areas of equilateral triangles formed on the side and diagonal of a square is 1:2. (or) State and prove “Pythagoras Theorem”. Prove that the sum of the square of the diagonals of a rhombus is equal to the sum of the square of its sides. 0 27. From an aeroplane 1000 m high, a man observes the angles of depression of two ships to be 60 and 0 45 . If the ships are on the opposite sides of the observer, find the distance between the ships. 28. The mean of the following frequency distribution is 26.5. If the total number of observations is 100, find the frequencies f1 and f2. Classes 0-10 11-20 21-30 31-40 41-50 Frequency f1 14 21 f2 22 29. A bucket of height 16 cm and made up of metal sheet is in the form of frustum of a right circular cone with radii of its lower and upper ends as 6 cm and 15 cm respectively. Calculate: (i) the height of the cone of which the bucket is a part. (ii) The volume of water which can be filled in the bucket. (iii) The slant height of the bucket. (iv) The area of the metal sheet required to make the bucket. 30. A and B jointly finish a piece of work in 15 days. When they work separately, A takes 16 days less than the number of days spent by B to finish the same piece of work. Find the number of days taken by B to finish the work. “Efficiency is the capacity to bring proficiency into expression”. -Swami Chinmayananda __________________________________ Sample Paper – 5 SECTION A [1 marks each] 2 1. For what value of k, does the equation 3x – 4 kx + 12 = 0 have equal roots? 2. Sides of two similar triangles are in the ratio 4: 9. What are the Areas of these triangles? 3. Evaluate 2 tan 30 0 ? 1 tan 2 30 0 4. Find the value of k, if the points (2, 3), (6, - 3) and (4, k) are collinear. 5. In the figure given below, AQ and AR are a tangent to the circle drawn from an external point A. CB is third tangent touching the circle at P. If AQ=15 cm, and CP=4cm.What is the length of AC. 6. A pair of dice is tossed once. What is the probability of getting a doublet (same number on both dice)? 7. For the polynomial 3x 5 x 1 , what is the sum of zeros? 8. Find the condition that if the linear equations lx my n and ax by c have unique solution. 9. State the Euclid‟s Division Lemma. 10. A right circular cylinder is shown in figure which encloses a sphere of radius r. Find the curved surface area of the cylinder. 2 8 Chandan’s Sample Papers : Sure Success and Excellent Marks SECTION B [2 marks each] 0 0 sin 50 cos ec 40 4 cos 50 0 cos ec 40 0 0 0 cos 40 sec 50 12. In given figure A, B and C are points on OP, OQ and OR respectively such that AB PQ and 11. Without using trigonometric table, evaluate: AC PR. Show that BC QR. 13. What is the probability that a leap year contains (a) 53 Sundays (b) 53 Sundays and 53 Mondays? (or) Two coins are tossed simultaneously. Find the probability of getting (a) two heads (b) at least one head. 14. If the distances of (x, y) from (5, 1) and (- 1, 5) are equal, prove that 3x = 2y 15. Find the sum of all numbers between 700 and 950 which are multiple of 8. SECTION C [3 marks each] 7 is an irrational numbers. 16. Prove that 17. In given figure XY and X Y are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersects XY at A and X Y at B. Prove that AOB 90 0 (or) In given figure PQR is a right angled triangle with PQ = 12 cm and QR = 5 cm. A circle with centre O and radius x is inscribed in PQR . Find the value of x. x 3 3x 2 x 2 by a polynomial g (x) , the quotient and remainder were x 2 and 2 x 4, respectively. Find g (x) . 18. On dividing 19. If cosθ + sinθ = √2 cosθ , show that cosθ – sin θ = √2 sinθ (or) 2 2 4 If sinθ + sin θ = 1, prove that cos θ + cos θ = 1 20. Solve x and y: 6( ax + by ) = 3a + 2b ; 6( bx – ay ) = 3b – 2a (or) 9 Chandan’s Sample Papers : Sure Success and Excellent Marks A man travels 370km partly by train and partly by car. If he covers 250km by train and the rest by car, it takes him 4 hours. But if he covers 130km by train and the rest by car, it takes him 18 minutes longer. Find the speed of the train and that of the car. 21. In given figure, ABCD is a square whose each side is 14 cm. APD and BPC are semicircles. Find the area of the shaded region. th 22. Which term of the A.P 1, 10, 19, 28, 37, ……is 153 more than its 25 term? 2 23. Solve by using Quadratic formula: 4x + 2(b – 3a)x – 3ab = 0 24. Use section formula to show that A (4, 6), B (7, 7), C (10, 10) and D (7, 9) are the vertices of a parallelogram. 25. Construct a triangle with sides 5 cm, 6 cm and 7 cm and then another triangle whose sides are 7/5 of the corresponding sides of the first triangle. SECTION D [6 marks each] 26. From a point on the ground 40m away from the foot of a tower, the angle of elevation of the top of 0 0 the tower is 30 . The angle of elevation of the top of a water tank (on the top of the tower) is 45 . Find the height of the tower and the depth of the tank. 27. From a point on the ground 40m away from the foot of a tower, the angle of elevation of the top of 0 0 the tower is 30 . The angle of elevation of the top of a water tank (on the top of the tower) is 45 . Find the height of the tower and the depth of the tank. (or) A 10m high flagstaff is fixed on the top of a tower .the angle of elevation of the top of the flag-staff as 0 observed from o point P on the ground is 60 ,the angle of depression of the point P from the top of 0 the tower is 45 . Find the height of the tower. 27. A metallic bucket is in the shape of a frustum of a cone mounted on a hollow cylindrical base given in the figure. If the diameters of two circulars ends of the bucket are 45cm and 25 cm, total vertical height is 30 cm and that of the cylindrical portion is 6 cm, find the area of the metallic sheet used to make the bucket. Also find the volume of water it can hold. (or) A toy is in the shape of a right circular cylinder with a hemisphere on one end and a cone on the other .The height and radius of the cylindrical part are 13cm and 5cm respectively. The radii of the hemispherical and conical parts are the same as that of the cylindrical part .Calculate the surface area of the toy if the height of the conical part is 12cm. (Л = 22/7). 28. Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of the corresponding sides. Using the above theorem, prove that the area of the equilateral triangle described on the side of a square is half the area of the equilateral triangle described on its diagonal. 29. If the mean of the following frequency distribution is 53, find the value of „f 1 and f2‟ Class 0-20 20-40 40-60 60-80 80-100 Total F 15 F1 21 F2 17 100 10 Chandan’s Sample Papers : Sure Success and Excellent Marks 30. Ramesh travels 300 km to his home partly by train and partly by bus. He takes 4 hours, if he travels 60 km by train and rest by bus .If he travels 100 Km by train and rest by a bus, he takes 10 minutes longer. Find speeds of train and the bus. “Wisdom is the assimilated knowledge in us, gained from an intelligent estimation and close study of our own direct and indirect experience in the world.” _______________________________________________________________ Sample Paper – 6 SECTION - A 1. Sate the Euclid‟s division lemma. 2. The graph of y= f(x) is given below. Find f(x). (10x1=10) Y X‟ -4 -1 2 X Y‟ 2 3. On dividing x + 7x + 3 by a polynomial g(x) the quotient and remainder were x+5 and -7 respectively. Find g(x). 4. What is the nature of roots of the quadratic equation x + 1 = 3? x 5. In the adjoining figure OACB is a quadrant of a circle with centre O and radius 7cm. If OD = 4cm, find the area of the shaded region O D DD B C A 6. In ∆ABC, AB = 6 √3, AC= 12cm and BC= 6cm. Find the angle B. 7. Write down the empirical relationship between the three measures of central tendency. 8. One card is drawn from a well-shuffled deck of 52 cards. Calculate the probability that the cards will not be an ace 0 9. Two tangents TP and TQ are drawn to circle with centre O from an external point T, and ∟ PTQ = 60 , find ∟OPQ. P O T Q 11 Chandan’s Sample Papers : Sure Success and Excellent Marks 10. Find the probability of getting 53 Tuesdays in a leap year. SECTION B 11. How many two-digit numbers are divisible by 7? 0 12. Evaluate sin70 + tan10 tan40 tan50 tan80 0 0 cos 20 2 cos 43 cosec 47 (5x2=10) 13. In the figure, ABCD is a trapezium in which AB || DC and 2AB = 3CD. Find the ratio of the areas of ∆AOB and ∆COD. A B O C C C D 14. Find the ratio in which the line segment joining the points (6, 4) and (1, -7) is divided by x-axis. 15. Cards numbered 3,4,5,6 ………, 17 are put in a box and mixed thoroughly. A card is drawn at random from the box. Find the probability that the card drawn bears (i) A number divisible by 3 or 5 (ii) A number divisible by 3 and 5. SECTION C (10x3=30) 2 16. Find the zeros of the quadratic polynomial x + 7x + 10 and verify the relationship between the zeros and the coefficients. (or) 4 3 2 Find all the zeroes of x – 5x + 3x + 15x -18, if two of its zeroes are √ 3 and -√3. 17. Prove that 7√ 5 is irrational. (or) Explain why 7 x11 x 13 + 13 and 7 x 6 x 5 x 4 x3 x 2 x 1 + 5 are composite numbers. 18. For which value of k will the following pair of linear equations have no solution? 3x + y = 1 ; (x-1) 2k – 1(x + y) = 1 – ky. (or) Solve : 6x + 3y = 6xy 2x + 4y = 5xy. 19. Determine the A.P whose 5th term is 15 and the sum of its 3rd and 8th terms is 34. (or) Find the sum of all three digit numbers which leave the remainder 2 when divided by 7. 20. Prove that cosA – sinA + 1 = cosecA + cotA cosA + sinA – 1 21. The line joining the points (2, 1) & (5, -8) is trisected at the points P & Q. If the point P lies on the line 2x – y +k = 0, Find the value of k. 22. If the points p(x, y) is equidistant from the points A(5, 1) and B(-1, 5), prove that x = 2. y 3 (or) Show that the points (-3, 2) (1, -2) & (9, -10) can never be the vertices of a triangle. 0 23. Draw a pair of tangents to a circle of radius 5cm which are inclines to each other at an angle of 60 . 24. In an equivalateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes. 25. Find the area of the designed region in fig given below between the two quadrants of radius 7cm each 12 Chandan’s Sample Papers : Sure Success and Excellent Marks SECTION D (5x6=30) 26. The cost of 5 oranges and 3 apples is Rs 25 and the cost of 3 oranges and 4 Apples are Rs 26, find the cost of an orange and an apple graphically. 27. In a triangle, if the square on one side is equal to the sum of the squares on the other two sides, prove that the angle, opposite to the first side is a right angle. Use the above theorem and prove the 2 2 2 following. In a ∆ABC, AD ┴ BC and BD = 3CD. Prove that 2AB = 2AC + BC (or) 2 2 In an equilateral triangle ABC, D is a point on side BC such that BD = 1 BC. Prove that 9AD = 7AB . 3 28. A man is standing on the deck of a ship, which is 8cm above of the water level. He observes the º angle of the elevation of the top of the hill as 60 and the angle of depression of the base of the hill as º 30 . Calculate the height of the hill from the water level. (or) º The angle of elevation of the top of a tower from a point A on the ground is 30 . On moving a distance º of 20m towards the foot of the tower to a point B, the angle of elevation increases to 60 . Find the height of the tower. 29. A farmer connects a pipe of internal diameter 20cm from a canal into a cylindrical tank in her field, which is 10m in diameter and 2m deep. if water flows through the pipe at the rate of 3km/hr, in how much time will the tank to be filled? 30. The median of the following data is 28.5.Find the missing frequencies x and y, if the total frequency is 60 Class interval 0-10 10-20 20-30 30-40 40-50 50-60 Frequency 5 X 20 15 Y 5 (or) Verify the relation Mode = 3median – 2 mean from the following data C.I 0-10 10-20 20-30 30-40 40-50 50-60 Frequency 5 8 20 15 7 5 Total 60 “You are successful and creative only when you see an opportunity in every difficulty.” ___________________________________________________________________________________ Sample Paper – 7 SECTION A [1 marks each] 1. For what value of k the system of linear equations: (k+1) x + 2y = 5, 3 x + (k-1)y =10 have unique solution? 2 2. The quadratic equation kx – 2 kx + 2 = 0 has real roots find the value of k. 3. In an A.P. if common difference is 3. Determine t5 – t7 . 4. In ABC , AD is bisector of A and Ad meets BC at D. If AB=5cm, AC=6cm and CD=3cm, find BC 5. A point P is 13 cm from the centre of a circle. If the radius of the circle is 5 cm, Calculate the length of the tangent drawn from P to the circle is: 6. Evaluate: Co sec39 2( Sin2 5 Sin2 85 ) . Sec51 7. Two cubes each of 10 cm edge are joined end to end. Find surface area of the resulting cuboids . 8. The mean of 30 numbers is 18. If 5 is added to every number. What will be the new mean? 9. The three vertices of a triangle are (4, 5), (6,8) and (8,1). Find the coordinates of its centroid . 10. Find the zeros of the quadratic polynomial x 2 7 x 12, SECTION B [2 marks each] 11. How many terms of the sequence 18, 16, 14… Should be taken so that their sum is zero. (or) 13 Chandan’s Sample Papers : Sure Success and Excellent Marks How many numbers between 20 and 200 are exactly divisible by 7? 12. By using Euclid‟s division algorithm, find the H.C.F. of the following numbers. 867 and 255. 13. If and are the zeroes of 3x 8 x 2, find the value of . 14. Show that the points (-1,-1), (2,3) and (8,11) are collinear. 15. Find the area of the triangle whose vertices are (2, 9), (-2, 1) and (6,3). 2 2 2 SECTION C 16. Show that [3 marks each] 5 is an irrational number. 17. Draw the graphs of the following equations: 2 x y 2 0; 4 x 3 y 24 0 Obtain the vertices of the triangle so obtained. Also determine the area. 18. Find the coordinates of the point equidistant from A(5,3), B(5,-5) and C(1,-5). (or) The line segment joining A(2,3) and B(6,-5) is intersected by the x-axis at a point P. Write down the coordinates of the point P. Find the ratio in which P divides AB. 19. A fast train takes 3 hours less than a slow train for a journey of 600 km. If the speed of the slow train is 10 km/hr less than that of the fast train, find the speeds of the two trains. 20. A bag contains 20 black, x white and y green balls. A ball is drawn at random from the bag. If the probability of getting a black ball is 4 and the probability of getting a white ball is twice the probability of 7 getting a green ball, find values of x and y. 21. A round table cover has six equal design as shown in the adjacent figure. If the radius of the cover is 28 cm, find the cost of making the design at the rate 2 of Rs 0.35 per cm . 1 3 tan 25 tan 40 tan 50 tan 65 tan 2 60 2 22. Evaluate: 4(Cos 2 29 Cos 2 61) (or) Prove that: (Cos Sec ) (Sin Co sec )2 7 tan 2 Cot 2 2 23. Construct a triangle similar to a given ABC such that each of its sides is 2 3 rd of the corresponding sides of ABC . It is given that BC = 6 cm, B 50and C 60 . 24. The perpendicular AD on the base BC of a ABC intersects BC at D so that DB = 3 CD. Prove that: 2 2 2 2AB = 2AC + BC . 25. A 90 cm tall boy is walking away from the base of a lamp post at a speed of 1.5m/s. If the lamp is 4.5m above the base, find the length of the shadow of the boy after 4 seconds. SECTION D [6 marks each] 26. Prove that the lengths of tangents drawn from an external point to a circle are equal. In the adjacent figure, PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length TP. 14 Chandan’s Sample Papers : Sure Success and Excellent Marks 27. Water flows through a circular pipe, whose internal diameter is 2 cm, at the rate of 0.7 m per second into a cylindrical tank, the radius of whose base is 40cm. By how much will the level of water in the cylindrical tank rise in half an hours (or) A bucket is in the form of a frustum of a cone. Its depth is 15 cm and the diameters of the top and bottom are 56 cm and 42 cm respectively. Find how many liters of water the bucket holds. Also find 2 the cost of the sheet used to make this bucket at the rate of Rs 0.75 per cm . 0 28. The angle of elevation of a cloud from a point 60m above a lake is 30 and the angle of depression of 0 the reflection of cloud in the lake is 60 . Find the height of the cloud. (or) 0 The angle of elevation of a jet plane from a point A on the ground is 60 . After a flight of 15 seconds, 0 the angle of elevation changes to 30 . If the jet plane is flying at a constant height of the speed of the jet plane. 29. Locate the median for the following distribution graphically and verify the results. 1500 3 m, find Income (in Rs) 0-30 30-40 40-50 50-60 60-70 70-100 frequency 10 15 30 32 8 5 30. A boat covers 32 km upstream and 36 km downstream in 7 hours. Also, it covers 40 km upstream and 48 km downstream in 9 hours. Find the speed of the boat in still water and that of the stream. “All disturbances and challenges rise not only from our relationship with others, but in our attitude to all other things and beings.” _______________________________________ Sample Paper-8 SECTION A [1 marks each] 1. Given H.C.F ( 306, 657 ) = 9, find L.C.M ( 306, 657 ) 2. Prove that 3 + 2 √5 is irrational. 3. For which value of „P‟ does the pair of equations has unique solutions. 4x + Py + 8 = 0 2x + 2y + 2 = 0 4. Find the discriminant of the quadratic equation 5. In fig. DE // BC. Find EC. A 1.5cm 1cm D E 3cm B C 6. Find the distance between the points (-5, 7) and (-1, 3). 7. Find the co-ordinates of the centre of a circle whose end points of the diameter are (3, -10) and (1, 4). 0 8 If tan 2A = cot ( A – 18 ), where 2A is an acute angle, find the value of A. 0 9. Find the length of the arc of a circle with radius 6cm if the angle of sector is 60 . 10. One card is drawn from a well shuffled deck of 52 cards. Calculate the probability that the card drawn will be an ace. SECTION B [2 marks each] 11.Use Euclid's division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m. (or) Show that any positive odd positive integer is of the form 8q + 1, or 8q + 3, 8q + 5, or 8q + 7,where q is some integer. 12. Solve 2x + 3y = 11 and 2x – 4y = - 24 and hence find the value of „m‟ for which y = mx + 3. 13. Solve: 1 1 3; x 0, 2 x x2 th 14. Find the 20 term from the last term of the A.P: 3, 8, 13….… 253. 15 Chandan’s Sample Papers : Sure Success and Excellent Marks 15. If cot θ = 7/8, evaluate (1 sin A)(1 sin A) (1 cos A)(1 cos A) SECTION C [3 marks each] 3 2 16. On dividing x – 3x + x + 2 by a polynomial g (x), the quotient and remainder where x–2 and –2x+4, respectively. Find g (x). 17. Solve the equation graphically x–y+1=0 and 3x+2y–12=0. determine the coordinates of the vertices of the triangle formed by these lines and the x – axis, and shade the triangular region. 18. A train travels 360 km at a uniform speed. If the speed had been 5km/hr more, it would have taken 1 hour less for the same journey. Find the speed of the train. (or) A motor boat whose speed is 18km/hr in still water takes 1 hour more to go 24km upstream than to return downstream to the same spot. Find the speed of the stream. 19.Find the sum of first 24 terms of the list numbers whose nth term is given by an = 3 + 2n. 20. Show that the points ( 1, 7 ), ( 4, 2 ), ( -1, -1 ) and ( -4, 4 ) are the vertices of a square. 21. Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are (0, -1), (2, 1) and (0, 3). 