# File

## Comments

## Transcription

File

S.No. Chapter Page 1. Real Numbers 1 2. Polynomials 7 3. Pair of Linear Equations in two Variables 13 4. Similar Triangles 5. Trigonometry 35 6. Statistics 45 T N OI P N WI [Class-X – Maths] 4 22 1. Euclid’s division lemma : For given positive integers ‘a’ and ‘b’ there exist unique whole numbers ‘q’ and ‘r’ satisfying the relation a = bq + r, 0 r < b. 2. Euclid’s division algorithms : HCF of any two positive integers a and b. With a > b is obtained as follows: T N OI Step 1 : Apply Euclid’s division lemma to a and b to find q and r such that a = bq + r , 0 r < b. P N WI Step 2 : If r = 0, HCF (a, b) = b Step 3 : if r 0, apply Euclid’s lemma to b and r. 3. The Fundamental Theorem of Arithmetic : Every composite number can be expressed (factorized) as a product of primes and this factorization is unique, apart from the order in which the prime factors occur. p , q 0 to be a rational number, such that the prime q factorization of ‘q’ is of the form 2m5n, where m, n are non-negative integers. Then x has a decimal expansion which is terminating. 4. Let x 5. Let x 6. p is irrational, where p is a prime. A number is called irrational if it cannot p be written in the form q where p and q are integers and q 0. p , q 0 be a rational number, such that the prime factorization q of q is not of the form 2m5n, where m, n are non-negative integers. Then x has a decimal expansion which is non-terminating repeating. 5 Page 1 [Class-X – Maths] 1. 2. 5 × 11 × 13 + 7 is a (a) prime number (b) composite number (c) odd number (d) none Which of these numbers always ends with the digit 6. (a) 4n (b) 2n (c) 6n (d) 8n where n is a natural number. 3. 4. 5. 6. 7. For a, b (a b) positive rational numbers ____ (a) Rational number (c) a b a (b) 2 T N OI (d) P N WI b a b is a irrational number 0 If p is a positive rational number which is not a perfect square then 3 p is (a) integer (c) irrational number (b) rational number (d) none of the above. All decimal numbers are– (a) rational numbers (b) irrational numbers (c) real numbers (d) integers In Euclid Division Lemma, when a = bq + r, where a, b are positive integers which one is correct. (a) 0 < r b (b) 0 r < b (c) 0 < r < b (d) 0 r b Which of the following numbers is irrational number (a) 3.131131113... (b) 4.46363636... (c) 2.35 (d) b and c both [Class-X – Maths] 6 Page 2 21 8. The decimal expansion of the rational number 7 2 5 4 will terminate after ___ decimal places. 9. 10. 11. 12. 13. 14. (a) 8 (b) 4 (c) 5 (d) never HCF is always (a) multiple of L.C.M. (b) Factor of L.C.M. (c) divisible by L.C.M. (d) a and c both The product of two consecutive natural numbers is always. (a) an even number (b) an odd number (c) a prime number (d) none of these Which of the following is an irrational number between 0 and 1 (a) 0.11011011... (c) 1.010110111... T N OI P N WI (b) 0.90990999... (d) 0.3030303... pn = (a × 5)n. For pn to end with the digit zero a = __ for natural no. n (a) any natural number (b) even number (c) odd number (d) none. A terminating decimal when expressed in fractional form always has denominator in the form of — (a) 2m3n, m, n 0 (b) 3m5n, m, n 0 (c) 5n 7m, m, n 0 (d) 2m5n, m, n 0 3 3 – 5 5 is 4 4 (a) An irrational number (b) A whole number (c) A natural number (d) A rational number 7 Page 3 [Class-X – Maths] 15. 16. If LCM (x, y) = 150, xy = 1800, then HCF (x, y) = (a) 120 (b) 90 (c) 12 (d) 0 Which of the following rational numbers have terminating decimal expansion? (a) (c) 91 2100 (b) 343 2 3 5 2 7 (d) 3 64 455 29 73 17. Solve 50. What type of number is it, rational or irrational. 18. Find the H.C.F. of the smallest composite number and the smallest prime number. 19. If a = 4q + r then what are the conditions for a and q. What are the values that r can take? 20. What is the smallest number by which 5 3 be multiplied to make it a rational number? Also find the number so obtained. 21. What is the digit at unit’s place of 9n? 22. Find one rational and one irrational no. between 23. State Euclid’s Division Lemma and hence find HCF of 16 and 28. 24. State fundamental theorem of Arithmetic and hence find the unique factorization of 120. 25. Prove that 26. Prove that 5 18 T N OI P N WI [Class-X – Maths] 1 2 5 2 7 is irrational number.. 3 is irrational number.. 8 Page 4 3 and 5. 27. Prove that 28. Find HCF and LCM of 56 and 112 by prime factorisation method. 29. Why 17 + 11 × 13 × 17 × 19 is a composite number? Explain. 