22. Prove the following identity: . (or) Without using trigonometric tables, evaluate the following: 0 0 2 0 2 0 0 0 0 cotθ.tan( 90 - θ ) – sec( 90 - θ ).cosecθ + sin 25 + sin 65 + √3( tan5 tan45 tan85 ) 23.Construct a triangle with sides 5cm, 6cm and 7cm, then another triangle whose sides are 7/5 of the corresponding sides of the first triangle. 0 24. A chord of a circle of radius 15cm subtends an angle of 60 at the centre. Find the area of the corresponding minor and major segments of the circle ( Use π = 3.14 and √3 = 1.73). (or) Find the area of the shaded region in figure, ABCD is a square of side 14 cm. 25 In the following frequency distribution, the frequency of the class –interval (40-50) is missing. It is known that the mean of the distribution is 52. Find the missing frequency. Wages No of workers 10-20 5 20-30 3 30-40 4 40-50 - 50-60 2 60-70 6 70-80 13 SECTION D 26. In a triangle, if the square of one side is equal to the sum of squares of the remaining two sides, prove that the angle opposite to the first side is a right angle. Using the above, do the following: ABC 2 2 is an isosceles triangle with AB = BC. If AB = 2AC , prove that ABC is a right triangle. 27.As observed from the top of a 75m high lighthouse from the sea-level, the angles of depression of 0 0 two ships are 30 and 45 . If one ship is exactly behind the other on the same side of the light house, find the distance between the two ships. (or) A tower is surmounted by a flag staff of height h. At a point on the plane, the angle of elevation of the bottom and top of the flag staff are α and β respectively. Prove that the height of the tower is h tan tan tan 28. Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. Use the above theorem, in the following, If the areas of two similar triangles are equal, prove that they are congruent triangles. 29. The radii of the ends of a frustum of a cone 45cm high are 28cm and 7cm. Find its volume and total surface area. (or) Water in a canal, 6m wide and 1.5m deep, is flowing with a speed of 10km/hr. how much area 16 Chandan’s Sample Papers : Sure Success and Excellent Marks will it irrigate in 30 minutes, if 8cm standing water is needed? 30. The distribution below gives the weights of 30 students of a class. Find the median weight of the students. Weight ( in kg ) Number of students 40 - 45 2 45 - 50 3 50 - 55 8 55 - 60 6 60 - 65 6 65 – 70 3 70 - 75 2 “If you’re not failing every now and again, it’s a sign you’re not doing anything very innovative” _________________________________________________ Sample Paper-9 SECTION A [1 marks each] 1. Write the condition to be satisfied by q, so that a rational number p has a non-terminating repeating q decimal expansion . 7 3 x 2 2 3 2 3. If , , are the zeroes of the cubic polynomial x 3x 5x 3 , find ; and b. cos A sin A 4. If sinA= 0.6, find cos A sin A th th 5. Find the difference between n term and the k term of an A.P whose first term is a and common difference is d . 2. Find the sum and the product of the quadratic polynomial 5 x 2 6. How many spherical bullets can be made out of a solid cube of lead whose edge measures 44cm,each bullet being 4cm in diameter. 7. In ∆ABC, D and E are the mid points of AB and AC respectively. Find the ratio of the areas of ∆ADE and ∆ABC. 8. A tangent PQ at a point P of a circle of radius 5cm meets a line through the centre O at a point so that OQ = 12cm. Find the length PQ. P OO Q O 9. It is given that in a group of 3 students, the probability of two students not having the same birthday is 0.992. What is the probability that the 2 students have the same birthday? 10. The wickets taken by a bowler in 10 cricket matches are as follows: 2,6,4,5,0,2,1,3,2,3. Find the mode of the data. SECTION – B 11. State the nature of the solutions of the following pair of linear equations: 3x + 5y – 8 = 0 8x – 11y + 3 = 0 12. Without using trigonometric tables, find the value of [2 marks each] cos 70 0 sec 70. cos ec 20 tan 70 cot 20 cos 57 cos ec33 sin 20 0 0 13. Read the values of y for which the distance between the points P(2, -3) and G(10,y) is 10 units. 14. 14. In the fig. if LM II CB and LN II CD, prove that AM AN AB AD B M A C L 17 Chandan’s N Sample Papers : Sure Success and Excellent Marks D 15. A bag contains 4 red, 5 black and 6 white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is ; a. White b. Red c. Not black d. Red or white. SECTION – C 16. Prove that [3 marks each] 3 is an irrational number. 17. If two zeroes of the polynomial 2 x 3x 3x 6 x 2 are 2 and 2 . Find the remaining zeroes of P(x); if any. 18. Draw the graph of the system of equations x + y = 5 and 2x – y + 2 = 0. Shade the region bounded by these lines and x axis. Find the area of the shaded region. th 19. Which term of the arithmetic progression 3,10,17,… will be 84 more than its 13 term? 4 20. 3 2 sin cos 1 1 sin cos 1 sin tan Prove that (or) tan A sec A 1 1 sin A tan A sec A 1 cos A Prove that 21. Find the area of the quadrilateral ABCD formed by the points A(-1,-2) B(1,0) C(-1,2) and D(-3,0). 22. The vertices of a triangle are A(3,4), B(7,2) and C(-2,-5). Find the length of the median through the vertex B. 23. Construct ∆ABC in which AB = 4cm, to ∆ABC such that B 1200 and BC = 5cm. Construct another ∆ ABC similar AB 5 AB . 4 24. In the adjoining figure, a circle touches all the four sides of a quadrilateral ABCD. Prove that AB + DC = AD + BC. B S A R R P C Q D 25. Find (i) the perimeter and (ii) area of the shaded region. If PQ=QR=RS=ST and PT= 28cm. P Q R S T 18 Chandan’s Sample Papers : Sure Success and Excellent Marks (or) Find the area of the unshaded region, the perimeter of the equilateral triangle is 42cm. SECTION – D [6 marks each] 26. Some students arranged a picnic. The budjet for food was Rs.600. Because five students of the group failed to go, the cost of each student got increased by Rs.4. How many students went for the picnic? 27. A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height h meters. At a point on the plane, the angle of elevation of the bottom of the flagstaff is α and that of the top of the flagstaff is β. Prove that the height of the tower is h tan mts. tan tan (or) The angles of elevation of the top of a tower from two points P and Q at distances of „a‟ and „b‟ respectively from the base and in the same straight line with it are complementary. Prove that the height of the tower is ab . 28. Find the missing frequencies in the following distribution. It is given that mean of the Frequency distribution is 50. Also find mode. Class 0 – 20 20 – 40 40 – 60 60 – 80 80 – 100 Total Frequency 17 F1 32 F2 19 120 29. Prove that the length of tangents drawn from an external point to a circle are equal. Using this theorem prove that, the opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the center of the circle. (or) Prove that the ratio of the similar triangle is equal to the square of ratio of their respective sides. Using the above, prove that area of equilateral triangle described on the side of a square is half of the equilateral triangle described on its diagonal. 30. A metallic bucket is in the shape of a frustum of a cone mounted on a hollow cylindrical base from the adjoining figure. If the diameter of two circular ends of the bucket are 35cm. and 15cm. the total vertical height is 20cm. and that of the cylindrical portion is 6cm, find the area of the metallic sheet used to make the bucket. Also find the volume of water it can hold. 35cm . 20cm. 6cm. “Teachers open the door. You enter by yourself.” ____________________________________________________ 19 Chandan’s Sample Papers : Sure Success and Excellent Marks SAMPLE PAPER - 10 SECTION: A [1 marks each] 1. If HCF of the numbers 306 and 657 is 9. Find LCM of the numbers. 2. Write down the general equation of cubic polynomial. 3. If the zeroes of a quadratic polynomial are -3 and 4. Find the polynomial. 4. Solve: 2x-y=1, x+2y=8 2 5. Discuss the nature of the roots of the equation: 2x +4x+1=0. 6. Find the sixth term of the A.P. 92, 75, 58, 41,……………… 7. A ladder 20 m long reaches a window of a house 16m above the ground. Determine the distance of the foot of the ladder from the house. 8. Find the co-ordinates of the mid point of the line segment joining the points (5,3) and (7,9). 9. Evaluate: Sin700 Co sec360 2Cos 430 C os ec470 Sin200 Sec540 tan100 tan 400 tan 500 tan 800 10. A card is drawn from an ordinary pack and a gambler bets that it is a spade or an ace. What are the odds against his winning this bet? SECTION B [2 marks each] 11. Find the value of k for which the points (3,2), (4,k) and (5,3) are collinear. 12. XP and XQ are two tangents to a circle with centre O from a point X outside the circle. ARB is tangent to circle at R. Prove that 13. Find the ratio in which the line segment joining (2, -3) and (5,6) is divided by x-axis. 14. Father's age is three times the sum of ages of his two children. After 5 years his age will be twice the sum of age of two children. Find the age of father. 15. Prove that: CosA Sin2 A SinA CosA 1 tan A CosA SinA (or) Without using trigonometric tables evaluate the following: SECTION C [3 marks each] 16. The distance between Mumbai and Pune is 192 km. Travelling by Deccan Queen, it takes 48 minutes less than another train. Calculate the speed of the Deccan Queen if the speed of the two trains differ by 20 km/hr. (or) 2 A farmer wishes to grow a 100 m rectangular vegetable garden. Since he has with him only 30 m barbed wire, he fences three sides of the rectangular garden letting compound wall of his house act as the fourth side-fence, Find the dimensions of his garden. 17. Find the circum centre of the triangle whose vertices are (-2 , -3 ), ( - 1 , 0 ) , ( 7, - 6 ). (or) Show that the points ( 2a , 4 a ) , ( 2a , 6 a ) and ( 2a + √3a , 5a ) are the vertices of an equilateral triangle. 18. Draw a circle of radius 3 cm. From any external point draw tangents to the circle without using its centre. 19. Solve the following system of equations graphically: 5 x – 6 y + 30 = 0, x + 4 y – 20 = 0. Also , find the vertices of triangle formed by the above two lines and the x-axis. 20. Find the probability of having 53 Sundays in a leap year and non leap year. 21. The mid point of the line segment joining (2a, 4) and (2, 3b) is (1, 2a +1). Find the values of a and b. 22. ABCD is a trapezium in which AB!! DC. the diagonals AC and BD intersect at o. prove that 23.Prove: AO BO CO DO 3 2 5 is an irrational number. 20 Chandan’s Sample Papers : Sure Success and Excellent Marks 24. A bag contains cards numbering from 3 to 117. A card is drawn randomly, find the probability that the card drawn is: (i) an odd number (ii) even number (iii) a perfect square number. 25. Find the four terms of A.P. , whose sum is 50 and in which the greatest number is 4 times the least. SECTION D [6 marks each] 26. The horizontal distance between two towers is 140 m. The angle of elevation of the top of the first O tower when seen from the top of the second tower is 30 . If height of second tower is 60 m, find the height of the first tower. (or) The angle of elevation of the top of a tower from a point on the same level as the foot of the tower is . On advancing p meters towards the foot of the tower, the angle of elevation become . Show that the height of the tower is h p.tan .tan . Also, determine the height of the tower if p = 150 tan tan meters, = 30 and = 60 27. In a right angled triangle, the square on hypotenuse is equal to the sum of squares on other two sides. Prove it. Use the above to prove the following: In a triangle ABC, AD is perpendicular on BC. 2 2 2 2. Prove that AB + CD =AC + BD (or) Prove the tangents drawn to a circle from an external point are equal in length. In the figure given o o below a circle is touching a triangle externally, prove that AP = 1 Perimeterof ABC 2 28.A tent is made in the form of a conic frustum surmounted by a cone. The diameter of the base and the top of the frustum are 14 m and 21 m and its height is 10 m. The height of the tent is 14 m. find the quantity of the canvas required. 29.The radii of the bases of two right circular solid cones of same height are r1 and r2 respectively. The cones are melted and recast into a solid sphere of radius R. show that the height of each cone is 4 R3 . given by h 2 r1 r2 2 30. Calculate the mean , median and mode of the following frequency distribution: C.I 0-20 20-40 40-60 60-80 f 4 5 3 1 Also construct a cumulative frequency curve the above data. 80-100 7 “Learn to see things backwards, inside out and upside down” Sample Paper-11 SECTION: A [1 marks each] 1. By using Euclidian algorithm find the H.C.F. of 365 and 125. 2. For what value of k the following system of linear equation has many solutions: 2x+ky=10; 4x–3y=10k+5 3. Following is the graph of the polynomial y = p(x).Find the zeros of p(x). 21 Chandan’s Sample Papers : Sure Success and Excellent Marks y 8 6 4 2 –2 –1 1 2 3 4 x 4. Which of the following are terminating decimals? 12 1 3 45 9 7 , , , , , . 50 120 33 170 128 625 2 5. If ABC PQR and area of ABC is 225 cm . AB= 10 cm, PQ = 15cm, find the length of PQ. 6. If the mean of 100 observations is 35. Later on it was observed that two numbers were copied wrong such that 35 as 53 and 82 as 28. Find the correct mean. 7. Find the coordinates of the centroid of a Triangle whose vertices are (4,5), (6,7) and (2, 4). 8. If tan A = 5 , find the value of Sin A + Cos A. 12 9. Radius of a circle is 8 cm. A point P is 17 cm away from the centre of this circle. What will be the length of the tangent drawn from this point P to the circle. 10. If sin 3θ =1 then what is the value of tan θ? SECTION: B [2 marks each] 11. Prove that the points ( 2,3), (4, 5) and (-4,6) are the vertices of a right angled triangle. 2 2 2 4 4 2 2 12. Using the quadratic formula, solve the equation: a b x – (4b – 3 a )x – 12a b =0 13.In the figure, tangents AP and AQ are drawn to a circle, with centre O from an external point A. Prove that PAQ 2OPQ P A O Q 2 14. Simplify : (1 + tan )(1 - sin )(1 + sin ) 15. The diagram shows the graph of y = x2 – 2x – 8. The graph crosses the x-axis at the point A, and has a vertex at B. 22 Chandan’s Sample Papers : Sure Success and Excellent Marks y A O B x 2 (a) Factorize: x – 2x – 8. (b) Write down the coordinates of each of these points (i) A; (ii) B. SECTION C [3 marks each] 16. Find the ratio in which the point (x, 1) divides the line joining the points (7,-2) and (-5, 6). Also find the value of x. (or) Prove that the point (3,3) is the centre of the circle passing through the points (6,2), (0,4) and (4,6). Also find its radius. 17.Two equal rectangles are intersecting each other in a circular field. If the dimensions of Rectangular courts are 20 m x 10 m. Find the area of the shaded region. 18. Solve for the following system of equations graphically: 2x + y - 3 = 0 2x - 3y - 7 = 0 19. A man has only 20 paise coins and 25 paise coins in his purse. If he has 50 coins in all totaling Rs. 11.25, how many coins of each does he have? (or) A train travels 360 km at a uniform speed .If the speed had been 5 km ∕h more, it would have taken 1 hour less for the same journey. Find the speed of the train. 20. The diameter of a sphere is 42 cm. It is melted and drawn into a cylindrical wire of 28 cm diameter. Find the length of the wire. 21. Find the sum of 51 terms of an A.P. whose second and third terms are 14 and 18 respectively. 22. Through the mid-point M of the side CD of a parallelogram ABCD, the line BM is drawn intersecting AC in L and AD produced in E. Prove that: EL = 2BL. 2 2 23. If -5 is a root of the quadratic equation 2x + px-15=0 and the quadratic equation p(x +x)+k=0 has equal roots, find the value of k. 24. Construct a circle of radius 3 cm. Draw tangent to this circle through any point on its circumference, without using the centre of the circle. 25. Tickets numbered from 1 to 35 are mixed thoroughly and a ticket is drawn. What is the probability of getting a number :which is odd? A. divisible by 5 or 7? B. neither divisible by 5 nor by 7? C. a prime number D. a perfect square? SECTION D [6 marks each] 0 26. From the top of a building 12 m high, the angle of elevation of the top of a tower is found to be 30 . From the bottom of the same building, the angle of elevation of the top of the tower is found to be 0 60 . Determine the height of the tower and the distance between the tower and the building. 27. A tent in the form of a right circular cylinder up to a height of 3 m and conical above it. The total height of the tent is 13.5 m and the radius of its base is 14 m. Find the cost of the cloth required to 2 make the tent at the rate of Rs. 80 per m . (or) A right triangle, whose two smaller sides are 15 cm and 20 cm, is made to revolve about its hypotenuse. Find the volume and the surface area of the double cone so formed. ( Use π = 3.14). 28. 4 men and 4 boys can do a piece of work in 3 days, while 2 men and 5 boys can finish it in 4 days. How long would it take 1 boy to do it? How long would it take 1 man to do it? 23 Chandan’s Sample Papers : Sure Success and Excellent Marks 29. State and prove the converse of Pythagoras theorem. Using this, prove the following: 2 2 2 2 In ∆PQR,QM ┴PR and PR -PQ =QR . Prove that QM =PM.MR. (or) State and prove “ Thales Theorem”. Using this theorem, prove the diagonals of a trapezium intersect each other in the same ratio. 30. If the mean of the following data is 52, find the missing frequency: Wages: (In Rs.) 10-20 20-30 30-40 40-50 50-60 60-70 70-80 No. of Workers 5 3 4 - 2 6 13 Also construct a cumulative frequency curve and find the median from the graph. “The starting point of all achievement is desire. Keep this constantly in mind. Weak desire brings weak results, just as a small amount of fire makes a small amount of heat.” -- Napoleon Hill – __________________________________________________________________________________ Sample Paper – 12 SECTION A [1 marks each] 1. Without doing actual division, determine whether 621 has a terminating or non-terminating decimal expansion. 1500 2. Give an example of polynomials p ( x ), g ( x ), q ( x ) and r ( x ), which will satisfy the division algorithm and deg p( x ) = deg q( x ). 2 3. One of the roots of the quadratic equation x – kx + 2 = 0 is 2, find k. 2 4. If cot θ = 5/8, evaluate 1 – sin θ 2 1 – cos θ 5. How many multiples of 4 lie between 10 and 250 ? 6. A protractor is in the shape of a semi-circle of radius 7cm. Find its perimeter. A O B 7. In fig. DE // BC, AD = 2 and AC = 18cm, find AE. AB 3 A D E B C 8. Given two concentric circles of radii a and b, where a > b. Find the length of a which touches the other. chord of larger circle O a P b M Q 24 Chandan’s Sample Papers : Sure Success and Excellent Marks 9. A letter of English alphabet is chosen at random. Calculate the probability that the letter so chosen is after the letter „u‟, in order. 10. Find the median when mean = 20 and mode = 18. SECTION B [2 marks each] 11. Find the value of k for which the following system of equations has infinitely many solutions. 2x + 3y = 4; ( k + 2 )x + 6y = 3k + 2 12. Without using trigonometric tables, find the value of : 0 2 0 2 0 2 0 sin39 – 3 ( sin 21 + sin 69 ) + 2sin 30 0 cos51 13. Find the point on the x-axis which is equidistant from ( 2, -3 ) and ( -2, 9 ) 2 2 14. ABC is an isosceles triangle with AC = BC. If AB = 2AC , prove that ABC is a right triangle. 15. Cards numbered 3, 4, 5, 6, ….., 17 are put in a box and mixed thoroughly. A card is drawn at random from the box. Find the probability that the card drawn bears ( i ) An even number ( ii ) A number divisible by 3 or 5. (or) Two black kings are removed from a pack of 52 cards and a card is drawn. Find the probability of getting ( i ) a spade ( ii ) a king . SECTION C [3 marks each] 16. Using Euclid‟s Algorithm, to find the H.C.F of 4052 and 12576. (or) n Check whether 12 can end with the digit 0 for any natural number n. 17. The graph of the polynomial P ( x ) is given. Find the zeros of the polynomial. Also find the quadratic polynomial which represents the graph. Y 4 3 2 1 X X‟ -4 -3 -2 -1 0 1 2 3 -1 -2 -3 4 Y‟ 18. Solve the following system of equations graphically. 3x + 2y + 4 = 0; 3x – 2y + 8 = 0 Also find the coordinates of the vertices of the triangle formed by the lines representing the above equations and y-axis. 19. A number of logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on. If there are 5 logs in the last row, find the number of rows and the total number of logs. 20. Prove that: cos A cos A 2sec A 1 sin A 1 sin A (or) 2 2 2 2 Prove that : ( 1 + tanAtanB ) + ( tanA – tanB ) = sec Asec B. 21. In a classroom. 4 friends are seated at the points A, B, C and D as shown in fig. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli, “Don‟t you think ABCD is a square?” Chameli disagrees. Using distance formula, find which of them is correct. 25 Chandan’s Sample Papers : Sure Success and Excellent Marks 10 9 8 7 6 5 4 3 2 1 B A C D 1 2 3 4 5 6 7 8 9 10 22. A median of a triangle divides it into two triangles of equal areas. Verify this result for Δ ABC whose vertices are A ( 4, -6 ), B ( 3, -2 ) and C ( 5, 2 ). 0 23. Draw a triangle ABC with side BC = 6cm, AB = 5cm and LABC = 60 . Then construct a triangle whose sides are 3/4 of the corresponding sides of the other. 24. Prove that the Parallelogram circumscribing a circle is a rhombus. 25. Find the area of the shaded region in fig. , where a circular arc of radius 6cm has been drawn with vertex O of an equilateral triangle OAB of side 12cm as centre. ***** *************** ********O******* **************** ********* ************ 6cm A 12cm B (or) The decorative block is made of two solids – a cube and a hemisphere. The base of the block is a cube with edge 5cm, and the hemisphere fixed on the top has a diameter of 4.2cm. Find the total surface area of the block ( Take π = 22 / 7 ) SECTION D [6 marks each] 26. The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers. (or) Rs1200 were distributed equally among a certain number of students. Had there been 8 more students each would have received Rs 5 less. Find the number of students. 0 27. A man on a cliff observes a boat at an angle of depression of 30 which is approaching the shore to the point immediately beneath the observer with uniform speed. Six minutes later, the angle of 0 depression of the boat found to be 60 . Find the time taken by the boat to reach the shore. (or) The angle of elevation θ of the top of a light house, as seen by a person on the ground, such that tanθ = 5/12 . When the person moves a distance of 240m towards the light house, the angle of elevation becomes φ such that tan φ =3/4. Find the height of the light house. 28. Prove that the ratio of the areas of two similar triangles is same as the ratio of the square of their corresponding sides. Using the above do the following: Let Δ ABC ~ Δ DEF and their areas be, 2 2 respectively, 64cm and 121cm . If EF = 15.4cm, find BC. 0 29. A metallic right circular cone 20cm high and whose vertical angle is 60 is cut into two parts at the middle of its height by a plane parallel to its base. If the frustum so obtained be drawn into a wire of diameter 1/16 cm, find the length of the wire. 26 Chandan’s Sample Papers : Sure Success and Excellent Marks 30. Find the missing frequencies f1 and f2 in the following frequency distribution table, it is given that the mean of the distribution is 56. C.I f 0 - 20 16 20 - 40 f1 40 - 60 25 60 - 80 f2 80 - 100 12 100 – 120 10 Total 90 “Look inside to find out where you‟re going, and it‟s better to do it before you get out of high school.” _______________________________________________________________________ Sample Paper – 13 SECTION A [1 marks each] 1. Write a quadratic equation whose roots are 3+3 and 3-3. 2. Write the prime factors of 4945. 3. If tanB = ¾, and A+B = 90, then find the value of cotA. 2 4. Find the zeroes of the quadratic polynomial x – 2x + 1. 5. A cylinder, a cone and a hemi-sphere have equal base and same height. What is the ratio of their volumes? 6. Give an example of polynomial p(x), g(x), q(x) and r(x), satisfying P(x)=g(x).q(x)+r(x), deg r(x) = 0 7. A die is thrown once. What is the probability of getting an even prime number? th 8. Find the 20 term of the sequence -2, 0, 2, 4 ------------. 9. Find the perimeter of the sector whose base radius is 14 cm and central angle is 120. 10. For what value of „k‟ the following pair of linear equations has infinitely many solutions? 10x + 5y – (k-5) = 0 and 20x + 10y – k = 0. SECTION – B [2 marks each] 11. Express sin 52 + cos 67 in terms of trigometric ratios of angles between 0 and 45. (or) Sin (A+B) =1/2 and cos (A+B) =1/2, 0<A+B≤90, A>B, find A and B. 12. How many three digit numbers are divisible by 7? 13. Find the values of y for which the distance between the points A (-3, 2) and B (4, y) is 7. 2 14. ABC is a triangle right angled at A and ADBC. Show that AC = BC.CD. 15. Two dice are thrown once. What is the probability that the sum of the two numbers appearing on the top of the dice is less than or equal to 12? SECTION – C [3 marks each] 16. Prove that 3 is irrational. (or) 2 Solve 8x – 77x + 45 = 0 by factorization. 17. Find a quadratic polynomial, whose zeroes are 2+5 and 2-5 2 2 18. Solve for x and y: 47x + 31y = 63; 31x + 47y = 15. th th 19. The third term of an AP is 16 and difference between 7 term and 5 is 12. Find AP. (or) Which term of the AP: 114, 109, 104, --------- is the first negative term? 20. Draw the graph of the following pair of linear equations: x + 3y = 6 and 2x – 3y = 12 and find the area of the region bounded by x = 0, y = 0 and 2x – 3y = 12. 21. Prove that sinA + cosA + sinA – cosA = 2______ 2 2 SinA – cosA sinA + cosA sin A – cos A (or) If 2 tan A = 1, find the value of 3 Cos A + 2 Sin A 2 Cos A – Sin A 22. For what value of „k‟ the points A (1, 5), B (k, 1) and C (4, 11) are collinear? 23. Construct a circle with radius 3 cm and draw two tangents from a point not lying on it. 24. If a student had walked 1 km/hr faster, he would have taken 15 minutes less to walk 3 km. find the rate of his walking. 25. Find the ratio in which the line segment joining the points A (3,-6) and B (5,3) is divided by x-axis. SECTION – D [6 marks each] 26. Prove that in a right triangle the square of the hypotenuse is equal to the sum of square of the other two sides. Using the result of this theorem prove that the sum of squares on the sides of a rhombus 27 Chandan’s Sample Papers : Sure Success and Excellent Marks is equal to the sum of squares on its diagonals. (or) State and prove Thales theorem. Using this theorem prove that the line segment drawn through the mid point of one side parallel to other side bisects the third side. 27. The shadow of a tower standing on a level ground is found to be 40m longer when the sun‟s altitude is 30 then when it is 60. Find the height of the tower. (or) A vertical tower stands on a horizontal plane is surmounted by a vertical flag staff of height 5 m. At a point on the ground the angles of elevation of the bottom and the top of the flag staff are respectively 0 0 30 and 60 . Find the height of the tower. 28. A cylindrical bucket 32cm high and with radius of base 18cm, is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24cm, find the radius and slant height of the heap. 29. The radii of the ends of a bucket 45cm high are 28cm and 7cm. find its volume and the total surface area. 30. Find the median from the following table: Marks No. of Students Below 10 15 Below 20 35 Below 30 60 Below 40 84 Below 50 94 Below 60 127 Below 70 198 Below 80 249 “Learning does not consist only of knowing what we must or we can do, but also of knowing what we could do and perhaps should not do.” Sample Paper – 14 SECTION A [1 marks each] 2 1. Find the positive p so that 4x -3px +9 has real roots. 2. If a = bq + r in division algorithm, give limits of r. 3. Find the value (s) of p for which the system of equations have exactly one solutions. px + 2y -5 = 0 3x + y - 1 = 0 th th 2 n 4. Find the first 4 terms of the sequence whose n term is n / 2 . 0 0 0 0 5. Express cos 75 + cot 75 in terms of angle between 0 and 45 . 6. If PT is a tangent to the circle whose center is O ,OP=10 cm and radius of the circle is 6cm, Find the length of tangent segment PT. 2 2 7. ABC Similar to DEF and their areas are respectively 64 cm and 121 cm . If EF= 15.4 cm, find BC. 8. In a leap year, find the probability of getting 53 Sundays. 0 9. Find the area of a sector of a circle with radius 6 cm if angle of the sector is 60 . 10. Convert the following data into more than frequency distribution. Class 0-20 20-40 40-60 60-80 21-23 80-100 No. of workers 40 51 64 SECTION B 38 5 7 [2 marks each] 11.Find a quadratic polynomial with the given numbers as the sum and product of its zeros respectively ¼, -1. 2 12. Prove that (1 + tan A) (1-Sin A) ( !+ SinA) =1. (or) 2 2 2 2 If x = a Sin B and y = b Tan B, prove that ( a /x - b /y ) = 1. 13. What point (s) on the X-axis are at a distance of 5 units from the point (5, -4)? 14. If AD and PM are medians of triangles respectively, where ABC similar to PQR, prove that AB/ PQ = AD/ PM. 15. What is the probability of getting a total of less than 12 in the throw of two dice? 28 Chandan’s Sample Papers : Sure Success and Excellent Marks SECTION C [3 marks each] 16. Prove that 3 is irrational. 17. Some students planned a picnic. The budget for food was Rs 500. But 5 of them failed to go and thus the food for each member increased by Rs 5. How many students attended the picnic.? 18.Using factorization, find the roots of the following quadratic equation: 2 2 2 2 2 (a+ b) x + 8 ( a – b ) x + 16(a – b) =0 x y 2; and (a-b)x+(a+b)y=a a b2 where a,b 0 a b tan 2 A cos ec 2 A 1 20.Prove that : 2 2 2 2 tan A 1 sec A cos ec A sin A cos 2 A 19 Solve: (or) 2 2 sin + tan = m and tan - sin = n, then prove that ( m – n ) = 4mn. 21. Show that the points (0,-2), (3,1), (0,4) and (-3,1) are the vertices of a square. Also, find the area of the square. (or) If the segment with the end points (3,4) and (14,-3) meets the X axis at P, in what ratio does P divide the segment? Also, find the coordinate of P. 22.The vertices of a ABC are A( 4,6), B(1,5) and C(7,2). A line is drawn to intersect side AB and AC at D and E respectively, such that AD/AB = AE/AC = ¼. Calculate the area of DEA. (or) The two vertices of a square are (-1,2) and (3,2). Find the coordinate of other two vertices. th 23. Construct a similar to a given triangle ABC with its sides 7/5 of the corresponding side of ABC. It is given that AB= 6cm, angle BC =7 cm, and angle CA = 8cm.Write the steps of contraction also. 24. If two tangents are drawn to a circle from an external point, then (i) they subtend equal angle at the center. (ii) they are equally inclined to the segment, joining the center to that point. 25. An ice cream cone consists of a right circular cone of height 14 cm and diameter of circular top is 5cm, It has hemisphere on the top with the same diameter as the circular top. Find the volume of ice cram in the cone. SECTION D 26. Find mean, median and mode of the following distribution. Marks: Number Of Children: [6 marks each] median for the following data:: Less Than 10 Less Than 10 Less Than 10 Less Than 10 Less Than 10 Less Than 10 Less Than 10 Less Than 10 0 10 25 43 65 87 96 100 27. State and prove Pythagoras Theorem. Using it, prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals. 28. From an aero plane vertically above a straight horizontal plane, the angles of depression of two consecutive kilometer stones on the opposite sides of an aero plane are found to be α and β. Show that the height of the aero plane is: tan α tan β tan α + tan β (or) From a window (60 meters high above the ground) of a house in a street, the angles of elevation and depression of the top and the foot of another house on opposite side of street are 600 and 450 respectively. Show that the height of the opposite house is 60(1+3) meters. 0 28. A metallic right circular cone 20cm height and whose vertical angle is 60 is cut into two parts at the middle of its height by a plane parallel to its base .If the frustum so obtained be drawn into a wire of diameter 1/16cm find the length of the wire (or) The height of a cone is 30cm; a small cone is cut off at the top by a plane parallel to the base .If its volume be 1/27 of the volume of the given cone at what height above the base is the section made. 29. Determine the vertices of a triangle formed by lines representing the equation using graph paper 4x-5y-20=0; 3x+5y-15=0 and y = 0 Find the area of the triangle formed by these lines. “Spirituality is neither the privilege of the poor nor the luxury of the rich. It is the choice of wise man.” _______________________________________________________________ 29 Chandan’s Sample Papers : Sure Success and Excellent Marks Sample Paper – 15 SECTION A [1 marks each] What is Euclid's division lemma. Find a quadratic polynomial, the sum and product of whose zeroes are –3 and 2, respectively. Find the value of k for the following quadratic equation, so that they have two equal roots : If , evaluate : Which term of AP : 3, 8, 13, 18, ......, is 78? How many lead balls, each of radius 1cm, can be made from a sphere of radius 8cm? 7. The areas of two similar triangles and are 25cm 2 and 49 cm2 respectively. If QR = 9.8 cm, find BC. 8. From an external point P, tangents PA and PB are down to a circle with centre O. If CD is the tangent to the circle at a point E and PA=14cm, find the perimeter of . 1. 2. 3. 4. 5. 6. 9. Find the probability that a number selected at random from the numbers 1 to 25 is not a prime number when each of the given numbers is equally likely to be selected. 10. A student draws a cumulative frequency curve for the marks obtained by 40 students of a class, as shown below. Find the median marks obtained by the students of the class. SECTION – B [2 marks each] 11. Without drawing the graphs, state whether the following pair of linear equations will represent intersecting lines, coincident lines or parallel lines : 2x – 3y = 5, 6y – 4x = 3 12. Without using trigonometric tables, evaluate the following :13. If the point P(x,y) is equidistant from the points A(5, 1) and B(–1, 5), prove that 3x = 2y. 14. In the given figure, and . Prove that is an isosceles triangle. 15. There are 35 students in a class of whom 20 are boys and 15 are girls. From these students one is chosen at random. What is the probability that the chosen student is a (i) boy (ii) girl? (or) A card is drawn at random from a well-shuffled pack of 52 cards. Find the probability that the card drawn is neither a red card nor a queen. SECTION – C [3 marks each] 16. Using Euclid's division algorithm, find the HCF of 12, 15 and 21. (or) Prove that is irrational. 17. Find all the zeros of the polynomial , it being given that two of its zeros are and . 30 Chandan’s Sample Papers : Sure Success and Excellent Marks 18. Solve the following system of linear equations graphically : and .Determine the vertices of the triangle formed by the lines representing the above equations and the y-axis. 19. The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference. 20. Prove that : 21. Observe the graph given below and state whether triangle ABC is scalene, isosceles or equilateral. Justify your answer. Also find its area. 22. Prove that the points A(–3, 0), B(1, –3) and C(4, 1) are the vertices of an isosceles right-angled triangle. Find the area of this triangle. 23. Draw a triangle ABC with side BC = 7cm, B = 45º, A = 105º. Then, construct a triangle whose sides are times the corresponding sides of ABC. 24. From a point P, two tangents PA and PB are drawn to a circle C(O, r). If OP = 2r, show that APB is equilateral. 25. The cost of fencing a circular field at the rate of Rs. 24 per meter is Rs. 5280. The field is to be ploughed at the rate of Rs. 0.50 per m 2. Find the cost of ploughing the field (Take ). (or) Metallic spheres of radii 6cm, 8cm and 10cm, respectively, are melted to form a single solid sphere. Find the radius of the resulting sphere. SECTION – D [6 marks each] 26. A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24km upstream than to return downstream to the same spot. Find the speed of the stream. (or) A train travels 360km at a uniform speed. If the speed had been 5km/h more, it would have take 1 hour less for the same journey. Find the speed of the train. 27. The angle of elevation of the top of a building from the foot of the tower is 30º and the angle of elevation of the top of the tower from the foot of the building is 60º. If the tower is 50m high, find the height of the building. (or) From the top of a 7m high building, the angle of elevation of the top of a cable tower is 60º and the angle of depression of its foot is 45º. Determine the height of the tower. 28. Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points then the other two sides are divided in the same ratio. (or) In the given figure, in ABC, DE || BC so that AD = 2.4cm, AE = 32cm and EC=4.8cm. Find AB. 31 Chandan’s Sample Papers : Sure Success and Excellent Marks 29. A solid is composed of a cylinder with hemispherical ends. If the whole length of the solid is 104 cm and the radius of each of its hemispherical ends is 7 cm, find the cost of polishing its surface at the rate of Rs. 10 per dm2. 30. Three sets of Hindi, English and Mathematics books have to be stacked in such a way that all the books are stored topic wise and the height of each stack is the same. The number of English books is 96, the number of Hindi books is 240 and the number of Mathematics books is 336. Assuming that the books are of the same thickness, determine the number of stacks of English, Hindi and Mathematics books. “Reach high, for stars lie hidden in your soul. Dream deep, for every dream precedes the goal.” ___________________________________________________________________ Sample Paper –16 SECTION – A [1 marks each] 1. There are three children in a family. Find the probability that there is one girl in the family. 2. Which term of the A.P: 5, 13, 21….. is 181 ? 2 3. Find the values of P for which the quadratic equation 9x + 3Px + 4 = 0 has real and equal roots. 4. Prove that √2 + √3 is irrational. 5 If K is the zero of P(x) = ax + b, find K. 6. How many spherical balls each of radius 1 cm can be made from a sphere of lead of radius 8 cm. 7. Prove that the tangents at the end of a diameter are parallel. 8. Verify that sin 3A = sin2A.cosA + cos2A.sinA, if A = 30˚. 9. The perimeters of two similar triangles are 24 cm and 16 cm. if one side of the first triangle is 12 cm, find the corresponding side of the other. 10. Find the value of Y if the mode of the following data is 25. 15,20,25,18,14,15,25,15,18,16,20,25,20,Y,18 SECTION- B [2 marks each] 2 11. Solve the equation 2x –7x + 3 = 0 by the method of completing the square. 