30. Check whether 5 × 6 × 2 × 3 + 3 is a composite number. 31. Check whether 14n can end with the digit zero for any natural number, n. 32. If the HCF of 210 and 55 is expressible in the form 210 × 5 + 55y then find y. 33. Find HCF of 56, 96 and 324 by Euclid’s algorithm. 34. Show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m. 35. Show that any positive odd integer is of the form 6q + 1, 6q + 5 where q is some integer. 36. Two milk containers contains 398 l and 436 l of milk. The milk is to be transferred to another container with the help of a drum. While transferring to another container 7l and 11l of milk is left in both the containers respectively. What will be the maximum capacity of the drum. 37. Show that 38. A sweet seller has 420 Kaju burfis and 130 Badam burfis. He wants to stack them in such a way that each stack has the same number and they take up the least area of the tray. 39. 2 7 is not rational number.. T N OI P N WI 7 is not a rational number.. i. What is the number of burfis that can be placed in each stack for this purpose? ii. Which value of the sweet seller is reflected in the question? An Army contingent of 616 members is to march behind an army band of 32 members in a parade, the two groups are to march in the same number 9 Page 5 [Class-X – Maths] of columns. What is the maximum number of columns in which they can march ? What values are reflected by the members? 40. In the morning Ankita and Salma walk around a rectangular park. Ankita takes 18 minutes to cover one round of the park while Salma 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, find i. After how many minutes will they meet again at the starting point? ii. What values of Ankita and Salma are reflected in the question? 1. b 2. c 3. a 4. c 5. c 6. b 7. a 8. b 9. b 10. a 11. b 12. b 13. d 15. c 17. 30, rational 18. 2 19. r, q whole no. 0 r < 4 20. 21. 1 23. 4 24. 2 × 2 × 2 × 3 × 5 28. HCF = 56, LCM = 112 30. Yes 31. No 32. HCF (210, 55) = 5, 5 = 210 × 5 + 55y y = – 19 33. 4 34. a = 3q + r 35. a = 6q + r 36. 17 litre 38. 10, Economic value 39. 8, Work in groups, Discipline etc. 40. 36 minutes, Health Awareness etc. P N I W [Class-X – Maths] T N OI 14. d 16. c 10 Page 6 5 3 , 2 1. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively. 2. A quadratic polynomial in x with real coefficient is of the form ax2 + bx + c, where a, b, c are real numbers with a 0. 3. The zeroes of a polynomial p(x) are precisely the x–coordinates of the points where the graph of y = p(x) intersects the x-axis i.e. x = a is a zero of polynomial p(x) if p(a) = 0. 4. A polynomial can have at most the same number of zeros as the degree of polynomial. 5. For quadratic polynomial ax2 + bx + c (a 0) T N OI P N WI Sum of zeros b a Product of zeros 6. c . a The division algorithm states that given any polynomial p(x) and polynomial g(x), there are polynomials q(x) and r(x) such that : p(x) = g(x).q (x) + r(x), g(x) 0 where r(x) = 0 or degree of r(x) < degree of g(x). 1. A real no. is a zero of the polynomial f(x) if (a) f() > 0 (b) f() = 0 (c) f() < 0 (d) none 11 Page 7 [Class-X – Maths] 2. 3. 4. 5. 6. 7. 8. 9. The zeroes of a polynomial f(x) are the coordinates of the points where the graph of y = f(x) intersects (a) x-axis (b) y-axis (c) origin (d) (x, y) If is a zero of f(x) then ____ is one of the factors of f(x) (a) (x – ) (b) (x – 2) (c) (x + ) (d) (2x – ) If (y – a) is factor of f(y) then ___ is a zero of f(y) (a) y (b) a (c) 2a (d) 2y If 1 is one of the zeroes of the polynomial x2 – x + k, the value of k is: (a) 0 (c) 1 (b) –2 (d) 3 (b) two points (d) four points T N I O P IN Cubic polynomial x = f(y) cuts y-axis at atmost (a) one point (c) three points W Polynomial x2 + 1 has ___ zeroes (a) only one real (b) no real (c) only two real (d) one real and the other non-real. If 1 is one of the zeroes of the polynomial ax2 – bx + 1 then. (a) a – b = 0 (b) a – b – 1 = 0 (c) a + 1 = b (d) a + b +1 = 0 If degree of polynomial f(x) is ‘n’ then maximum number of zeroes of f(x) would be – (a) n (b) 2n (c) n + 1 (d) n – 1 [Class-X – Maths] 12 Page 8 10. 11. 12. 13. If 2 is a zero of both the polynomials, 3x2 + ax – 14 and 2x – b then a – 2b = ___ (a) –2 (b) 7 (c) –8 (d) –7 If zeroes of the polynomial ax2 + bx + c are reciprocal of each other then (a) a = c (b) a = b (c) b = c (d) a = – c The zeroes of the polynomial h(x) = (x – 5) (x2 – x–6) are (a) –2, 3, 5 (b) –2, –3, –5 (c) 2, –3, –5 (d) 2, 3, 5 If are the zeroes of the polynomial x2 + x + 1, then + = (a) –1 (c) 1 (b) T N OI (d) 0 2 P N WI 14. If and are the zeroes of the polynomial 2x2 – 7x + 3. Find the sum of the reciprocal of its zeroes. 