12. If 7 cosec θ - 3 cot θ = 7, then prove that 7 cot θ – 3 cosec θ = 3. (or) Prove that (1+ cot A + tan A )(sin A – cos A) = secA. 13. ABCD is the rectangle whose vertices are A(0,0), B(a,0), C(a,b), D(0,b). Show that it bisect each other and are equal. the diagonals of 14. Find the value of P if the mean of the following distribution is 20. x 15 17 19 20+P 23 f 2 3 4 5P 6 15. Construct two tangents to a circle of radius 3cm from a point on the concentric circle of radius 6cm. (or) Construct a triangle similar to a given triangle with sides 5 cm, 12 cm, 13 cm and whose sides are 3/5 of the corresponding sides of the given triangle. SECTION-C [3 marks each] 16. Show that any positive odd integer is of the form 6q+1 or 6q+3 or 6q+5 where q is some integer. 32 Chandan’s Sample Papers : Sure Success and Excellent Marks (or) 2 If the sum of the zeros of the polynomial (a+1) x + (2a+3) x + (3a+4) be –1, find the product of its zeros. 2 17. Sum of the areas of two squares is 468 m . If the difference of their perimeters is 24 m, find the sides of the two squares. 18. Find the vertices of the triangle, the mid-points of whose sides are (3,1), (5,6) and (-3,2). 19. Mizna is walking along the path joining (-2,3) and (2,-2) while Fathima is walking along the path joining (0,5) and (4,0). Represent and discuss this situation graphically. (or) A 90% acid solution is mixed with a 97% acid solution to obtain 21 liters of a 95 % solution. Find the quantity of each of the solution to get the resultant mixture. 2 2 20. If 7 Sin θ + 3 Cos θ=4, find the value of Secθ + Cosecθ. 21. Find the lengths of the medians of the triangle whose vertices are (1,-1), (0,4) and (-5,3). 22. The vertices of a ∆ABC are (3,0), B(0,6) and C (6,9) and DE divides AB and AC in the same ratio 1:2. Prove that area of ∆ABC = 9(area of ∆ADE). 23. In fig. ABC, points P and Q lies on AB and AC respectively. If PQ || BC, Prove that the median AD bisect PQ. 24. Two circles with radii a and b (a > b) touch each other externally. Find the length of the common tangent AB. 2 25. The area of an equilateral triangle is 17300 cm . With each vertex of the triangle as centre, a circle is drawn with a radius equal to half the length of the side of the triangle. Find the area of the triangle not included in the circles. ( π = 3.14 and √3 =1.73 ) (or) A solid composed of a cylinder with hemi spherical ends . The whole height of the solid is 19cm and 3 the radius of the cylinder is 3.5cm. Find the weight of the solid if 1cm of the metal weighs 4.5g. SECTION- D [6 marks each] 33 Chandan’s Sample Papers : Sure Success and Excellent Marks 26. In Birla auditorium the number of rows was equal to the number of seats in each row. When the number of rows was doubled and the number of seats in each row was reduced by 10, the total no. of seats increased by 300. How many rows were there? 27 Two pillars of equal height stand on either side of a roadway which is180 m wide From a point on the roadway between the pillars, the angles of elevations of the top of the pillars are 60˚ and 30˚. Find the height of the pillars and the position of the point. 2 2 28. BL and CM are the medians of ∆ ABC, right angled at A. Prove that 4(BL + CM ) = 5 BC 2 (or) State and prove the converse of Pythagoras theorem. Using it prove that triangle PQR is right 2 angled if QS =PS×SR and QS⊥PR. 29. Water is flowing at the rate of 15 km per hour through a pipe of diameter 14 cm into a rectangular tank which is 50 m long and 44 m wide. Find the time in which the level will rise by 21cm. (or) A right circular cone is divided into two portions by a plane parallel to the base and passing through a point, which is 1/3 of the height from the top. Find the ratio of the smaller cone to that of the remaining frustum of the cone. 30. For the following frequency distribution draw the less than give and using it find the median. Marks obtained 50-60 60-70 70-80 80-90 90-100 No. of students 4 8 12 6 6 “If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” _________________________________________________________________________ Sample Paper –17 SECTION – A [1 marks each] 1. If cos θ = 4 0 and θ + φ = 90 , find the value of sin φ. 5 2. Find the quadratic polynomial, the sum and product of whose zeros are 3. 4. 5. 6. 7. 8. 9. 10. 2 3 2 and respectively. 2 5 Find the discriminant for the equation 9x – 12x + 4 = 0. 2 If one root of the equation 3x + 11x + k = 0 is the reciprocal of the other, find the value of k. 0 If tan 2A = cot ( A – 18 ), where 2A is an acute angle, find the value of A. A bag contains 8 red, 2 black and 5 white balls. One ball is drawn at random. What is the probability that the ball drawn is neither black nor red? State Euclid‟s Division Lemma. 2 The curved surface area of a cylinder is 1760 cm and its base radius is 14 cm, find the height of the cylinder? Both the ogives (less than and more than) for a data intersect at P(30, 23). Find the median for the data. Following is the graph of the polynomial y = p(x).Find the zeros of p(x). 34 Chandan’s Sample Papers : Sure Success and Excellent Marks SECTION – B [2 marks each] 11. Anand Patil started working in a firm in 1995 at an annual salary of Rs. 5000 and received an increment of Rs. 200 each year. In what year did his annual salary will reach Rs. 7000? 12. If 4 sin θ = 3 cos θ, find the value of 5 sin + 7 cos . 7 sin + 5 cos (or) Prove the following identity: 2 2 2 2 ( Sin A + Cosec A ) + ( Cos A + Sec A ) = 7 + tan A + Cot A A 13. In the given figure, AD AE and ADE = ACB. DB EC D E Prove that ∆ ABC is isosceles. B 14. Find the value of x for which the distance between the points P(2, –3) and Q(x, 5) is 10 units. 15. A jar contains 54 marbles each of which is blue, green or white. The probability of selecting a blue marble at random is 1 4 , and the probability of selecting a green marble at random is . How many 3 9 white marbles does the jar contain? SECTION – C [3 marks each] 16. Using Euclid‟s division algorithms find the H C F of 84, 90 and 120. 2 17. Find the values of k for which the quadratic equation x – 2x(1 + 3k) + 7(3 + 2k) = 0 has real and equal roots. (or) 1 2 4 , x ≠ –1, –2, –4 x 1 x 2 x 4 2 18. Find the zeros of the polynomial f(x) = 4 3x + 5x 2 3 , and verify the relationship between the Solve for x: zeros and its coefficients. 2 4 2 4 4 4 19. Prove that: 2 sec θ – sec θ – 2 cosec θ + cosec θ = cot θ – tan θ 20 Three numbers are in A.P. If the sum of these numbers is 27 and their product is 648, find the numbers. (or) Sum of first 7 terms of an A.P. is 20 and the sum of next 7 terms is 17. Find the A.P. 20. Determine the ratio in which the point P(m, 6) divides the join of A(–4, 3) and B(2, 8). Also find the value of m. (or) If (x, y) be on the line joining the two points (1, –3) and (–4, 2), prove that x + y + 2 = 0. 21. Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are (0, –1), (2, 1) and (0, 3). Find the ratio of the area of the triangle formed to the area of the given triangle. 35 Chandan’s Sample Papers : Sure Success and Excellent Marks C 22. Construct a ∆ ABC in which AB = 5.5 cm, BC = 4 cm and ABC, each of whose sides are B = 750. Construct a triangle similar to ∆ 3 times the corresponding sides of ∆ ABC. 5 23. On a square handkerchief, nine circular designs each of radius 7 cm are made. Find the area of the remaining portion of the handkerchief. 24. In the figure ∆ ABC is a right triangle, right angled at B. AD and CE are the two medians drawn from A and C respectively. If AC = 5 cm and AD = A 3 5 cm, find the length of CE. 2 E B D C SECTION – D [6 marks each] 25. Form a pair of linear equations in two variables using the following information and solve it graphically. Five years ago, Sagar was twice old as Vijay. Ten years later Sagar‟s age will be ten years more than Vijay‟s age. Find their present ages. What was the age of Sagar when Vijay was born? A 26. State and prove the converse of Pythagoras theorem. Use the above theorem to prove the following: 2 In the figure, AD BC. If AD = BD × DC, Prove that ABC is a right triangle. C B D (or) Prove that the lengths of two tangents drawn from an external point to a circle are equal. Use the above theorem to prove the following: A circle is touching the side BC of ∆ ABC at P and touching AB and AC produced at Q and R respectively. Prove that: AQ = 1 (Perimeter of ∆ ABC) 2 B A R C P Q 28. As observed from the top of a 75m high lighthouse from the sea-level, the angles of depression 0 0 of two ships are 30 and 45 . If one ship is exactly behind the other on the same side of the light house, find the distance between the two ships. (or) 0 The angle of elevation of a jet plane from a point A on the ground is 60 . After a flight of 15 0 seconds, the angle of elevation changes to 30 . If the jet plane is flying at a constant height of 1500 3 m, find the speed of the jet plane in km/h. 29.The radii of the ends of a frustum of a cone 45cm high are 28cm and 7cm. Find its volume and total surface area. 36 Chandan’s Sample Papers : Sure Success and Excellent Marks (or) Water in a canal, 6m wide and 1.5m deep, is flowing with a speed of 10km/hr. How much area will it irrigate in 30 minutes, if 8cm standing water is needed? 30. The median of the following data is 20.75. Find the missing frequencies x and y if the total frequency is 100. Class Interval Frequency 0–5 5 – 10 10 – 15 15 – 20 20 – 25 25 – 30 30 – 35 35 – 40 7 10 x 13 y 10 14 9 Work without faith and prayer is like an artificial flower without fragrance. There is no destiny beyond and above ourselves; we are ourselves the architects of our future. - Pujya Gurudev ______________________________________________ Sample Paper –18 SECTION – A 4 3 [1 marks each] 2 1. What must be added to polynomial f(x) = x +2x -2x +x-1 so that the resulting polynomial is exactly 2 divisible by x -4x+3. 2. Radius of a circle is 8 cm. A point P is 17 cm away from the centre of this circle. What will be the length of the tangent drawn from this point P to the circle. 3. Find the distant of a point (x, y) from the origin of the co-ordinate axis. 0 4. A pendulum swings through an angle of 30 and describes an arc 8.8 cm in length. Find the length of the pendulum. ?(Take π = 22/7 ) th 5. Find the sum of n terms of an AP whose n term is given by an=5-6n. 6. Which measure of central tendency is given by the x – coordinate of the point of intersection of the „more than‟ ogive and „less than‟ ogive? 7. A die is thrown once. What is the probability of getting a number between 3 and 6. 8. The common difference of an A.P. is 4. Find the value of a60 - a55 9. The sum and product of the zeroes of a quadratic polynomial are and –3 respectively. What is the quadratic polynomial. 10. The lengths of two cylinders are in the ratio 3 : 1 and their diameters are in the ratio 1 : 2. Calculate the ratio of their volumes. SECTION- B [2 marks each] 11. Without using trigonometric table, evaluate the following: 2 cos 65 0 tan 20 0 sin 90 0 tan 5 0 tan 35 0 tan 60 0 tan 55 0 tan 85 0 0 0 sin 25 cot 70 Or 1 1 tan A tan B tan A , tan B and tan( A B) 2 3 1 tan A. tan B , Find A+B If 12. A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball from the bag is thrice that of a red ball, find the number of blue balls in the bag. 13. Prove that 3 2 3 is an irrational number. 14. If the heights of two cones are in the ratio of 1:3 and their diameters are in the ratio of 3:5,find the ratio of their volumes 15. Prove that the tangents drawn at the ends of a diameter of a circle are parallel. SECTION – C [3 marks each 16. A spiral is made up of successive semicircles, with centre alternately at A and B, starting with centre at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, . . ….. as shown in fig. What is the total length of such a spiral made up of thirteen consecutive semicircles?(Take π = 22/7 ) 17. Solve by using Quadratic formula: a 2b 2 x 2 b 2 x a 2 x 1 0 (or) 37 Chandan’s Sample Papers : Sure Success and Excellent Marks Solve the equation by the method of completing the square: 5x 6 x 2 0 18. Show that the points A (2, -2), B (14, 10), C (11, 13) and D (-1, 1) are the vertices of a rectangle. (or) Determine the ratio in which the points (6,a) divides the join of A (-3,-1) and B (-8,9). Also find the value of “a”. 19. Draw a triangle ABC with side BC = 7cm, B = 45º, A = 105º. Then, construct a triangle whose sides 2 4 are 3 times the corresponding sides of ABC . sinA 1 - cosA 1 sinA cosA 20. Prove the identity: cosA - 1 sin A 21. How many terms of the AP: 24, 21, 18, ….. must be taken so that their sum is 78 ? (or) th Find the sum of first 24 terms if the n term is given by an = 9 – 5n 2 2 2 2 22. Solve for x and y : (a – b)x + (a + b)y = a – 2ab – b ; (a + b)(x + y) = a + b (or) 6( ax + by ) = 3a + 2b ; 6( bx – ay ) = 3b – 2a 23. A cylindrical container is filled with ice cream, whose radius is 6cm and height 15cm.The whole ice cream is distributed among 10 children in equal cones having hemi-spherical top. If the height of the conical portion is 4 times the radius of its base. Find the radius of the base of the cone. 24. In fig OACB is a quadrant of a circle with centre O and radius 3.5 cm. If OD = 2 cm, find the area of the shaded region. ( use 22 / 7 ) 25. A shopkeeper buys a certain no. of books for Rs.1200.If he had bought 10more books for the same amount, each book would have cost him Rs.20 less. Find the original no. of books he had purchased. SECTION – D [6 marks each] 23. From an aero plane vertically above a straight horizontal road, the angles of depression of two consecutive milestones on opposite sides of the aero plane are observed to be α and β.Show that the height of the aero plane above the road is tan α.tan β/(tan α+tan β). (or) A vertical tower stands on a horizontal plane and is surmounted by a flagstaff of height h. At a point on the ground , the angle of the elevation of the bottom of the flagstaff is and that of the top of the flagstaff is . Prove that the height of the tower is h tan / (tan - tan). 24. A cylindrical bucket 32cm high and with radius of base 18cm, is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24cm, find the radius and slant height of the heap. (or) The radii of the ends of a bucket 45cm high are 28cm and 7cm. find its volume and the total surface area. that the ratio of the areas of two similar triangles is equal to the ratio of the squares of the corresponding sides. 25. Show that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their PQR corresponding sides. Two triangles ABC and PQR are similar. If area ( ABC ) = 4 area ( ) and BC = 12 cm. Find QR 26. A person on tour has Rs. 360 for his expenses> If he extends his tour for 4 days, he has to cut down his daily expenses by Rs. 3. Find the original duration of the tour. ˚ 27. .In the figure, PQR is a right angled triangle with PQ =12cm, QR = 5cm and ∟Q =90 .A circle with centre O and radius x is inscribed in triangle PQR. Find the value of x P ox Q R “Temper brings you to trouble. Pride keeps you there.” __________________________________________________________ Sample Paper –19 SECTION – A [1 marks each] 2 1. Verify whether x= - 3 is a zero of the polynomial p(x) = x + 7x + 12. Also find the value of p(–2) 2. Given H.C.F ( 210, 55 ) = 5, find their L.C.M 38 Chandan’s Sample Papers : Sure Success and Excellent Marks 3. Prove that sec A (1 – sin A)( sec A + tan A) = 1. 4. A cylinder, a cone and a hemisphere are of equal base and have the same height. Find the ratio of the volumes? 5. A card is drawn at random from a well-shuffled deck of 52 cards. Find the probability that it is neither an ace nor a queen. 6. Which term of the sequence is the first negative term113,109,105,101,…. . 7. In ABC, DE || BC and AD/DB = 2/3 and AE = 3.7 cm. Find the value of AC. 8. If TP and TQ are two tangents to a circle with centre O so that POQ=110, find the value of PTQ. 9. If the system of equations 2x+3y=7 and 2ax+(a+b)y=28 has infinitely many solutions, prove that b=2a 10. Find the arithmetic mean of 1, 2, 3, 4……., n. SECTION – B [2 marks each] 2 11. Find k so that the equation (k + 1)x + 2kx + 4 = 0 has sum of the roots equal to the product of the roots. 12. If sin 63 - 1 ,find k k sec 2 13. Find the probability that a number selected at random from the numbers 1,2,3….,35 is a) a multiple of 7 b) a multiple of 3 or 5 14. Find the area of the triangle whose vertices are (a, b+c ), ( b, c+a ) and ( c, a+b) 2 15. In the trapezium ABCD, AB is parallel to CD and AB= 2 CD. If the area of AOB = 84 cm , find the area of COD, where O is the meeting point of the diagonals SECTION – C [3 marks eah] 16. Draw a circle of radius 3cm. Draw tangents to the circle from a point P which is 6 cm away from the centre of the circle. 17. Prove that 5 + 2 3 is irrational. 18. If the point (x,y) is equidistant from the points (a+b, b-a) and (a-b,a+b), prove that bx=ay. (or) Determine the ratio in which the point (-6, a) divides the join of A (-3,-1) and B(-8,9). Also find the value of a. 19. Solve for x and y : 2 2 1 x 3y 6 3 2 0 x y Hence find a where y = ax – 4. (or) uv 6 uv 5 Solve for u and v : uv u v 0; u v 6 uv cos ec cot 2 cos ec 2 2 cos ec cot 1 20. Prove that cos ec cot 1 cos sin 1 sin 1 cos sin cos (or) 21. A party of tourists booked a room in a hotel for Rs.1200. Three of the members failed to pay as they had no cash with them. As a result, each of the remaining people had to pay Rs.20 more. How many tourists were there in all? 4 3 2 2 22. Find the quotient and remainder when –10y + 21y – 21y + 18 is divided by 2y – 3y + 4. Hence verify the division algorithm 23. If the points A(6,1), B(8,2), C(9,4) and D(p,3) are the vertices of a parallelogram taken in order, find the value of p 24. If the diagonals of a quadrilateral divide each other proportionally, prove that it is a trapezium. 25. Construct a triangle similar to a given triangle ABC with its sides 4/3 times of the corresponding side 0 0 of triangle ABC. It is given that BC = 7cm, B =45 , and A = 105 SECTION – D [6 marks each] 26.(a) State and prove Thales theorem. Use this theorem to answer the following. 39 Chandan’s Sample Papers : Sure Success and Excellent Marks (b) Two poles of height „a‟ and „b‟ (b > a) are „c‟ meters apart. Prove that the height „h‟ in meters of the point of intersection of the lines joining the top of each pole to the foot of the opposite pole is ab . ab 27. The mean of the following frequency distribution is 26.5. If the total number of find the frequencies f1 and f2. Classes Frequency 0-10 f1 11-20 14 21-30 21 31-40 f2 observations is 100, 41-50 22 25. A metallic toy is in the shape of a hemisphere of radius 3.5cm mounted over a cylinder of height 3cm. It is melted to form a cone of height 4 cm. Find the radius of the cone. (or) An oil funnel of tin sheet consists of a cylindrical portion 10cm long attached to a frustum of a cone. If the total height is 22 cm, diameter of the cylindrical portion is 8cm and the diameter of the top of the funnel is 18cm, find the area of the tin required to make the funnel. 26. From a window (60 meters high above the ground) of a house in a street, the angles of elevation and depression of the top and the foot of another house on opposite side of street are 600 and 450 respectively. Show that the height of the opposite house is 60(1+3) meters. (or) A person standing on the bank of a river observes that the angle of elevation of the top of a tree o standing on the opposite bank is 60 . When he moves 40 m away from the bank, he finds the angle of elevation to be 30. Find the height of the tree and the width of the river. 27. (a) Solve graphically the following pair of equation 2x – y = 2; 4x – 4y= 8. (b) Find „a‟ if y = ax + 15 Use graph to answer the following: (c) Write the co-ordinate of point where the lines meet the x-axis. (d) Find the co-ordinates of the vertices of the triangle formed by these two lines and x –axis. (e) Shade the above triangle and find its area. TRY HARD TO GET WHAT YOU LIKE, OR YOU WILL BE FORCED TO LIKE WHAT U GET _________________________________________________________________________ Sample Paper – 20 1. 2. 3. 4. 5. 6. 7. SECTION – A [1 marks each] Find the HCF of 96 and 404 by prime factorization method. Hence, find their LCM. For what value of k will the following pair of linear equations have infinitely many solutions? kx + 3y - ( k - 3 ) = 0 ; 12x + ky - k = 0. Find the values of k so that the given quadratic equation has equal roots: kx(x - 2) + 6 = 0. Determine the AP whose third term is 5 and seventh term is 9. A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28 m long . Find the height of the tower. 2 D is a point on the side BC of a triangle ABC such that ADC = BAC. Show that CA =CB.CD. The length of a minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes. 8. In triangle ABC, right-angled at B if tan A = 1 , find the value of : sin A cos C + cos A sin C. 3 9. One card is drawn from a well shuffled deck of 52 cards. Find the probability of getting : (i) the jack of hearts (ii) the queen of diamonds. 10. Savita and Hamida are friends . What is the probability that both will have (i) different birthdays (ii) the same birthday ? (Ignoring a leap year) SECTION-B [2 marks each] 2 11. If the sum of first n terms of an AP is 4n - n , find first 4 terms of this AP. 12 Find the value of k for which the points A (8, 1) , B (k, -4), C (2, -5) are collinear. (or) Find the values of y for which the distance between the points P(2, -3) and Q (10, y) is 10 units. 13. Without using trigonometric tables evaluate: 40 Chandan’s Sample Papers : Sure Success and Excellent Marks sin 2 63 sin 2 27 tan 48 tan 23 tan 42 tan 67 . cos 2 17 cos 2 73 14. Two concentric circles are of radii 5 cm and 3 cm . Find the length of the chord of the larger circle which touches the smaller circle. 15.The table below shows the daily expenditure on food of 25 households in a locality. Find the mean daily expenditure on food by step-deviation method. Daily expenditure in Rs. No. of households 100-150 150-200 200-250 250-300 300-350 4 5 12 2 2 SECTION-C 3 [3 marks each] 2 16. On dividing x – 3x + x + 2 by a polynomial g(x), the quotient and remainder were x– 2 and -2x + 4, respectively. Find g(x). 17. Check whether the pair of equations 2x + y – 6 = 0 and 4x – 2y – 4 = 0 is consistent. If so, solve graphically. 18. A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Determine the speed of the stream and that of the boat in still water. (or) The ratio of incomes of 2 persons is 9 : 7 and the ratio of their expenditures is 4 : 3. If each one of them manages to save Rs 2000 per month , find their incomes. 19. The first and the last terms of an AP are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum ? 2 2 20. Prove that cos A sin A 1 = cosec A + cot A using the identity cosec A = 1 + cot A. cos A sin A 1 (or) (cosecA – sinA )( secA – cosA) = 1 . tan A cot A 21. Do the points (3, 2) ,(-2, -3) and (2, 3) form a triangle ? If so, name the type of triangle formed. 22. Find the coordinates of the points of trisection of the line segment joining the points A (2, -2) and B (-7, 4). 23. Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that PTQ = 2OPQ . 24. Construct a triangle ABC with side BC = 7 cm, B = 45°, A = 105°. Then construct a triangle whose sides are 4 times the corresponding sides of ABC . 3 25. A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank in her field, which is 10 m in diameter and 2 m deep. If water flows through the pipe at the rate of 3 Km/hr, in how much time will the tank be filled ? (or) A container shaped like a right circular cylinder having diameter 12 cm and height 15cm is full of icecream. The ice-cream is to be filled into cones of height 12 cm and diameter 6 cm, having a hemispherical shape on the top. Find the number of such cones which can be filled with ice-cream. SECTION D [6 marks each] 1 1 11 ; x 4,7 x 4 x 7 30 26. Find the roots of the following equation: (or) Two water taps together can fill a tank in 9 3 hours. The tap of larger diameter takes 10 hours less 8 than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank. 27. The shadow of a tower standing on the level ground is found to be 40 m longer when the sun‟s altitude is 30 than when it is 60 . Find the height of the tower. 41 Chandan’s Sample Papers : Sure Success and Excellent Marks 28. Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. And further , if in triangle ABC, the line segment XY is parallel to side AC and it divides the triangle into two parts of equal areas. Find the ratio AX . AB (or) a. The perpendicular from A on side BC of a triangle ABC intersects BC at D such that DB = 3 CD. 2 2 2 Prove that 2AB = 2AC + BC . b. In an equilateral triangle ABC, D is a point on side BC such that BD = 1 2 2 BC. Prove that 9AD = 7 AB . 3 29. A metallic right circular cone 20 cm high and whose vertical angle is 60 is cut into 2 parts at the middle of its height by a plane parallel to its base. If the frustum so obtained be drawn into a wire of diameter 1 cm, find the length of the wire. 16 30. During the medical check-up of 35 students of a class, their weights were recorded as follows: Weight in kg Less than 38 Less than 40 Less than 42 Less than 44 Less than 46 Less than 48 Less than 50 Less than 52 No of students 0 3 5 9 14 28 32 35 Draw a less than type ogive for the given data. Hence, obtain the median weight from the graph and verify the result by using the formula. of median. Inspiration could be called inhaling the memory of an act never experienced. ___________________________________________________________________ Sample Paper – 21 SECTION – A [1 marks each] 1. Use Euclid‟s division algorithm to find HCF of 420 and 130 2. In the given figure the graph of a polynomial p(x) is given. Find the zeroes of the polynomial. 3. For what value of k will the following pair of linear equation have no solution? 3x+y=1 (2k-1)x+(k-1)y=2k+1 4. If A, B C are interior angles of a ABC, then show that 5. If sin (A-B) = 1 1 , cos (A+B) = , 2 2 sin BC A cos 2 2 0 A B90 0, A>B, find A and B. 6. If the perimeter and the area of a circle and numerically equal then what is the radius of the circle. 7. If tangent PA and PB from a point P to a circle with centre O are inclined to each other at angle 80 then what is the value of POA ? 0 42 Chandan’s Sample Papers : Sure Success and Excellent Marks 8. Express sinA in term of cotA. 9. Savita and Hamida are friends, what is the probability that both will have the same birthdays in a non-leap year. 10. What is the value of the median of the data given in the following figure. 11. 12. a) b) SECTION – B [2 marks each] Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively. One card is drawn from a well shuffled deck of 52 cards calculate the probability that the card will not be an ace be an ace 4 3 y 14 x 13. Find the solution of the pair of equations 3 4 y 23 x 14. If the point A(6,1), B(8,2),C(9,4) and D(p,3) are vertices of a parallelogram, taken in order, find the value of p. 15. Find a quadratic polynomial, the sum and product of whose zeroes are -3 and 2 respectively. (or) 3 2 2 What are quotients and remainder, when x -3x +5x-3 is divided by x -2. SECTION – C [3 marks each] 16. A train travels 360km at a uniform speed. if the speed had been 5km/h more, It would have taken 1 hour less for the same journey. Find the speed of the train. 17. Show that 2 is an irrational number. 2 18. Find the roots of the quadratic equation 3x -2 6 x+2=0 19. Draw a circle of radius 3cm. Take two points P and Q on one of its extended diameter each at a distance of 7cm from its centre. Draw tangents to the circle from these two points P and Q. 20. Prove that (cos ecA sin A)(sec A cos A) 1 tan A cot A Or Evaluate a) b) sin 2 63 sin 2 27 cos 2 17 cos 2 73 tan 48 tan 23 tan 42 tan 67 21. If the areas of two similar triangles are equal, prove that they are congruent. Or 43 Chandan’s Sample Papers : Sure Success and Excellent Marks PQ is a chord of length 8cm of circle of a radius 5cm. The tangents at P and Q intersects at point T. find the length of TP and TQ 22. Find a relation between x and y such that the point (x,y) is equidistant from points (7,1) and (3,5). 23. Find the area of the triangle formed by joining the midpoints of the sides of a triangle whose vertices are (0,-1),(2,1) and (0,3). 24. ABC and AMP are two right angled triangles right angled at B and M respectively. prove that (i) ABC AMP (ii) CAMP=PABC 25. Find the area of the segment AYB shown in figure, If radius of the circle is 21cm and (use = AOB 120 22 ) 7 (or) Find the area of the shaded region whose ABCD is a square of side 10cm and semicircles are drawn with each side of the square as diameter (use =3.14) SECTION – D [6 marks each] 26. In a triangle, if square of one side is equal to the sum of the squares of other two sides, then angle opposite the first side is a right angle. Using the converse of above theorem determines the length of an altitude of an equilateral triangle of side 2a. 27. Given the linear equation 2x+3y-8=0, write another linear equation in two variable such that the geometrical representation of the pair so formed is intersecting lines. and solve that equation so obtained with given equation graphically. 28. The angles of depression of the top and bottom of an 8m tall building from the top of multi-storied building are 30 and 45 respectively. Find the height of the multi-storied building and distance between the two buildings. 29. A metallic right circular cone 20cm height and whose vertical angle is 60 is cut into two part at the middle of its height by a plane parallel to its base. if the frustum so obtained is drawn into a wire of diameter 1 cm, find the length of the wire. 16 (or) 44 Chandan’s Sample Papers : Sure Success and Excellent Marks A solid iron pole consists of cylinder of height 220cm and base diameter 24cm, which is surmounted 3 by another cylinder of height 60cm and radius 8cm. find the mass of the pole, given that 1cm of iron has approximately 8g mass. (Use =3.14) 30. The median of the following data is 525. find the values of x and y, if the total frequency is 100. Class interval 0-100 100-200 200-300 300-400 400-500 500-600 600-700 700-800 800-900 900-1000 Frequency 2 5 X 12 17 20 Y 9 7 4 (or) Consider the following distribution and find mean using step deviation method. Class interval frequency 50-52 15 53-55 110 56-58 135 59-61 115 62-64 25 It is easier to tone down a wild idea than to think up a new one. ______________________________________________________________________ Sample Paper – 22 SECTION – A 1. Use Euclid‟s division algorithm to find the HCF of 867 and 255. 2. Find the zeroes of the polynomial x²-3 3. Find the discriminant of the equation 3x 2 2 x [1 marks each] 1 0 3 4. For which values of „p‟ does the pair of equation given below have unique solution? 4x+py+8=0 2x+2y+2=0 5. sin(A+B)= sinA+ sinB , Is it true or false. Justify your answer with an example. 6. ABC and BED are two equilateral triangles such that “D” is the mid point of BC. Find the ratio of areas of triangles ABC and BED. 7. Observe the given figure and find P 80 3.8 33 7.6 63 60 6 12 45 Chandan’s Sample Papers : Sure Success and Excellent Marks 8. Metallic spheres of radii 6cm, 8cm and 10cm respectively are melted to form a single solid sphere. Find the radius of the resulting sphere. 9. What is the empirical relationship between the three measures of central tendency? 10. If P(E)=0.05 what is the probability of „not E‟? SECTION-B 11. How many two digit numbers are divisible by 3? 12. If tan (A+B) = 3 and tan (A-B)= 1 [2 marks each] ,0<A+B90,A>B ,Find A and B 3 13. Find the values of „y‟ for which the distance between the points P(2,-3) and Q(10,y) is 10 units. 14. ABD is right triangle, right angled at A and ACBD, Show that AC² =BC×DC (or) PQ is a chord of length 8cm of circle of a radius 5cm. The tangents at P and Q intersects at point T. find the length of TP and TQ 15. A piggy bank contains hundred 50 P coins, fifty Re 1 coins, twenty Rs 2 coins and ten Rs 5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, what is the probability that the coin will not be an Rs 5 coins? SECTION-C [3 marks each] 16. A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in Rupees) was 3 more then twice the number of articles produced on that day. If the total cost of production on that day was Rs 90, find the number of articles produced and the cost of each article. 17. Show that 3+ 2 5 is irrational. 18. Solve 2x+3y=11 and 2x-4y=-24 and hence find the value of ‟m‟ for which y=mx+3. 19. Prove that the area of the triangle BCE described on one side Bc of a square ABCD as base is one half the area of the similar triangle ACF described on the diagonal AC as base. 20. Prove that (cos ecA sin A)(sec A cos A) 1 tan A cot A (or) 5 cos 60 4 sec 30 tan 45 sin 2 30 cos 2 30 2 Evaluate 2 2 21. If the points A(6,1),B(8,2), C(9,4) and D(p,3)are the vertices of a parallelogram taken in order find the value of „p‟. 22. Construct an isosceles triangle whose base is 8cm and altitude 4cm and then another triangle whose sides are 1 1 times the corresponding side of the isosceles triangle.(with steps of construction) 2 23. Find the coordinates of the points of trisection of the line segment joining (4,-1) and (-2,-3). 24. How many terms of the AP 24,21,18…………….must be taken so that their sum is 78? (or) In the sum of first seven terms of an AP is 49 and that of seventeen terms is 289. Find the sum of first n terms. 25. In the given figure a square OABC is inscribed in a quadrant OPBQ. If OA=20cm. Find the area of shaded region(use =3.14) (or) 46 Chandan’s Sample Papers : Sure Success and Excellent Marks Find the area of the shaded region whose ABCD is a square of side 24cm and semicircles are drawn with each side of the square as diameter. SECTION-D [6 marks each] 26. Prove that the ratio of the areas of two similar triangles is equal to the square of their corresponding sides. Using the above theorem prove that two similar triangles of equal area, are congruent 27. Given the linear equation 2x+3y-5=0, write another linear equation in two variable such that the geometrical representation of the pair so formed is intersecting lines. And solve equation so obtained with given equation graphically. 28. The following type gives production yield per hectare of wheat of 100 farms of a village Production yield in(kg/ha) 50-55 55-60 60-65 65-70 70-75 Number of farms 2 8 12 24 38 Change the distribution to a more than type distribution and draw its ogive. 75-80 16 (or) The following table shows the ages of the patients admitted in a hospital during a year Age in years Number of Patient 5-15 6 15-25 11 25-35 21 35-45 23 45-55 14 55-65 5 Find the mode and mean of the data given above. Compare and interpret the two measures of central tendency. 29. A container, opened from the top and made of metal sheet, is in the form of a frustum of a cone of height 16cm with radii of its lower and upper ends as 8cm and 20cm respectively. Find the cost of milk which can completely fill the container, at the rate of Rs 20/litre. Also find the cost of metal sheet 2 used to make the container ,if it cost Rs 8 /100cm (Use =3.14) (or) A solid iron pole consists of cylinder of height 220cm and base diameter 24cm, which is surmounted 3 by another cylinder of height 60cm and radius 8cm. find the mass of the pole, given that 1cm of iron has approximately 8g mass. (Use =3.14) 30. Two poles of equal heights are standing opposite each other on either side of the road, which is 80m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60 and 30 respectively. Find the height of the poles and distances of the point from the poles. To be a man of knowledge one needs to be light and fluid. Sample Paper – 23 SECTION – A [1 marks each] 1. 5 7 13 17 17 is a composite number because ……………………… 2. If H.C.F.(26,91)=13, find the L.C.M.(26,91). kx2 2 2 x 1 0 has equal roots 0 0 0 4. Express sin 72 cos 81 in terms of trigonometric ratios of angles between 0 and 3. Find the value of k if 0 45 5. How many terms of the AP 6, 11 ,5....... will give the sum zero. 2 6. If each side of an equilateral triangle is „2a‟ units, what is the length of its altitude? 47 Chandan’s Sample Papers : Sure Success and Excellent Marks 7. In the given figure DE is parallel to BC and AD : DB 2 : 3 determine ar (ADE) : ar (ABC ) 8. A circle is inscribed in ABC having sides AB = 8 Cm ,BC = 10 Cm , and AC= 12 Cm as shown in the figure find AD , BE and CF 9. One letter is selected at random from the word „UNNECESSARY‟. Find the probability of selecting an E _ 10. If the mean of n observations _ _ x1 , x2 , x3 .......xn is x then find _ _ ( x1 x) ( x2 x) ( x3 x).....( xn x) SECTION B [2 marks each] 11. Find the value of k for which the system of equations has infinite number of solutions 4x y 3 8 x 2 y 5k 12. Without using trigonometric table find the value of Sin500 Co sec 400 4Cos500 Co sec 400 0 0 Cos 40 Sec50 13. Determine the ratio in which the point (6, a) divides the join of A(3,1) and B(8,9) also find the value of a 14. In the figure ,AB parallel to DE and BD parallel to EF , is CD 15. In a single throw of two dice , find the probability of getting i) Two heads ii) At least one heads SECTION C 2 CF CA justify your answer [3 marks each] 16. Find the HCF of 426 and 576 using Euclid‟s division algorithm (or) Prove that no number of the type 4k 2 be perfect square 17. m and n are zeros of m n m.n 10 ax 2 5x c . Find the values a and c if 18. Draw the graph of x y 1 0 and 3x 2 y 12 0 and show that there is a unique solution .Calculate the area bounded by these lines and x-axis 19. A polygon has 10 sides .The lengths of the sides starting with the smallest form an AP .If the perimeter of the polygon is 420 Cm and the length of the longest side is twice that of the shortest side .Find the first term and the common difference of the AP 20. Prove that (sin A cos ecA) 2 (cos A sec A) 2 7 tan 2 A cot 2 A Or cos A sin A 1 Prove that cos ecA cot A , Using the identity cos A sin A 1 cos ec 2 A 1 cot 2 A 21. Observe the graph and state whether the quadrilateral ABCD is a parallelogram Justify your answer . 