15. If are the zeroes of the polynomial p(x) = x2 – a (x + 1) – b such that ( + 1) ( + 1) = 0 then find value of b. 16. If are the zeroes of the polynomial x2 – (k + 6) x + 2 (2k – 1). Find 1 k if . 2 17. If (x + p) is a factor of the polynomial 2x2 + 2px + 5x + 10 find p. 18. Find a quadratic polynomial whose zeroes are 5 3 2 and 5 3 2 . 19. If 1 and – 2 are respectively product and sum of the zeroes of a quadratic 5 polynomial. Find the polynomial. 13 Page 9 [Class-X – Maths] 20. If one of the zero of the polynomial g(x) = (k2 + 4) x2 + 13x + 4k is recoprocal of the other, find k. 21. If be the zeroes of the quadratic polynomial 2 – 3x – x2 then find the value of + (1 + ). 22. Form a quadratic polynomial, one of whose zero is 2 of zeroes is 4. 23. If sum of the zeroes of kx2 + 3k + 2x is equal to their product. Find k. 24. If one zero of 4x2 – 9 – 8kx is negative of the other, find k. 25. Find the quadratic polynomial whose zeroes are 2 and –3. Verify the relation between the coefficients and the zeroes of the polynomial. 26. If one zero of the polynomial (a2 + a) x2 + 13x + 6a is reciprocal of the other, find value (s) of a. 27. –5 is one of the zeroes of 2x 2 + px – 15. Quadratic polynomial p(x2 + x) + k has both the zeroes equal to each other. Then find k. 28. What should be subtracted from the polynomial 2x3 + 5x2 – 14x + 10 so that the resultant polynomial is a multiple of (2x – 3). 29. If f(x) = 2x4 – 5x3 + x2 + 3x – 2 is divided by g(x), the quotient is q(x) = 2x2 – 5x + 3 and r(x) = – 2x + 1 find g(x). 30. If (x – 2) is one of the factors of x3 – 3x2 – 4x + 12, find the other zeroes. 31. If and are the zeroes of the polynomial x2 – 5x + k such that – = 1, find the value of k. 32. If are zeroes of quadratic polynomial 2x2 + 5x + k, find the value of 21 k, such that ( )2 – = . 4 33. Obtain all zeroes of x4 – x3 –7x2 + x + 6 if 3 and 1 are zeroes. 34. If the two zeroes of the polynomial 2x4 – 2x3 – px2 + qx – 6 are –1 and 2, find p and q. 5 and the sum T N OI P N WI [Class-X – Maths] 14 Page 10 35. If 2 3 and 2 3 are two zeroes of x4 – 4x3 – 8x2 + 36x – 9 find the other two zeroes. 36. What must be subtracted from 8x4 + 14x3 – 2x2 + 7x – 8 so that the resulting polynomial is exactly divisible by 4x2 + 3x – 2. 37. When we add p(x) to 4x4 + 2x3 – 2x2 + x – 1 the resulting polynomial is divisible by x2 + 2x – 3 find p(x). 38. Find a and f if (x4 + x3 + 8x2 + ax + f) is a multiple of (x2 + 1). 39. If the polynomial 6x4 + 8x3 + 17x2 + 21x + 7 is divided by 3x2 + 1 + 4x then r(x) = (ax + b) find a and b. 40. If Honesty and Dishonesty are the zeroes of the polynomial ax2 + bx + c, a 0 and are the reciprocal to each other then 41. 42. T N OI i. Find a relation between a and c ii. What values from the question should be adopted by the people in their life? P N WI An NGO distributed K number of Books on moral-education to the students. If K is the zero of polynomial x2 – 100x – 20000 then i. How many books were distributed by the NGO? ii. Write any three moral values which you should adopt in your life by reading such books. If 3 is one of the zero of the polynomial x3 – 12x2 + 47x – 60 and the remaining two zeroes are the number of plants, planted by the two students then i. Find the total number of plants, planted by both the students. ii. What value is reflected in this question? 1. b 2. a 3. a 4. b 15 Page 11 [Class-X – Maths] 5. a 6. c 7. b 8. c 9. a 10. d 11. a 12. a 13. a 14. 15. 1 16. k = 7 17. p = 2 18. x2 – 10x + 7 19. x 21. k = – 5 23. 25. x2 + x – 6 27. p 7, k 29. g(x) = x2 – 1 31. k = 6 33. –2, –1 34. 1, –3 35. ± 3 36. 14x – 10 37. 61x – 65 38. r(x) = 0 a 1 x f 7 0 a 1 and f 7 39. r (x) = x + 2 = ax + b a = 1 and b = 2 40. a = c, Honesty 41. (i) 200 (ii) Honesty, Discipline etc. 42. (i) 9 (ii) Love towards nature, plantation etc. 2 2x 1 5 1 1 7 3 20. 2 22. x2 – 4x – 1 2 3 24. 0 7 4 P N I W [Class-X – Maths] T N OI 26. 5 28. 7 30. –2, 3 32. k = 2 16 Page 12 1. The most general form of a pair of linear equations is : a1x + b1y + c1 = 0 a2x + b2y + c2 = 0 Where a1, a2, b1, b2, c1, c2 are real numbers and a12 + b12 0, a22 + b22 0. 2. 