22. Find the area of the triangle formed by joining the mid points of the sides of the sides of triangle whose vertices are (4,7), (8,7), (10,13) BC 6cm , B 300 , A 1200 then 4 construct a triangle whose sides are times the corresponding sides of 3 ABC 23. Draw a triangle ABC with side 24. Prove that the opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle 25. A regular hexagon is circumscribed by a circle of radius 14 Cm .Find the area of the shaded region . (or) A square ABCD is inscribed in a circle of radius 10 units .Find the area of the circle , not included in the square (use 3.14 ) SECTION D [6 marks each] 48 Chandan’s Sample Papers : Sure Success and Excellent Marks 26. A train covers a distance of 90 Km at uniform speed .Had the speed been 15 Km /hr more . it would have taken 30 minutes less for the journey find the original speed of the train (or) A party of tourists booked a room in a hotel for Rs 1200 , three of the members failed to pay as a result others had to pay Rs 20 more (each) . How many tourists were there in the party 600 .After a flight of 15 seconds 0 , the angle of elevation changes to 30 .If the jet plane is flying at a constant height of 1500 3m , 27. The angle of elevation of a jet plane from a point A on the ground is find the speed of the jet plane (or) From a building 60m high the angle of depression of the top and bottom of lamp post are 300 and 600 respectively .Find the distance between lamp post and building ,also find the difference of height between building and lamp post 28. Prove that in a right triangle , the square of the hypotenuse is equal to the sum of the squares of the other two sides Using the above solve the following L and M are the mid points of AB and BC respectively of ABC , right angled at B prove that 4LC 2 AB 2 4BC 2 29. A building is in the form of a cylinder surmounted by a hemispherical vaulted dome ,the building 3 contains 17.7 m of air and its internal diameter is equal to the height of the cylindrical part find the height of the building (use 22 ) 7 30. Find the median of the following data Class interval 110-119 120-129 Frequency 5 25 130-139 40 140-149 60 150-159 40 160-169 25 170-179 5 Life changes when we change. ____________________________________________________ Sample Paper – 24 SECTION - A [1 marks each] If the nth tern of an AP is (2n + 1), find the sum of first n terms of the AP. Find the probability that a number selected from the numbers 1 to 25 is not a prime number when each of given number is equally likely to be selected. 3 If tan A + cot B, prove that A + B = 90º 4 Find the HCF of 96 and 404 by the prime factorization method. Hence , find the LCM. 5 Give examples of polynomials P(x), G(x), Q(x) and R(x) which satisfy the division algorithm and deg R(x) = 0. 6 Find the values of k for the quadratic equation kx (x – 2) + 6 = 0 have equal roots. 7 Let ABC~ DEF and their area be respectively 64 cm² and 121 cm². If EF = 15. 4 cm, find BC. 8 From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. Find the radius of the circle. 9 2 cubes each of volume 64 cm³ and joined end to end. Find the surface area of the resulting cuboid. 10 Write the formula of Mode and Median. 1 2 SECTION- B [2 marks each] 11 A bag contains 5 red, 8 white and 7 black balls. A ball is drawn at random from the bag. Find the probability that the drawn ball is (i) Not black (ii) neither red or not white. 12 Find the sum of all multiple of 9 lying between 300 and 700. 13 In OPQ, right-angled at P, OP = 7cm and OQ – PQ = 1cm. Determine the values of sin Q and cos Q. 14 In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes. 15 Find the value of „k‟ if the points A(4, 2), B (4, k) and C ( 6, - 3) are collinear. SECTION – C 16 Solve the following system of linear equations graphically: 2x – 5y + 4 = 0 ; 2x + y – 8 = 0 Also, find the points where the lines meet the y – axis. 17 Prove that: 1 _ 1 = 1 [3 marks each] _ 1 49 Chandan’s Sample Papers : Sure Success and Excellent Marks sec x – tan x cos x cosx secx + tan x (or) sec² 54º _- cot² 36º + 2 sin² 38º . sec² 52º - sin² 45º cosec² 57º - tan² 33º If the point P (x, y) is equidistant from the points A (5, 1) and B (- 1, 5), prove that 3x = 2y. The line joining the points (2, 1) and (5, - 8) is trisected at the point P and Q. If Point P lies on the line 2x – y + k = 0, find the value of K. (or) The line segment joining the points (3, -4) and (1, 2) is trisected at the point P and Q. If the coordinate of P and Q ( p, 2) and (5/3, q) respectively, find the value of p and q. Draw a circle of diameter 7 cm. From a point P outside the circle at a distance of 6 cm from the centre of circle, draw two tangents to the circle and measure s their lengths. Prove that the parallelogram circumscribing a circle is a rhombus. Find the zeros of the quadratic polynomial 3x² - x – 4 and verify the relationship between the zeroes and the coefficient. Prove 3 is irrational number. IN Fig. OACB is a quadrant of a circle with centre O and radius 3. 5 cm If OD = 2 cm. , find the area of the (i) quadrant OACB (ii) shaded region. A Evaluate: 18 19 20 21 22 23 24 O B 25 For which values of a and B does the following pair of linear equations have an infinite number of solutions? 2x + 3y = 7; (a – b) x + (a =b) y = 3a + b - 2 SECTION – D [6 marks each] 26 From a building 60 meters high the angle o depression of the top and bottom of lamppost are 30º and 60º respectively. Find the distance between lamppost and building. Also find the difference of height between building and lamppost. 27 A tent is in a shape of a right circular cylinder up to a height of 3 m and conical above it. The total height of the tent is 13.5 m and radius of base is 14 m. Find the cost of cloth required to make the tent at the rate of Rs. 80 per sq. m. (or) A container shaped like a right circular cylinder having diameter 12 cm and height 15 cm is full of ice cream. The ice cream is to be filled into cones of height 12 cm and diameter 6 cm, having a hemispherical shape on the top. Find the number of such cones which can be filled with ice cream. 28 Prove that in a triangle, a line drawn parallel to one side to intersect the other two sides in distinct points, divides the two sides in the same ratio. Using above prove that the quadrilateral ABCD is a trapezium if the diagonal AC and BD of the quadrilateral ABCD intersect each other at O such that AO = BO OC OD 29 A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km down stream. Determine the speed of the stream and that of the boat in still water. (or) In a class test, the sum of Sonal‟s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 less in English, the product of their marks would been 210. Find her marks in the two subjects. 30 The mean of the following frequency distribution is 57.6 and sum of the observations is 50. Find the missing frequencies x and y. CI 0-20 20-40 40-60 60-80 80-100 100-120 frequency 7 x 12 y 8 5 “A good book is the best of friends, the same to-day and forever” Sample Paper - 25 50 Chandan’s Sample Papers : Sure Success and Excellent Marks SECTION –A [1 marks each] 1. A wire is in the form of a circle of radius 7 cm. It is rebent into a square form. Find the length of the side of the square. 2. Write down two events which have probability 1. 3. The lengths of the tangents drawn to a circle from a point outside the circle are always equal. Is it true? 2 2 4. If triangles ABC and DEF are similar, area of ABC = 9 cm , area of DEF = 25 cm and DE = 6 cm, find the length of AB. 5. For an acute angle A, value of sin A lies between 0 and 1. Is it true? 6. How many terms are there in the AP 8, 12, 16 ………. 96? 7. The diameter of a garden roller is 1.4m and it is 2m long. How much area will it cover in 5 revolutions? 8. Both the ogives (less than and more than) for a data intersect at P (30,15). Find the median for the data. 9. Without performing the actual division, state whether non-terminating repeating decimal. 10. If will represent a terminating decimal or a then find the mean. SECTION- B [2 marks each] 11. In a lottery there are 10 prizes and 25 blanks. What is the probability of getting a prize? 12. Find x so that the line segment with end points A (x, y) and B (3. 0) is divided at (2,1) in the ratio 3 : 1. Or Find the relation between x and y such that the point (x, y) may lie on the line joining the points (3, 4) and (- 5, - 6). 13. If 14. Find the distance between the points : R(a + b, a - b) and S(a - b, - a - b) 15. A bag contains 5 red balls. 8 white balls. 4 green balls and 7 black balls. If one ball is drawn at random, find the probability‟ that it is black. SECTION – C [3 marks each] 16. Determine graphically the vertices of a triangle, the equations of whose sides are (or) Solve for x and y 17. One fourth of a herd of camels were seen in the forest. Twice the square root of the herd gone to mountain and the remaining 15 camels were seen on the bank of a river. Find the total number of camels. 18. The second and third terms of an AP are 2 and 22 respectively. Find the sum of its first 30 terms. 19. Prove that 20. Points (1. 2), (3, - 4) and (5, - 6) lie on the circumference of a circle. Find the coordinates of its centre. (or) Using the formula of area of a triangle, show that the points (4, 3). (5. 1) and (1. 9) are collinear. 21. Three consecutive vertices of a parallelogram ABCD are A (1, 2), 13 (1, 0) and C (4, 0). Find the fourth vertex. 22. In what ratio is the line segment joining points P (4, 3) and Q (2,- 6) divided by the x-axis? Also, find the coordinates of the point of intersection. (or) Find the circumventer of the triangle whose vertices are (0, - 3), (7, 0) and (4, 7). 23. The perimeter of a sector of a circle with central angle 90° is 25 cm. Find the area of the minor segment of the circle. 2 24. In the given figure, DE // BC and CD // EF. Prove that AD =AB x AF 25. Find the area of the sector of a circle with radius 4 cm and of angle 30. Also find the area of the corresponding major sector. 51 Chandan’s Sample Papers : Sure Success and Excellent Marks SECTION – D [6 marks each] 26. A circular grassy plot of land, 42 m in diameter, has a path 3.5 m wide running round it on the outside. Find the cost of gravelling the path at Rs 4 per square meter. 27. A vertical tower is surmounted by a flagstaff of height h metres. At a point on the ground, the angles of elevation of the bottom and top of the flagstaff are respectively. Prove 28. If a line is drawn parallel to one side of a triangle, prove that the other two sides are divided in the same ratio. Use the above to prove the following. 29. A circus tent is cylindrical upto a height of 6 m and conical above it. If its diameter is 105 in and the slant height of the conical portion is 50 m, find the total area of canvas required to built it. (or) The height of a cone is 42 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume is 1/27 of the volume of the given cone, at what height above the base. 30.The mean of the following distribution is 18 and the sum of all frequencies is 64. Compute the missing frequencies f1 and f2. (or) Draw a less than type ogive for the following data and estimate the median from it. An essential aspect of creativity is not being afraid to fail Sample Paper - 26 SECTION - A [1 marks each] k the quadratic equation kx 5x k 0 has equal roots ? 2 2. If the sum of the squares of zeroes of quadratic polynomial f ( x) x 8 x k is 40 , find the value of k . 5 sin 3 cos 3. If 5 tan 4 , find the value of 5 sin 2 cos 2 1. For what values of 4. What is the probability of getting a number less than 7 in a single throw of a die ? 5. If the sum of n terms of an AP be 3n n and its common difference is 6, then find the first term.. 6. The length of tangent from a point A to a circle, of radius 3 cm is 4 cm. Find the distance of A from the centre of the circle. 7. Write the empirical relationship between the three measures of central tendency, namely, Mean, Mode and Median. 8. If the surface areas of two spheres are in the ratio 4 : 9 , then find the ratio of their volumes. 2 9. Without actually performing the long division, write whether the rational number 13 has a 6250 terminating decimal or a non terminating repeating decimal. 10. In figure, AB QR. Find the length of RB, if AB = 3 cm. 52 Chandan’s Sample Papers : Sure Success and Excellent Marks P 4 c m A B 3 cm Q R 9 cm SECTION – B [2 marks each] 11. Find the zeroes of the quadratic polynomial x zeroes and its coefficients 12. In figure, 2 7 x 12 , and verify the relationship between the QR QT and 1 2 . Show that PQS ~ TQR . QS PR T P T Q S R 13. Two dice, one blue and one grey, are thrown at the same time. Write down all possible outcomes. What is the probability that the sum of the two numbers appearing on the top of the dice is (i) 8? (ii) Less than or equal 12 14. If the points A(6,1) , B(8,2) , C (9,4) and D( p,3) are the vertices of a parallelogram, taken in order, find the value of p. 15. Meena went to a bank to withdraw Rs.2000. She asked the cashier to give her Rs.50 and Rs.100 notes only. Meena got 25 notes in all .Find how many notes of Rs.50 and Rs.100 she received. SECTION – C [3 marks each] 16. Prove that 6 2 is irrational. 17. In an A.P, if the 12th term is -13 and the sum of first four terms is 24, what is the sum of first 10 terms? ABC with side BC 6cm , AB 5cm and ABC 600 . Then construct a 3 triangle whose sides are of the corresponding sides of the triangle ABC . 4 18. Draw a triangle 53 Chandan’s Sample Papers : Sure Success and Excellent Marks 19. The median of triangle divides it into two triangles of equal areas. Verify this result for ABC whose vertices are A(4,6), B(3,2) and C (5,2). 20. Four horses are tethered at four corners of a square shaped grass field. If the length of the rope with which the horses are tethered is 10 meter and the length of each side of the square is 40 m. Find the area grazed by the horses. 21. Find the coordinates of the points of trisection (i.e., points dividing in three equal parts) of the line segment joining the points A(2. 2) and B(7,4) 22. QR is a chord of length 8cm of a circle of radius P. Find the length of PQ 5cm . The tangents at Q and R intersect at a point Q P O R 23. Two trains leave a railway station at the same time. The first train travels due west and the second train due north. The first train travels 5km/ h faster than the second train. If after two hours, they are 50km apart, find the average speed of each train 3 0 1 0 and cos( x y ) ; 0 x y 90 and x y , find x and y . 2 2 2 2 2 2 25. Show that: sin cos ec cos sec 7 tan cot 24. If cos(3 y 2 x) SECTION – D [6 marks each] 26. Form a pair of linear equations in two variables using the following information and solve it graphically: Aftab tells his daughter “Seven years ago, I was seven times as old as you were then. Also three years from now, I shall be three times as old as you will be”. Find their present ages. What was the age of Aftab when his daughter was born? 27. If the angle of elevation of a cloud from a point h meters above a lake is and that of depression of its reflection is , prove that the distance of the cloud from the point of observation is 2h sec tan tan . 28. If the mean of the following distribution is 19.92, find the missing frequencies f1 and f 2 Class 4-8 8-12 12-16 16-20 20-24 24-28 28-32 32-36 Number of 2 15 25 18 12 3 f1 f2 Students Total 100 XY is parallel to the side AC of AX ABC and XY divides the triangular region into two parts of equal areas. Find the ratio . AB 29. State and Prove Area Theorem. In given figure, the line segment X 54 Chandan’s Sample Papers : Sure Success and Excellent Marks B 30. A shuttle cock used for playing badminton has the shape of frustum of a cone mounted on a hemisphere. The external diameters of the frustum are 5cm and 2cm, the height of the entire shuttle cock is 7cm .Find its external surface area. It is better to have enough ideas for some of them to be wrong, than to be always right by having no ideas at all. ________________________________________________________________________ SAMPLE PAPER – 27 SECTION – A [1 mark each] 1. State the fundamental theorem of Arithmetic. 2. When does a parabola curve points downwards? 3. Find the mode of the data: 14,12,13,16,14,12,13,14,16 4. The perimeter of a sector of radius 3.5cm is 17.5cm. Find the length of its arc. 2 5. Two triangles are said to be similar. If area of the larger triangle is 121cm and the area of the smaller 2 triangle is 81cm . The side of the larger triangle is 14.4cm, find the side of the smaller triangle. 6. What is the standard form of a linear equation? th 7. Find the 24 term from the end of an AP 13,16,19,22… 8. Find the probability that a card drawn is a king from a pack of 52 cards. 9. Why mean is not suitable for finding the maximum number of TV programs watched by 200 families in a survey ? 10. Write the theoretical formula for calculating median of a distribution. SECTION – B [2 marks each] 11. Find the maximum number of biscuits that can be kept in 32 orange boxes and 24 red boxes. 12. Prove that tangent is perpendicular to the radius at point of contact. 13. A book has 5 lessons of economics , 15 lessons of geography , 7 lessons of history and 3 lessons of politics. Find the probability that the read lesson is: a). not history. B).neither politics nor geography 14. Prove (secA + tanA – 1) (secA – tanA + 1) = 2tanA. (or) 3 Prove that : (1 + tanA + cotA) (secA – cosA) = sin A + sinA.tanA. 2 15.The area of segment cut off from a circle of radius 12cm is 56cm . The sector so formed has a right angle in the centre. Find the area of remaining part of the circle which is not included in the sector. SECTION – C [3 marks each] 4 3 16.If the zeroes of the polynomial 3x – 2x + 6x – 14 are in AP , find the zeroes of the polynomial and verify the relationship between the zeroes and coefficients. 17.Find the median of the following distribution: CI 10-14 14-18 18-22 22-26 26-30 F 8 12 21 6 3 18. If the median of through the vertex A of a triangle is 12 units and the vertices of the midpoint of the triangle is (1,3) , B(2,-4) . Find the vertices of A and C. 19. If the points (-3,5) (2,-6) and (k,4) are collinear , find the value of k. 20.In a square of side 28cm , 4 quadrants are drawn simultaneously each of whose radius is 3.5cm , find the area of the remaining region. th th th 21.If the p term of an AP is 1/q , and the q term of an AP is 1/p. Show that the pq term is (or) th If the sum of m terms is same as the su of n terms show that the sum of (m+n) is 0. 22. A solid sphere of diameter 3.5m is melted and recast into small cylindrical balls each of radius 0.7m and height 1.6m . Find how many such balls can be obtained. 23. Find the total internal area of frustum of height 12cm and radii of its ends 8cm and 3cm. 2 2 24. D is a point on side BC of an equilateral triangle ABC such that DC = ¼ BC. Prove that AD = 13DC . (or) A point O in the interior of a rectangle ABCD is joined with each of the vertices A , B , C and D. Prove 2 2 2 2 that OB + OD = OA + OC . 55 Chandan’s Sample Papers : Sure Success and Excellent Marks 25. One fourth of a herd of camels was seen in the forest. Twice the square root of the camels had gone to mountains and the remaining 15 camels were seen on the bank of a river. Find the total number of camels. SECTION-D [6 marks each] 26. Prove that the tangents drawn from an external point to a circle are equal. Using the above result 0 solve this: ABCD is a quadrilateral circumscribing a circle and <D = 90 such that the circle touches side AB, BC, CD, AD at P,Q,R and S. If BC=25cm, BP=8cm, CD=33cm. Find the radius of the circle. (or) Prove that if a line parallel to one side of a triangle to intersect other two sides at two distinct points , then the other two sides are divided in the same ratio. Using above prove that in a trapezium ABCD , AB║ EF║ DC and E and F are points on the non-parallel sides , BF/FC = ¾ , show that 7FE = 10AB. 27. 