3. T N OI The graph of a pair of linear equations in two variables is represented by two lines; P N WI (i) If the lines intersect at a point, the pair of equations is consistent. The point of intersection gives the unique solution of the equation. (ii) If the lines coincide, then there are infinitely many solutions. The pair of equations is consistent. Each point on the line will be a solution. (iii) If the lines are parallel, the pair of the linear equations has no solution. The pair of linear equations is inconsistent. If a pair of linear equations is given by a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 (i) (ii) a1 b 1 the pair of linear equations is consistent. (Unique a2 b2 solution). a1 b c 1 1 the pair of linear equations is inconsistent a2 b2 c2 (No solution). 17 Page 13 [Class-X – Maths] (iii) a1 b c 1 1 the pair of linear equations is dependent and a2 b2 c2 consistent (infinitely many solutions). 1. 2. 3. 4. 5. 6. Every linear equation in two variables has ___ solution(s). (a) no (b) one (c) two (d) infinitely many a1 b c 1 1 is the condition for a2 b2 c2 (a) intersecting lines (b) parallel lines (c) coincident lines (d) none T N OI For a pair of linear equations in two variables to be consistent and dependent the pair must have P N I (a) no solution (c) infinitely many solutions W (b) unique solution (d) none of these Graph of every linear equation in two variables represent a ___ (a) point (b) straight line (c) curve (d) triangle Each point on the graph of pair of two lines is a common solution of the lines in case of ___ (a) Infinitely many solutions (b) only one solution (c) no solution (d) none of these If the system of equations 6x – 2y = 3 and Kx – y = 2 has unique solution then K is (a) K = 3 (b) K = 4 (c) K 3 (d) K 4 [Class-X – Maths] 18 Page 14 7. 8. 9. 10. 11. 12. 13. One of the common solution of ax + by = c and y-axis is _____ (a) c 0, b (b) b 0, c (c) c b , 0 (d) c 0, b For x = 2 in 2x – 8y = 12 the value of y will be (a) –1 (b) 1 (c) 0 (d) 2 The pair of linear equations is said to be inconsistent if they have (a) only one solution (b) no solution (c) infinitely many solutions. (d) both a and c On representing x = a and y = b graphically we get ____ T N OI (a) parallel lines (c) intersecting lines at (a, b) P N WI (b) coincident lines (d) intersecting lines at (b, a) A motor cyclist is moving along the line x – y = 2 and another motor cyclist is moving along the line x + y = 2. Tell whether the y (a) move parallel (b) move coincidently (c) collide somewhere (d) none For 2x + 3y = 4, y can be written in terms of x as— (a) y 4 2x 3 (b) y 4 3x 2 (c) x 4 3y 2 (d) y 4 2x 3 For what value of p, the pair of linear equations 2x + py = 8 and x + y = 6 has a unique solution x = 10, y = –4. (a) –3 (b) 3 (c) 2 (d) 6 19 Page 15 [Class-X – Maths] 14. 15. 16. The point of intersection of the lines x – 2y = 6 and y-axis is (a) (–3, 0) (b) (0, 6) (c) (6, 0) (d) (0, –3) Graphically x – 2 = 0 represents a line (a) parallel to x-axis at a distance 2 units from x-axis. (b) parallel to y-axis at a distance 2 units from it. (c) parallel to x-axis at a distance 2 units from y-axis. (d) parallel to y-axis at a distance 2 units from x-axis. If ax + by = c and lx + my = n has unique solution then the relation between the coefficients will be ____ (a) am lb (b) am = lb (c) ab = lm (d) ab lm T N OI P N WI 17. Form a pair of linear equations for : If twice the son’s age is added to father’s age, the sum is 70. If twice the father’s age is added to the son’s age the sum is 95. 18. Amar gives 9000 to some athletes of a school as scholarship every y month. Had there been 20 more athletes each would have got 160 less.. Form a pair of linear equations for this. 19. Write a pair of linear equations which has the unique solution x = –1 and y = 4. How many such pairs are there? 20. What is the value of a for which (3, a) lies on 2x – 3y = 5. 21. Solve : x + 2y – 8 = 0 22. 2x + 4y = 16 Dinesh is walking along the line joining (1, 4) and (0, 6), Naresh is walking along the line joining (3, 4,) and (1,0). Represent on graph and find the point where both of them cross each other. [Class-X – Maths] 20 Page 16 23. Solve the pair of linear equations x – y = 2 and x + y = 2. Also find p if p = 2x + 3 24. Check graphically whether the pair of linear equations 3x + 5y = 15, x – y = 5 is consistent. Also check whether the pair is dependent. 25. For what value of p the pair of linear equations (p + 2) x – (2 p + 1)y = 3 (2p – 1) 2x – 3y = 7 has unique solution. 26. Find the value of K so that the pair of linear equations : (3 K + 1) x + 3y – 2 = 0 (K2 + 1) x + (k–2)y – 5 = 0 is inconsistent. 