2 women and 5 men can do a piece of work in 4 days. The same work can be 3 women and 6 men can do the work in 3 days. Find the time in which 1 woman and 1 man can do it alone. 0 28. Construct a triangle in which AB = 6.4cm , <BAC = 60 , BC = 4.2cm. Construct another triangle which has 7/5 of the sides of the given triangle Write the steps of construction. 29.Solve the following pair of linear equations graphically: 7x – 6y= 19 3x – 5y = 8 0 30. From the foot of the mountain , the angle of elevation of its summit is 45 . After ascending 1km 0 0 towards the cliff , at an inclination of 30 , the angle changes to 60 . Find the height o the mountain. No one travels so high as he who knows not where he is going. ___________________________________________________________________________________ SAMPLE PAPER – 28 SECTION – A [1 mark each] 1. If sum of the squares of zeros of the quadratic polynomial f(x) = x2 –8x + k is 40, find the value of k. 2. If the system of equations 3x + y = 1 and (2k – 1) x + (k – 1)y = 2K +1 is inconsistent, then find the value of k. 3.Find the sum of n terms of the series √2 + √8 + √18 + √32 + … 4. Find the value of √6 + √6 + √6 + … 5.If angles, A,B,C, of a ΔABC form an increasing AP, then find the value of Sin B 6. If ABC and DEF are similar triangles such that angle A = 470 and angle E = 830, then find angle C. 7. If four sides of a quadrilateral ABCD are tangential to a circle, then prove: AB + CD = BC + AD 8. The probability of guessing the correct answer to a certain test questions is x/12, if the probability of not guessing the correct answer to this question is 2/3 then find x. 9. If a cone is cut into two parts by a horizontal plane passing through the mid point of its axis, then find the ratio of the volumes of the upper part and the cone. 10.By which kind of graphs (Graphically) how we can obtain mean, mode, median. SECTION – B [2 marks each] 11. Find the area of a triangle, two sides of which are 8 cm and 11 cm and the perimeter in 32 cm. 12.If √3 tanθ = 3 sinθ, find the value of sin2θ – cos2θ 13.Find the distance between the points (a cos 350 , 0) and (0, a cos 550) 14. Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the centre. 15. Savita and Hamida are friends. What is the probability that both will have (i) the same birthday? (ii) Different birthdays? (Ignoring a leap year) SECTION – C [3 marks each] 16. Find the greatest number which divides 2011 and 2623 leaving remainders 9 and 5 respectively. (or) Prove that one of every three consecutive positive integers is divisible by 3. 17. A boat covers 32km upstream and 36 km downstream in 7 hours. Also, it covers 40km upstream and 48 km downstream in 9 hours. Find the speed of the boat in still water and that of the stream. (or) 8 men and 12 boys can finish a piece of work in 10 days while 6 men and 8 boys can finish it in 14 days. Find the time taken by one man alone and that by one boy alone to finish the work. 18. Find four number in A.P. whose sum is 20 and the sum of whose squares is 120. 56 Chandan’s Sample Papers : Sure Success and Excellent Marks 19. Fin the coordinates of the circumcentre of the triangle whose vertices are (8,6), (8,-2) and (2,-2). Also find its circum radius. 20. Determine the ratio in which the line 3x+y–9 =0 divides the segment joining the points (1,3) and (2,7) 21. The square ABCD is divided into five equal parts, all having same area. The central part is circular and lines AE,GC,BF and HD lie along the diagonals AC and BD of the square. If AB = 22cm , find (i) the circumference of the central part. (ii) the perimeter of the part ABEF. 22. Draw a triangle ABC with side BC = 7 cm, angle B = 45, angle A = 105, then construct a triangle whose sides are 4/3 times the corresponding side of ABC. 23. If two sides and a median bisecting the third side of a triangle are respectively proportional to the corresponding sides and the median of another triangle, then the two triangles are similar. 24.Prove that the line segments joining the mid points of the sides of a triangle form four triangles, each of which is similar to the original triangle. 25. Raghav buys a shop for Rs.1,20,000. He pays half of the amount in cash and agrees to pay the balance in 12 annual installments of Rs.5000 each. If the rate of interest is 12% and he pays with each installment the interest due on the unpaid amount, find the total cost of the shop. . SECTION – D [6 marks each] 26. (i) If the price of a book is reduced by Rs5, a person can buy 5 more books for Rs300. find the original list price of the book (ii) Students of a class are made to stand in rows, if one student is extra in a row, there would be 2 rows less. If one student is less in a row there would be 3 rows more, find the number of students in the class. 27. A man standing on the deck of a ship, which is 10m above water level. He observes the angle of elevation of the top of a hill as 60 and the angle of depression of the base of the hill as 30. calculate the distance of the hill from the ship and height of the hill . 28. Show that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two Sides. triangle ABC is an obtuse triange, otuse angles at B if AD perpendicular to CB show that AC2 = AB2 + BC2 + 2 BC X BD. 29.A circus tent is cylindrical to a height of 3m and conical above it. If its base radius is 52.5m and the slant height of the conical portion is 53m, find area of the canvas needed to make the tent. (use=22/7). (or) A wooden toy is conical at the top, cylindrical in the middle and hemispherical at the bottom. If the height and radius of the cylindrical portion are both equal to 21 cm and the total height of the toy is 70 cm, find the cost of painting its surface at a rate of Re 0.70 per sq.cm. 30. Find the median from the following data : Marks Number of Students Below 10 12 Below 20 32 Below 30 57 Below 40 80 Below 50 92 Below 60 116 Below 70 164 Below 80 200 The good ideas are all hammered out in agony by individuals, not spewed out by groups. ___________________________________________________________________________________ SAMPLE PAPER - 29 SECTION-A [1 mark each] 57 Chandan’s Sample Papers : Sure Success and Excellent Marks 1. If HCF(24,x)=6 and LCM(24,x)=144 then find the value of x. 3 2 -1 -1 - -1 2. If α, β, γ are the zeroes of P(x)= 3x +x -10x -8 , find α +β +γ . 3. For what values of β will the system of linear equations β+3y =β -3; 12x +βy =β have a unique solution? 4. Prove that Sin (B C) A Cos , where A,B and C are interior angles of ∆ABC. 2 2 (or) ˚ If sinA =1/2 and A+B=90 ,then what is the value of cotB. 2 2 ˚ 2 ˚ 5. Evaluate 5cos 60˚ +4sec 30 - tan 30 2 ˚ 2 ˚ Sin 30 + cos 30 6.Find AD, if AB=3cm, BC =6cm and CD = 7cm. C D B A 7.Find the ratio of the areas of a square and triangle in the figure given below where M is the mid point of AB. M A C B D 2 8. If -2 is one of the zero of the quadratic polynomial x -kx-8, find the other zero of the polynomial. 9. Three coins are tossed once , find the probability of getting at least one head. 58 Chandan’s Sample Papers : Sure Success and Excellent Marks 10. For what value of x , the mode will be 2 in the following data. 0, 2, 4, 3, 3, 4, 2, 3, x, 3, 4, 0, 2, 2. SECTION-B 11.In an A.P prove that t2 + t8 [2 marks each] =2 t5 (or) Prove that tp + t p+2q = 2 t p+q. 12. It is known that a box of 550 bulbs contains 4 ℅ defective bulbs. One bulb is taken out at random from the box. Find the probability of getting a good bulb. 2 2 13. Solve ax +by = a +b bx –ay = 0 14. The two vertices of a triangle are (6,7) and (4,-5). If the centroid of a triangle is origin , find the coordinates of the third vertex. (or) Find the value of p for which the points (-5,1), (1,p) and (4,2) are collinear. 15. If p and q are real and p≠ q, then show that the roots of the equation 2 (p-q)x +5(p+q)x – 2(p-q) = 0 are real and unequal. SECTION –C [3 marks each] 16.Draw the graph 2x-y = 6; and 2x –y +2 =0.Shade the region bounded by these lines and x-axis . Find the area of the shaded region. 2 17. Find the roots of the equation 2x -5x + 3 =0 by the method of completing the square. 18. Show that 3 -√5 is an irrational number. 2 19. If -4 is a root of the quadratic equation x +p x – 4 =0 and the quadratic equation 2 x +p x + k =0 has equal roots, find the value of k. 20. Construct a circle whose radius is equal to 4cm. Let P be a point whose distance from its centre is 6cm. Construct two tangents to it from P. 21. Prove that cotA +cosecA -1 1 +cosA ------------------- = -----------cotA –cosecA +1 sinA 22.Three vertices of a parallelogram ABCD are (0,0) (a,0) and (b,c). Find the co-ordinates of the fourth vertex. 23. .If PA and PB are two tangents to the circle whose centre is O, then prove that the quadrilateral AOBP is cyclic. 24. ABCD is a square whose each side is 14cm. find the area of the shaded region. A C B D 25. In the given figure , base BC of a triangle ABC is bisected at D and ∟ADB ,∟ADC are bisected by DE and DF respectively , meeting AB in E and AC in F. Show that EF║ BC 59 Chandan’s Sample Papers : Sure Success and Excellent Marks A E B F D C SECTION-D [6 marks each] ˚ 26.State and prove Pythagoras Theorem. In triangle ABC;∟B =90 and BD ┴ AC; AD=4cm and DC =9cm Find BD. A D D B C (or) State and prove Basic Proportionate Theorem and hence show that the diagonals of a trapezium divide each other proportionally. 27.From a point on the ground 40m away from the foot of a tower , the angles of elevation of the top of ˚ ˚ the tower and top of the water tank,which is fixed at the top nof the tower are respectively 30 and 45 . Find the height of the tower and the depth of the water tank. 28.A cylindrical bucket 32cm high and 18cm of radius of the base ,is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24cm, find the radius and slant height of the heap. 29.A two digit number is such that the product of the digits is 20. If 9 is subtracted from the number , the digits interchange their places . Find the number. 30.Find the mean, median and mode of the following data. Class 0-10 10-20 20-30 30-40 40-50 50-60 Frequency 12 14 21 24 13 16 What is now proved was once only imagined. ________________________________________________ SAMPLE PAPER–30 SECTION – A [1 mark each] 1. Find the HCF of 6 and 20 by prime factorisation. 2. Out of three equations which two of them have infinite many solutions : 3x – 2y = 4, 6x + 2y = 4, 9x – 6y = 12. 2 3. Find the nature of the roots of quadratic equation : 2x – 4x + 3 = 0. 4. Is 310 is a term of the A.P. 3, 8, 13, 18, …….? 2 2 5. The areas of two similar triangles ABC and DEF are 64 cm and 121 cm respectively. If EF = 13.2 cm, then find BC. 6. Show that the tangent lines at the end points of a diameter of a circle are parallel. 0 7. If sin 3A = cos ( A – 6 ) where 3A and ( A – 6 ) are acute angles, then find the value of A. 8. A chord of a circle of radius 7 cm subtends a right angle at the centre. Find the area of minor segment. 9. Calculate the mode of the following data : 4, 6, 7, 9, 12, 11, 13, 9, 13, 9, 9, 7, 8. 10. Two coins are tossed once. What is the probability of getting exactly one head ? SECTION – B [2 marks each] 60 Chandan’s Sample Papers : Sure Success and Excellent Marks 2 11. Find the zeros of the quadratic polynomial x – 2x – 8 and verify the relationship between the zeros and the coefficients. cos 67 0 tan 40 0 12. Evaluate : sin 90 0. 0 0 sin 23 cot 50 13. If two black kings and two red aces are removed from a deck of 52 cards and then well shuffled. One card is selected from the remaining. Find the probability of getting : (i) an ace of heart (ii) a king (iii) a red card (iv) a black queen. 14. The P( 2, – 3 ) is midpoint of A(1, 4 ) and B(x, y), find the value of x and y. 15. Prove that 2 is an irrational number. SECTION – C [3 marks each] 16. Solve the system of linear equations graphically : 2x + y = 6, x – 2y = – 2. Also, find the co-ordinates of the points where the lines meet the x-axis. 17. Solve for x and y : 47x + 31y = 63, 31x + 47y = 5. th 18. If the sum of first 14 terms of an A.P. is 1050 and its first term is 10, find the 20 term. 2 2 2 19. In acute triangle ABC acute angled at B. If AD BC, prove that AC = AB + BC – 2BC.BD. 20. Construct a triangle ABC similar to a given triangle with sides 6 cm, 7 cm and 8 cm and whose sides are 2/3 times the corresponding sides of the given triangle. 21. Prove that : cos A sin A sin A cos A . 1 tan A 1 cot A 22. The inner circumference of a circular track is 220 m. The track is 7 m wide. Calculate the cost of putting up a fence along the outer circle at a rate of Rs. 2 per meter. (or) The wheels of a car are of diameter 80 cm each. How many complete revolutions does each wheel make in 10 minutes when the car is travelling at a speed of 66 km per hour? 23. Find a point on x-axis which is equidistant from the points ( 7, 6 ) and ( – 3, 4 ). 24. Find the value of p for which the points ( – 1, 3 ), ( 2, p ) and ( 5, – 1 ) are collinear. 25. If A(– 5, 7 ), B(– 4, – 5), C(– 1, – 6) and D(4,5) are the vertices of a quadrilateral, find the area of the quadrilateral ABCD. SECTION – D [6 marks each] 26.In a flight of 600 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 200 km/h and the time increased by 30 minutes. Find the original duration of the flight. 27.Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. Using the above result, find the value of x if DE // BC in ABC and AD = x, DB = x – 2, AE = x + 2 and EC = x – 1. 28.A person standing on the bank of a river observes the angle of the elevation of the top of a tree 0 standing on the opposite bank is 60 .When he moves 40 m away from the bank, he finds the angle of 0 elevation to be 30 . Find the height of the tree and the width of the river. 29.A circus tent is cylindrical to a height of 3 m and conical above it. If its base radius is 52.5 m and the slant height of the conical portion is 53 m, find area of the canvas needed to make the tent.(use =22/7). (or) A wooden toy is conical at the top, cylindrical in the middle and hemispherical at the bottom. If the height and radius of the cylindrical portion are both equal to 21 cm and the total height of the toy is 70 cm, find the cost of painting its surface at a rate of Re 0.70 per sq.cm. 30.The distribution below gives the weights of 30 students of a class. Find the mean weight of the students. Weight ( in kg ) Number of students 40 - 45 2 45 - 50 3 50 - 55 8 55 - 60 6 60 - 65 6 65 - 70 3 70 - 75 2 Life is "trying things to see if they work". _____________________________________________________________________________________________________________________ SAMPLE PAPER – 31 SECTION- A [1 mark each] If H.C.F.(26,91)=13, find the L.C.M.(26,91). 2 If -2 is one of the zero of the quadratic polynomial x -kx-8, find the other zero of the polynomial. For what value of „k‟, the numbers 3k+2, 4k+3 and 6k-1 are the consecutive terms of an AP? For what value of „a‟, the following pair of equations will have a unique solution? 4x + 3y = 3 and 8x + ay =5 5. If each side of an equilateral triangle is „2a‟ units, what is the length of its altitude? 1. 2. 3. 4. 61 Chandan’s Sample Papers : Sure Success and Excellent Marks 6. If Sin (A + 2B) = √3 ∕ 2 and Cos (A + 4B) = 0, find A and B. 7. Two concentric circles are of radii „a‟ cm and „b‟ cm. Find the length of the chord of the larger circle which touches the smaller circle. 8. A solid cylinder of radius „r‟ cm and height „h‟ cm is melted and changed into a right circular cone of radius „4r; cm. Find the height of the cone. 9. What is the probability of a prime number in the factors of the number 20? 10. The median can graphically be found from (a) Ogive (b) histogram (c) Frequency curve (d) none of these SECTION – B [2 marks each] 2 11. Find the value of „k‟ for which the quadratic equation (k+1) x + (k+4) x + 1 = 0 has equal roots. 12. If 3 tan A = 4, find the value of 5 sin A – 3 cos A 5 sin A + 2 cos A 13. Find the value of p for which the points (-1, 3), (2, p) and (5, -1) are collinear. (or) If the point P(x, y) is equidistant from the points A (5,1) and B(-1, 5), prove that 3x = 2y. 14 . Prove that the intercept of a tangent between two parallel tangents to a circle subtends a rightangle at the centre. 15. One card is drawn from well-shuffled deck of 52 cards. Find the probability of getting (i) a king or a spade (ii) a king and a red card SECTION – C [3 marks each] 16. Show that for any odd positive integer to be a perfect square, it should be of the form 8k +1 for some integer k. 4 3 3 17. Obtain all the zeroes of the polynomial 3x + 6x - 2x – 10x + 5, if two of its zeroes are √5 / √3 and -√5 / √3 18. Solve the following system of equations graphically: 3x – 5y = 19, 3y -7x + 1 = 0. Does the point (4, 9) lie on any of these lines? Write its equation. 19. Find the sum of all multiples of 13 lying between 100 and 999. (or) 2 th If the sum of first n terms of an A.P. is given by S n = 4n – 3n, find the n term of the A.P. 20. Without using trigonometric table evaluate the following 2 2 Sec 39◦ + 2 tan 17◦ tan 38◦ tan 60◦ tan 52◦ tan 73◦ - 3(sin 31 + sin 59) Cosec 51◦ √3 (or) Prove that cot A + cosec A – 1 = 1 + Cos A cot A - cosec A + 1 Sin A 21. Prove that the centroid of triangle ABC whose vertices A(x1,y1), B(x2,y2) and C(x3, y3) are given by ( x1+ x2+ x3, y1+ y2+ y3 ) 3 3 (or) In what ratio is the line segment joining the points (-2,-3) and (3,7) divided by the y-axis? Also, find the coordinates of the point. 22. Show that the points A(5,6), B(1,5), C(2,1) and D(6,2) are the vertices of a square. 2 2 23. In an equilateral triangle PQR, the side QR is trisected at S. Prove that 9 PS = 7 PQ 24. Construct a triangle with sides 5 cm,6 cm and 7 cm and then construct another triangle whose sides are 7/5 of the corresponding sides of the first triangle. 25. PQRS is a diameter of a circle of radius 6 cm. The lengths PQ, OR and RS as diameters. Find the perimeter of the shaded region. SECTION- D [6 marks each] 26. Abdul traveled 300 Km by train and 200 Km by taxi, it took him 5 hous 30 minutes. But if he travels 260 Km by train and 240 Km by taxi, he takes 6 minutes longer. Find the speed of the train and that of the taxi. 62 Chandan’s Sample Papers : Sure Success and Excellent Marks 27. 28. 29. 30. (or) Two pipes running together can fill a cistern in 2 8/11 minutes. If one pipe takes 1 minute more than the other to fill the cistern, find the time in which each pipe would fill the cistern. If the radii of the ends of a bucket,45 cm high, are 28 cm and 7 cm, find the capacity and surface area. If the median of the distribution is 28.5, find the values of x and y. Class interval Frequency 0-10 5 10-20 x 20-30 20 30-40 15 40-50 3 50-60 Y Total 50 If the angle of elevation of the cloud from a point h m above a lake is α and the angle of depression of its reflection in the lake is β, prove that the height of the cloud is h(tan β + tan α) tan β – tan α (or) The angle of elevation of a jet plane from a point A on the ground is 60◦. After a flight of 15 seconds, the angle of elevation changes to 30◦. If the jet plane is flying at a constant height at a constant height of 1500√3 m, find the speed of the jet plane. The ratio of areas of similar triangles is equal to the ratio of the squares on the corresponding sides. Prove. Using the above theorem, prove that the area of the equilateral triangle described on the side of a square is half the area of the equilateral triangle described on the diagonal. Do not seek to follow in the footsteps of the men of old; seek what they sought. Sample Paper : 32 SECTION - A [ 1 Marks] 1. If the H C F of 309 and 657 is 9, find their L C M. 2. If α and β are the zeros of the polynomial 3x2 – 5x + 7, find the value of 1 1 . 