27. Given the linear equation x + 3y = 4, write another linear equation in two variables such that the geometrical representation of the pair so formed is (i) intersecting lines (ii) parallel lines (iii) coincident lines. 28. Solve x – y = 4, x + y = 10 and hence find the value of p when y = 3 x –p 29. Determine the values of a and b for which the given system of linear equations has infinitely many solutions: T N OI P N WI 2x + 3y = 7 a(x + y) – b(x – y) = 3a + b – 2 30. The difference of two numbers is 5 and the difference of their reciprocals is 31. 1 . Find the numbers 10 Solve for x and y : x 1 y 1 x 1 y 1 8; 9 2 3 3 21 Page 17 2 [Class-X – Maths] 32. Solve for x and y x + y = a + b ax – by = a2 –b2 33. Solve for x and y 139x 56 y 641 56x 139 y 724 34. Solve for x and y 5 1 2 x y x y 15 5 2 x y x y 35. T N OI Solve the following system of equations graphically x + 2y = 5, 2x – 3y = – 4. P N WI Write the coordinates of the point where perpendicular from common 1 solution meets x-axis. Does the point – , 1 lie on any of the lines? 2 36. Draw the graph of the following pair of linear equations x + 3y = 6 and 2x – 3y = 12 Find the ratio of the areas of the two triangles formed by first line, x = 0, y = 0 & second line x = 0, y = 0. 37. Solve for x and y 1 12 1 2 2 2x 3 y 7 3x 2y 7 4 2 for 2x + 3y 0 and 3x – 2y 0 2x 3 y 3 x 2 y [Class-X – Maths] 22 Page 18 38. Solve for p and q p q p q 2, 6, p 0, q 0. pq pq 39. 40. On selling a T.V. at 5% gain and a fridge at 10% gain, a shopkeeper gains Rs. 2000. But if he sells the T.V. at 10% gain & the fridge at 5% loss, he gains Rs. 1500 on the transaction. Find the actual price of the T.V. and the fridge. 2 x 3 y 2, 4 x 9 y 1 ; x 0, y 0 41. If from twice the greater of two numbers, 20 is subtracted, the result is the other number. If from twice the smaller number, 5 is subtracted, the result is the greater number. Find the numbers. 42. In a deer park the number of heads and the number of legs of deer and visitors were counted and it was found that there were 39 heads and 132 legs. Find the number of deers and visitors in the park, using graphical method. 43. 44. T N OI P N WI A two digit number is obtained by either multiplying the sum of the digits by 8 and adding 1; or by multiplying the difference of the digits by 13 and adding 2. Find the number. How many such numbers are there. 1 In an examination one mark is awarded for every correct answer and 4 mark is deducted for every wrong answer. A student answered 120 questions and got 90 marks. How many questions did he answer correctly? 45. A boatman rows his boat 32 km upstream and 36 km down stream in 7 hours. He can row 40 km upstream and 48 km downstream in 9 hours. Find the speed of the stream and that of the boat in still water. 46. In a function if 10 guests are sent from room A to B, the number of guests in room A and B are same. If 20 guests are sent from B to A, the number of guests in A is double the number of guests in B. Find number of guests in both the rooms in the beginning. 47. In a function Madhu wished to give Rs. 21 to each person present and found that she fell short of Rs. 4 so she distributed Rs. 20 to each and 23 Page 19 [Class-X – Maths] found that 1 were left over. How much money did she gave and how w many persons were there. 48. A mobile company charges a fixed amount as monthly rental which includes 100 minutes free per month and charges a fixed amount there after for every additional minute. Abhishek paid 433 for 370 minutes and Ashish paid 398 for 300 minutes. Find the bill amount under the same plan, if Usha use for 400 minutes. 49. Ravi used 2 plastic bags and I paper bag in a day which cost him 35 while Mahesh used 3 plastic bags and 4 paper bags in a day which cost him 65. 50. i. Find the cost of each bag. ii. Which bag has to be used and what value is reflected by using it? T N OI Two schools want to award their selected students on the values of Discipline and Punctuality. First school wants to award these values to its 3 and 2 students respectively, while second wants to award for the same values to its 4 and 1 students respectively. If the amount of award for each value given by both the school are same and total amount spent by each school is 600 and 550 respectively then. P N WI i. Find the award money for each value. ii. Which more values of the students may be chosen for the award? 