3. The system of equations 3x – 4y + 7= 0, kx + 3y – 5 = 0 is inconsistent. Find the value of k. 4. Find the coordinate of the point at which the line 3x + 2y = 12 intersects the x-axis. 5. Find the 10th term of the A.P. 2, 8, 18, 32 ,........ 1 6. If 3 sin2 θ = 2 , find the value of θ. 4 7. Find the distance between the points (a cos 350, 0) and (0, a cos 650) A 8. In Δ ABC, DE // BC, so that AD = 2.4 cm, DB = 3.2 cm, and AC = 9.6 cm, then find EC? D E B C 0 9. Find the perimeter of a sector of a circle of radius 14 cm and central angle 60 . 10. In a throw of a pair of dice, what is the probability of getting a sum more than 7. SECTION – B [ 2 Marks] th 11. Find the 12 term from the end of the A.P. 3, 8, 13, .............., 253 12. If 3 cos θ – 4 sin θ = 2 cos θ + sin θ, find tan θ. OR If 3 tan 2x = cos 600 + sin 450 cos 450, find the value of x. 13. The diagonal BD of a parallelogram ABCD D C 63 Chandan’s Sample Papers : Sure Success and Excellent Marks intersects the segment AE at the point F, where E is any point on the side BC. Prove that: DF × EF = FB × FA E F A B 14. Find the value of k, if the point P (0, 2) is equidistant from (3, k) and (k, 5). 15. From a pack of 52 playing cards jacks, queens, kings and aces of red colour are removed. The remaining cards are shuffled and one card is taken at random. Find the probability that the card drawn is (i) a face card (ii) neither queen nor king. SECTION – C [ 3 Marks] 16. In a morning walk three persons step off together, their steps measure 80 cm, 85 cm and 90 cm respectively. What is the minimum distance each should walk so that they can cover the distance in complete steps? 1 1 1 1 17. Solve: OR ab x a b x If – 5 is a root of the quadratic equation 2x2 + px – 15 = 0 and the quadratic equation p(x2 + x) + k = 0 has equal roots, find the value of k. 18. If α and β are the zeros of the polynomial f(x) = 3x2 – 6x + 4, find a quadratic polynomial whose zeros are (α + β) and (α – β). 19. Prove that: (sin θ + sec θ)2 + ( cos θ + cosec θ)2 = (1 + sec θ cosec θ)2 OR If x = r Sin A cos C, y = r sin A sin C and z = r cos A, prove that r2 = x2 + y2 + z2. 20. The sum of the third and the seventh terms of an A.P. is 6 and their product is 8. Find the sum of first 16 terms of the A.P. 21. The vertices of a Δ ABC are (1, 2), (3, 1) and (2, 5). Point D divides AB in the ratio 2 : 1 and P is the mid-point of CD. Find the coordinates of the point P. OR The line joining the points (2, 1) and (5, – 8) is trisected at the points P and Q. If the point P lies on the line 2x – y + k = 0, find the value of k. 22. The area of a triangle is 5. Two of its vertices are (2, 1) and (3, – 2). The third vertex lies on y = x + 3. Find the third vertex. Page 2 of 4 23. Let ABC be a right triangle in which AB = 7 cm and B = 900 and BC = 5 cm. BD is the perpendicular from B on AC. The circle through B, C, D is drawn. Construct tangents from A to this circle. 24. In the adjoining figure ABC is a right angled triangle, B = 900, AB = 28 cm and BC = 21 cm. With AC as diameter a semicircle is drawn and as BC as radius a quadrant is drawn. Find the area of the shaded region. 25. In figure, 64 Chandan’s Sample Papers : Sure Success and Excellent Marks ABC and DBC are two triangles on the same base BC. If AD intersect BC at 0. ar. ABC AO Prove that: ar. DBC DO SECTION – D [ 6 Marks] 26. Determine graphically the vertices of the triangle, the equations of whose sides are given below: 2x – y + 1 = 0; x – 5y + 14 = 0; x – 2y + 8 =0 27. State and prove the Pythagoras theorem. Using the above theorem prove the following: In an isosceles triangle ABC with AB = AC, BD is perpendicular from B to the AC. Prove that: BD2 – CD2 = 2 CD . AD 28. A boy is standing on the ground and flying a kite with a string of 150 m, at an angle of elevation of 300. Another boy is standing on the top of a 25 m tall building and is flying his kite at an elevation of 450. Both the boys are opposite sides of both the kites. Find the length of the string in metres, correct to two decimal places that the second boy must have so that the two kites meet. OR Two pillars of equal height stand on either side of a roadway which is 150 m wide. From a point on the roadway between the pillars the elevations of the top of the pillars are 600 and 300. Find the height of the pillars and the position of the point. 29. A right triangle, whose sides are 15 cm and 20 cm, is made to revolve about its hypotenuse. Find the volume and surface area of the double cone so formed. (Use π = 3.14) OR A conical vessel of radius 6 cm and height 8 cm is completely filled with water. A sphere is lowered into the water and its size is such that when it touches the sides, it is just immersed as shown in the figure. What fraction of water overflows? 30. Compute the missing frequencies f 1 and f 2 in the following data if the mean is 9 166 and the sum of the observations is 52. 26 65 Chandan’s Sample Papers : Sure Success and Excellent Marks Classes Frequency 140 – 150 5 150 – 160 f1 160 - 170 20 170 – 180 f2 180 – 190 6 190 – 200 2 Total 52 “There are only two choices: either be a history reader or a history maker.” __________________________________________________________________________ QUESTION BANK (Set of all important questions most frequently asked) Exercise :1 1. Solve following system of linear equations for given variables: a) c) e) g) i) 1 1 1 1 = -1 , + =8 2x y x 2y 3 3 x + 2y = , 2x + y = 2 2 5x y x y =4 , + =4 3 6 8 4 x 7 + y = 0.8 , = 10 2 x ( y / 2) 1 1 1 1 + =3 , =5 7x 6 y 2x 3y 5 1 10 5 2 2 = , + = x 1 y 1 2 x 1 y 1 2 2 2 2 1 3 + = , + =0 3y 6 x x y x y d) + = 3 , 2x – y = 4 2 4 6 8 f) 4x + =15 , 6x =14 y y b) h) 7(y+3) -2(x+2) = 14 , 4(y-2) + 3( x-3)=2 j) 2 ( 3u – v ) = 5uv , 2(u+3v) = 5uv x y x y =2 , =6 xy xy 6 7 1 1 m) 149 x -330 y = -511 , -330 x +149y = -32 n) =3, =0 x y x y 2( x y ) 3( x y ) 17 2 3 5 1 o) + = , + =2 5 3x 2 y ) (3 x 2 y) (3x 2 y ) (3 x 2 y) k) l) 2. Solve following system of equations by method of cross multiplication:- x y a b , a x – b y = a2- b2 b) x + y = a + b , a x – b y = a 2 - b 2 a b x y c) = , a x + b y =a 2 + b 2 d) (a – b) x + (a + b) y = a 2 -2ab - b 2 , (a + b) (x + y) = a 2 + b 2 a b a) 3. Find the value of „k‟ in each case applying the given conditions:a) Find the value of k for which the following system of linear equations has a unique solution : 2x+5y=7, 3x–ky=5 66 Chandan’s Sample Papers : Sure Success and Excellent Marks b) Find value of k for which the following system of equations has no solution : kx+2y=1, 5x–3y = -2. c) Determine the value of k for which following system of equations has infinite solutions : (k – 3) x + 3 y = k , k x + k y =12. d) Determine value of k for which system of equations 3 x – k y =5, 2 x + 7 y = -6 has no unique solution. e) Find value of p for which given system has an infinite solution : 2x+3y = 4 , ( k + 2 ) x + 6 y = 3k + 2 4. The age of a father is 3 years more than 3 times the son‟s age .3 years hence the age of father will be 10 years more than twice the age of the son . Find their present ages . 5. The sum of a two digit number and the number obtained by reversing the order of digits is 121.The two digits differ by 3 .Find the number. 6. The age of a father is equal to the sum of the ages of his 5 children .After 15 years sum of the ages of the children will be twice the age of the father .Find the age of the father. 7. Students of a class are made to stand in rows. If 4 students are extra in a row , there would be two rows less .If 4 students are less in a row ,there would be four more rows .Find the number of students in the class. 8. Points A and B are 90 km apart from each other. A car starts from A and another from B at the same time .If they go in the same direction , they meet in 9 hours and if they go in opposite direction ,they meet in 9 hours .Find their speeds. 7 9. Ramesh travels 300 km to his home partly by train and partly by bus. He takes 4 hours , if he travels 60 km by train and rest by bus .If he travels 100 km by train and rest by a bus,he takes 10 minutes longer . Find speeds of train and the bus. 10. A man rowing at the rate of 5 km/hr in still water takes thrice as much time in rowing 40 km up the river as in 40 km down . Find the rate at which the river flows. 11. A takes 3 hours more than B to walk 30 km . But if a doubles his pace , he is ahead of B by 3 hours 2 .Find their speeds of walking. 12. A train covered a certain distance at a uniform speed .If the train would have been 6 km/h faster ,it would have taken 4 hours less than the scheduled time. And if the train were slower by 6 km/h it would have taken 6 hours more than the scheduled time .Find the length of the journey. 13 On selling a tea set at 5 % loss and a lemon set at 15 % gain ,a crockery seller gains Rs. 7 .If he sells the tea set at 5 5 gasin and the lemon set at 10 % gain ,he gains Rs. 13 .Find the actual price of the tea set. 14. In an examination paper ,one mark is awarded for every correct answer while 1 mark is deducted 4 for every wrong answer .A student answered 120 questions and got 90 marks .How many questions did he answer correctly . 15. A man sold a chair and a table together for Rs. 1520 thereby making a profit of 25% on chair and 10% on table. By selling them together for Rs. 1535 he would have made a profit of 10% on the chair and 25% on the table . Find cost price of each. 16. A boat goes 35 km upstream and 55 Km downstream in 12 hrs. It can go 30 Km upstream and 44 Km downstream in 10 hrs . Find the speed of the stream and that iof the boat in still water . ___________________________________________________________________ Exercise : 2 1. Find the area of the triangle formed by these lines and the x-axis. In fig. AD be a pole. Find the angle of elevation of the top of the pole from the point B. A 0 60 0 30 D 0 60 B C 2. Find the quadratic polynomial which represents the graph from the fig. given below. 67 Chandan’s Sample Papers : Sure Success and Excellent Marks Y 4 3 2 1 X X‟ -4 -3 -2 -1 0 1 2 3 -1 -2 -3 4 Y‟ 3. In fig. P ( -3, 3 ) is the mid-point of the line segment AB. Find the coordinates of A and B. Y B P A O X 4. From the graph given below state whether the triangle ABC is scalene, isosceles or equilateral. Justify your answer. Also find its area. Y 2 C X‟ 1 -5 -4 -3 -2 -1 B 0 1 2 3 4 5 6 7 X -1 -2 -3 -4 A 68 Chandan’s Sample Papers : Sure Success and Excellent Marks 5. What is the value of the median of the data using the graph given below of less than ogive and more than ogive. Y 50 40 35 C.f 30 20 10 0 10 20 C.I 30 40 50 60 X 6. In a classroom. 4 friends are seated at the points A, B, C and D as shown in fig. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli, “ Don‟t you think ABCD is a square ?” Chameli disagrees. Using distance formula, find which of them is correct. 10 9 8 7 6 5 4 3 2 1 B A C D 1 2 3 4 5 6 7 8 9 10 7. In fig. what are the angles of depression from the observation positions P and Q of the object. P Q 0 60 0 45 A B C 69 Chandan’s Sample Papers : Sure Success and Excellent Marks 8. In fig. what are the angles of depression of A and B from the observation point P. P 0 A 120 C 9. 10. 11. 12. 13. 14. 45 0 B 2 3 3 If α and β are the zeros of the polynomial x – 5x + 7, find α + β . 3 2 „1‟ and – 3 are the zeros of the polynomials x – ax – 13x + b, find the values of a and b. 2 2 2 2 If x = k sinA cosB, y = k sinA sinB and z = K cosA, then prove that x + y + z = k Solve the equation 1 + 6 + 11 + 16 + ……… + x = 148. Which term of the sequence 20, 19 ¼ , 18 ½, 17 ¾, ………… is the first negative term th th th If m times the m term of an A.P is equal to n times its n term, then show that its ( m + n ) term is zero. A 15 .In fig. DE // BC and AD : DB = 5 : 4. Find ar ( ΔDFE ) ar ( ΔCFB ) D E F B C 16. Two poles of height „a‟ and „b‟ are „c‟ metres apart. Prove that the height „h‟ metres of the point of intersection of the lines joining the top of each pole to the foot of the opposite pole is ab . a+b C A a b h B F D 17. A solid cone of height 12 cm and base radius 6 cm has top 4 cm removed as shown in the fig. Find the whole surface area of the remaining solid cone 4cm 12cm 6cm 18. A conical vessel of radius 6 cm and height 8 cm is completely filled with water. A sphere is lowered into the water and its size is such that when it touches the sides, it is just immersed. What fraction of water overflows? 70 Chandan’s Sample Papers : Sure Success and Excellent Marks 6cm 8cm 19. An oil funnel made of tin sheet consists of a cylindrical portion 10cm, long attached to a frustum of a cone. If the total height is 22cm, diameter of the cylindrical portion is 8cm, and the diameter of the top of the funnel is 18cm, find the area of the tin sheet required to make the funnel. 20. A right triangle whose sides are 15cm and 20cm is made to revolve about its hypotenuse. Find the volume and surface area of the double cone so formed. 21. The angle of elevation of the top of the tower from a point on the same level as the foot of tower is α , on advancing P metres towards the foot of the tower, the angle of elevation becomes β . Show that the height h of the tower is given by h = ptanβ tanα tanβ – tanα 0 0 Also determine the height of the tower when P = 150m, α = 30 and β = 60 . 22. From the top of a light house, the angles of depression of two ships on the opposite sides of it are observed to be α and β. If the height of the light house be h meters and the line joining the ships passes through the foot of the light house, show that the distance between the ships is h ( tan α + tan β ) meters. tan α tan β 23. A round balloon of radius „a‟ subtends an angle θ at the eye of the observer while the angle of elevation of its centre is φ . Prove that the height of the centre of the balloon is a sin φcosecθ/2 . 24. Two trains leave a railway station at the same time. The first train travels due west and the second due north. The first train travels 5km/hr faster than the second train. If after two hours, they are 50km apart, find the average speed of each train. 25. Find the value of „f‟, if the mean of the following distribution is 244. Class 200-220 220-240 240-260 260-280 280-300 Frequency 14 9 4 f 5 26. Find the mean salary of 60 workers of a factory from the following table. Salary (in Rs.) No. of workers 3000 16 4000 12 5000 10 6000 8 7000 6 8000 4 9000 3 10000 1 Total 60 27. The following table gives weekly wages in rupees of workers in a certain commercial organization. The frequency of class 49-52 is missing. It is known that the mean frequency distribution is 47.2. Find the missing frequency. Weekly Wages (Rs.) Number of workers 40-43 31 43-46 58 46-49 60 49-52 ? 52-55 27 Exercise :3 1. Find the smallest number which when increase by 17 is exactly divisible by both 520 and 468. 2. Find the smallest number which leaves remainders 8 and 12 when divided 3. Suppose you have 108 green marbles and 144 red marbles. You decide to separate them into packages of equal number of marbles. Find the maximum possible number of marbles in each package. 71 Chandan’s Sample Papers : Sure Success and Excellent Marks 4. Find the greatest number that will divide 55, 127 and 175, so as to leave the same remainder in each case. 5. Find the greatest possible rate at which a man should walk to cover a distance of 70 km and 245 km in exact number of days? 6. Three bells chime at an interval of 18, 24 and 32 minutes respectively. At a certain time they begin to chime together. What length of time will elapse before they chime together again? 7. Given that HCF (306, 657) = 9, find LCM (306, 657) n 8. Check whether 6 can end with the digit 0 for any natural number n. 9. Explain why 7 X 11 X 13 + 13 and 7 X 6 X 5 X 4 X 3 X 2 X 1 +5 are composite numbers. 10.There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point? 11. Prove that 3 is irrational. 2 is irrational. 13. Prove that 3 + 2 5 is irrational. 12. Show that 3 14. In a school there are two sections- A & B of class 10. There are 32 students in section A and 36 in section B. Determine the minimum numbers of books required for there class library so that they can be distributed among all the students 15. Find the greatest number of six digits exactly divisible by 24, 15 and 36. SOME IMPORTANT 6 MARKS QUESTIONS 1. An oil funnel made of tin sheet consists of a cylindrical portion 10cm, long attached to a frustum of a cone. If the total height is 22cm, diameter of the cylindrical portion is 8cm, and the diameter of the top of the funnel is 18cm, find the area of the tin sheet required to make the funnel. 2. A right triangle whose sides are 15cm and 20cm is made to revolve about its hypotenuse. Find the volume and surface area of the double cone so formed. 3. The angle of elevation of the top of the tower from a point on the same level as the foot of tower is α , on advancing P metres towards the foot of the tower, the angle of elevation becomes β . Show that the height h of the tower is given by h = ptanβ tanα tanβ – tanα 0 0 Also determine the height of the tower when P = 150m, α = 30 and β = 60 . 4. From the top of a light house, the angles of depression of two ships on the opposite sides of it are observed to be α and β. If the height of the light house be h metres and the line joining the ships passes through the foot of the light house, show that the distance between the ships is h ( tan α + tan β ) meters. tan α tan β 5. A round balloon of radius „a‟ subtends an angle θ at the eye of the observer while the angle of elevation of its centre is φ . Prove that the height of the centre of the balloon is a sin φcosecθ/2 . 6. Two trains leave a railway station at the same time. The first train travels due west and the second due north. The first train travels 5km/hr faster than the second train. If after two hours, they are 50km apart, find the average speed of each train. 7. A man on the top of a tower observes a car moving at a uniform speed coming directly towards the foot of the tower. If it takes 12 seconds for the angle of depression to change from 30 to 60, how soon after this, will the car reach the tower? 8. 26.State and prove Pythagoras Theorem. Using it, prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals. 9. A circus tent is cylindrical to a height of 3 m and conical above it. If its base radius is 52.5 m and the slant height of the conical portion is 53m, find its capacity and the area of the canvas needed to make the tent. ( use = 22/7) o 10. Point A is 45 . After going up a distance of 600 meters towards the top of the cliff at an inclination of o o 30 , it is found that the angle of elevation is 60 . Find the height of the cliff. (or) An aeroplane, when 3000 m high, passes vertically above another aeroplane. At an instant when the 0 0 angles of elevation of the two aeroplanes from the same point on the ground are 60 and 45 respectively. Find the vertical distance between the two aeroplanes. ******************************************************************************************* 72 Chandan’s Sample Papers : Sure Success and Excellent Marks