1. d 2. c 3. c 4. b 5. a 6. c 7. a 8. a 9. b 10. c 11. c 12. d [Class-X – Maths] 24 Page 20 13. b 14. d 15. b 16. a 17. Father’s age = x years, Son’s age = y years x + 2y = 70, 2x + y = 95 18. No. of athletes = x, No. of athletes increased = y 19. Infinite 20. 21. Infinite 22. (2, 2) 23. (2, 0) P = 7 24. No 25. p 4 26. k 1, k 28. (7, 3), 18 29. a = 5, b = 1 30. –5, –10 or 10, 5 31. (7, 13) 32. x = 1, y = b 33. (2, –1) 34. (3, 2) 35. (1, 2), (1, 0), yes 36. (6, 0), 1 : 2 1 3 19 2 T N OI P N WI 37. (2, 1) 40. 1 1 2, 4 (4, 9) 42. 27, 12 43. 41 or 14(2) 44. 96 45. 2 km/hr, 10km/hr. 46. 100, 80 47. Rs. 101, 5 38. 39. 20000, 41. 15, 10 49. 1 Rs. 298, Rs. 2 Rs. 448 (i) 15, 5, (ii) Environment friendly 50. (i) 100, 150 48. 100000 (ii) Honesty, regularity etc. 25 Page 21 [Class-X – Maths] 1. Similar Triangles : Two triangles are said to be similar if their corresponding angles are equal and their corresponding sides are proportional. 2. Criteria for Similarity : in ABC and DEF (i) AAA Similarity : ABC ~ DEF when A = D, B = E and C = F (ii) SAS Similarity : P N I W ABC ~ DEF when (iii) 3. T N OI AB BC and B E DE EF SSS Similarity : ABC ~ DEF , AB AC BC . DE DF EF The proof of the following theorems can be asked in the examination : (i) Basic Proportionality Theorem : If a line is drawn parallel to one side of a triangle to intersect the other sides in distinct points, the other two sides are divided in the same ratio. (ii) The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. (iii) Pythagoras Theorem : In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. [Class-X – Maths] 26 Page 22 (iv) 1. 2. Converse of Pythagoras Theorem : In a triangle, if the square of one side is equal to the sum of the squares of the other two sides then the angle opposite to the first side is a right angle. ABC ~ DEF. If DE = 2 AB and BC = 3cm then EF is equal to _______. (a) 1.5 cm (b) 3 cm (c) 6 cm (d) 9 cm In DEW, AB || EW if AD = 4 cm, DE = 12cm and DW = 24 cm then the value of DB = ____ (a) 4 cm (b) 8 cm (c) 12 cm (d) 16 cm 3. A O P IN Q W c b D Q f e O O B T N I a C E d F In the figure the value of cd = ________ 4. (a) ae (b) af (c) bf (d) be If the corresponding medians of two similar triangles are in the ratio 5 : 7, then the ratio of their sides is : (a) 25 : 49 (b) 7:5 (c) 1:1 (d) 5:7 27 Page 23 [Class-X – Maths] 5. 6. AD is the bisector of A. If BD = 4cm. DC = 3cm and AB = 6 cm. AC is equal to (a) 4.2 cm (b) 4.5 cm (c) 4.8 cm (d) 5 cm In the figure, ABC is similar to ______ 16 cm B A 53° 53° cm 36 cm 24 C T N OI D 7. (a) BDC (c) CDB W DBC (d) CBD AMB ~ CMD. Also 2ar (AMB) = ar (CMD) the length of MD is (a) (c) 8. P N I (b) (b) 2 MB 2 MB If ABC ~ QRP and (d) 2 MD 2 MD ar ABC 9 , AB = 18 cm, BC = 15 cm what ar PQR 4 will be PR. (a) 10 cm (b) 9 cm (c) 4 cm (d) 18 cm [Class-X – Maths] 28 Page 24 9. 10. 11. In ABC, D and E are points on side AB and AC respectively such that DE || BC and AD : DB = 3 : 1. If EA = 3.3 cm then AC = (a) 1.1 cm (b) 4.4 cm (c) 4 cm (d) 5.5 cm ABC and BDE are two equilateral triangles such that D is the midpoint of BC. Ratio of the areas of triangles ABC and BDE is— (a) 2:1 (b) 1:2 (c) 4:1 (d) 1:4 In ABC, DE || BC. In the figure, the value of x is ______ A x x+3 T N OI D P N WI x+1 B 12. 13. (a) 1 (c) 3 E It is given that ABC ~ PQR, with x+5 C (b) –1 (d) –3 ar (PRQ ) BC 1 then ar (BCA ) is equal to QR 3 (a) 9 (b) 3 (c) 1 3 (d) 1 9 The altitude of an equilateral triangle, having the length of its side 12cm is (a) 12 cm (b) 29 Page 25 6 2 cm [Class-X – Maths] (c) 14. 15. 16. 6 cm (d) 6 3 cm A right angled triangle has its area numerically equal to its perimeter. The length of each side is an even number & the hypotenuse is 10 cm. What is the perimeter. (a) 26 cm (b) 24 cm (c) 30 cm (d) 16 cm The perimeters of two similar triangles ABC and PQR are respectively 36 cm and 24 cm. If PQ = 10cm then AB = ........ (a) 10 cm (b) 20 cm (c) 25 cm (d) 15 cm T N OI In figure ABC ~ APQ. If BC = 8 cm, PQ = 4cm AC = 6.5 cm, AP = 2.8 cm, find AB and AQ. P N WI B P A Q C 1 1 CD . Prove that CA2 = AB2 + BC 2 3 2 17. In ABC, ADBC and BD 18. If ABC is an equilateral triangle such that AD BC then prove that AD2 = 3 DC2 19. An isosecles triangle ABC is similar to triangle PQR. AC = AB = 4 cm, RQ = 10 cm and BC = 6 cm. What is the length of PR? What type of triangle is PQR ? [Class-X – Maths] 30 Page 26 20. In the adjoining figure, find AE. A E 8 cm 4 cm 6 cm B 21. In PQR, DE || QR and DE T N OI P N WI E Q 22. 23. R In triangles ABC and PQR if B = Q and the value of AB BC 1 then what is PQ QR 2 PR ? AC ABC is a right angled at B, AD and CE are two medians drawn from A and C respectively. If AC = 5 cm and AD 24. D ar PQR 1 QR . Find . 4 ar PDE P D 3 cm C 3 5 cm. Find CE. 2 In the adjoining figure DE || BC. What is the value of DE. 31 Page 27 [Class-X – Maths] A 10 cm D E 2c m B C 3 cm 25. Legs (sides other then the hypotenuse) of a right triangle are of lengths 16 cm and 8 cm. Find the length of the side of the largest square that can be inscribed in the triangle. 26. In the following figure, DE || AC and T N OI P N WI BE BC . Prove that DC || AP EC CP A D B 27. E C P Two similar triangles ABC and PBC are made on opposite sides of the same base BC. Prove that AB = BP. [Class-X – Maths] 32 Page 28 28. In a quadrilateral ABCD, B = 90°, AD2 = AB2 + BC2 + CD2. Prove that ACD = 90°. D C B A 29. In figure DE || BC, DE = 3 cm, BC = 9 cm and ar (ADE) = 30 cm2. Find ar (trapezium BCED). A T N OI D E 3 cm P N WI B C 9 cm 30. Amit is standing at a point on the ground 8m away from a house. A mobile network tower is fixed on the roof of the house. If the top and bottom of the tower are 17m and 10m away from the point. Find the heights of the tower and house. 31. In a right angled triangle PRQ, PR is the hypotenuse and the other two sides are of length 6cm and 8cm. Q is a point outside the triangle such that PQ = 24cm RQ = 26cm. What is the measure of QPR? 32. PQRS is a trapezium. SQ is a diagonal. E and F are two points on parallel sides PQ and RS respectively. Line joining E and F intersects SQ at G. Prove that SG × QE = QG × SF. 33. Two poles of height a metres and b metres are apart. Prove that the height of the point of intersection of the lines joining the top of each pole to the ab foot of the opposite pole is metres.. a b 33 Page 29 [Class-X – Maths] D B O bm am h x C L y A 34. Diagonals of a trapezium PQRS intersect each other at the point O, PQ||RS and PQ = 3 RS. Find the ratio of the areas of triangles POQ & ROS. 35. In a rhombus, prove that four times the square of any sides is equal to the sum of squares of its diagonals. 36. ABCD is a trapezium with AE || DC. If ABD is similar to BEC. Prove that AD = BC. 37. In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then prove that the angle opposite to the first side is a right angle. 38. If BL and CM are medians of a triangle ABC right angled at A. Prove that 4 (BL2 + CM2) = 5 BC2. 39. ABCD is a rectangle in which length is double of its breadth. Two equilateral triangles are drawn one each on length and breadth of rectangle. Find the ratio of their areas. 40. In figure DE || BC and AD : DB = 5 : 4. Find T N OI P N WI ar DEF ar CFB A E D F C B [Class-X – Maths] 34 Page 30 41. Four friends went to a restaurant. They ordered four triangular shaped pizzas with different topings. One of them told to others they should divide their pizzas in four equal parts so that all can taste all the four pizzas but the three fail to do so. Fourth friend took a straw and he make a point at mid of each side and join all mid points and cut it by a knife into four equal parts. Prove that 42. i. ar (DEF ) 1 ar (ABC ) 4 if D, E, F are the midpot of sides AB, BC & CA of a triangular pizza. ii. Which value is described in this equation? A poster competition on “Save Earth” is organised by Directorate of Education. Medals in the shape of similar triangles were distributed to first and second position holder, first prize in form of ABC and second in the form of PQR. T N OI P N I A W 12 B SAVE EARTH 15 9 C i. Find the value of x and y. P 8 SAVE EARTH Q ii. Y x R Which mathematical concept was used in this question? 35 Page 31 [Class-X – Maths] iii. What is the ratios of areas of triangle ABC and PQR. iv. What moral values of a child was shown in this poster competition. 43. From a village A to city B road passes through a mountain C. It was constructed so that AC CB, AC = 2x km and CB = 2 (x + 7) km. It was proposed that a 26 km long highway will be constructed which will connect city B to village A directly. If a person goes to city B from village A through this highway by bus then what distance was reduced in his journey. Write the values of this question. 44. Ashok has a quadrilateral shaped field of rice. He divide it into two by one diagonal and distribute it to one son and one daughter. Each get the rice in proportion to the area which they got. Both don’t know how much area they got. Ashok using second diagonal distribute the rice in such ratio. ar (ABC ) AO ar (DBC ) DO i. Prove the above relation T N OI P N WI A O B ii. 45. C D Which value of Ashok was shown in this questions? Two advertising balloons showing “Pulse Polio” and “Save Girl Child" were tied by two wires ML and NR such that they makes an angle 40° with the ground. A person standing at the point K. Length of wire ML is ‘a’ metre, MN is ‘b’ metre and NK is ‘c’ metre. i. Find the length of wire NR when both the balloons are on a same straight line. ii. What moral values are shown in this question. [Class-X – Maths] 36 Page 32 save girl child L Pulse polio R x 40° M 40° b N K c 1. c 2. b 3. a 4. d 5. b 6. d 7. a 8. a 9. b 10. c 11. c 12. a 13. d 14. b 15. d 16. AB = 5.6 cm, AQ = 3.25 cm 18. APQ ~ ABC 19. 20 cm , isosceles triangle 3 20. 55 cm 21. 16 : 1 22. 1 2 23. 1 25 – 6 5 cm 4 24. 2.5 cm 25. 16 cm 3 29. 240 cm2 30. 9 m, 6 m T N OI W P N I 37 Page 33 [Class-X – Maths] 31. 90° 34. 9:1 39. 4:1 40. 25 81 41. (ii) Cooperation, healthy competition, intelligency, quick decision 42. ar (ABC ) 9 (i) x = 10, y = 6, ar (PQR ) 4 (ii) similar triangle (iii) Awareness about pollution, Saving of water, Energy conservation 43. (i) 8 Km (ii) Saving of time, Economic saving, Save energy. 44. (i) Intelligency, Self decision, Responsibility. 45. (i) x ac b c (ii) Save girl child, health awareness. T N OI P N WI [Class-X – Maths] 38 Page 34 1. Trigonometric Ratios : In ABC, B = 90°, for angle ‘A’ Perpendicular Hypotenuse sin A C Base Hypotenuse cos A tan A A W Base cot A Perpendicular sec A cosec A 2. T N I O P IN Perpendicular Base Hy te n po us e Base Perpendicular B Hypotenuse Base Hypotenuse Perpendicular Reciprocal Relations : sin 1 cosec , cosec 1 sin cos 1 sec , sec 1 cos 39 Page 35 [Class-X – Maths] tan 3. 1 tan , cot , cot Quotient Relations : tan 4. 1 cot sin cos cos sin Indentities : sin2 + cos2 = 1 sin2 = 1 – cos2 and cos2 = 1 – sin2 1 + tan2 = sec2 tan2 = sec2 – 1 and sec2 – tan2 = 1 1 + cot2 = cosec2 cot2 = cosec2 – 1 and cosec2 – cot2 = 1 5. Trigonometric Ratios of Some Specific Angles A 0° 30° sin A 0 1 2 cos A W tan A 0 cosec A 45° Not defined sec A 1 cot A Not defined [Class-X – Maths] 90° 3 2 1 2 1 2 0 1 3 T N I 1 O P IN 1 60° 3 2 1 3 2 1 2 2 2 3 2 2 2 3 1 3 40 Page 36 1 3 Not defined 1 Not defined 0 6. Trigonometric Ratios of Complementary Angles sin (90° – ) = cos cos (90° – ) = sin tan (90° – ) = cot cot (90° – ) = tan sec (90° – ) = cosec cosec (90° – ) = sec Note : In the following questions 0° 90° 1. 2. 3. 4. If x = a sin and y = a cos then the value of x2 + y2 is _______ (a) a (c) 1 (b) a2 (d) 1 a (b) 1 (d) 20° T N OI P N WI The value of cosec 70° – sec 20° is _____ (a) 0 (c) 70° If 3 sec – 5 = 0 then cot = _____ (a) 5 3 (b) 4 5 (c) 3 4 (d) 3 5 (b) 1 (d) 2 2 If = 45° then sec cot – cosec tan is (a) (c) 0 2 41 Page 37 [Class-X – Maths] 5. 6. 7. 8. If sin (90 – ) cos = 1 and is an acute angle then = ____ (a) 90° (b) 60° (c) 30° (d) 0° The value of (1 + cos ) (1 – cos ) cosec2 = _____ (a) 0 (b) 1 (c) cos2 (d) sin2 TRY is a right-angled isosceles triangle then cos T + cos R + cos Y is _____ (a) 2 (c) 1 (c) 10. 2 2 (d) 1 T N OI 1 2 If sec + tan = x, then sec = (a) 9. 2 (b) x 2 1 W x x P N I (b) 2 1 2x (d) x x 2 1 2x 2 1 x The value of cot sin cos is _______ 2 2 (a) cot cos2 (b) cot2 (c) cos2 (d) tan2 If sin – cos = 0, 0 90° then the value of is _____ (a) cos (b) 45° (c) 90° (d) sin [Class-X – Maths] 42 Page 38 sin 11. (a) sin cos (d) tan (a) sec2 + tan2 (b) sec – tan (c) sec2 – tan2 (d) sec + tan In an isosceles right-angled ABC, B = 90°. The value of 2 sin A cos A is _____ (a) (b) 1 P N WI 2 2 If T N OI 1 (c) 2 sin 20 sin 70 2 2 2 cos 69 cos 21 15. sin 1 sin is equal to 1 sin 12. 14. (b) cot (c) 13. can be written as 2 1 sin 1 2 (d) 2 sec 60 then K is ______ K (a) 1 (b) 2 (c) 3 (d) 4 If tan 1 7 , then 2 2 2 2 cosec sec cosec sec (a) 3 4 (b) 5 7 (c) 3 7 (d) 1 12 43 Page 39 [Class-X – Maths] 3 , write the value of cos P. 5 16. In PQR, Q = 90° and sin R 17. If A and B are acute angles and sin A = cos B then write the value of A + B. 18. If 4 cot = 3 then write the value of tan + cot 19. Write the value of cot2 30° + sec2 45°. 20. Given that 16 cot A = 12, find the value of 21. If = 30° then write the value of sin + cos2 . 22. If 1 tan 23. Find the value of if 24. If and are complementary angles then what is the value of 2 sin A cos A sin A cos A 2 then what is the value of . 3 T N OI 3 tan 2 3 0. P N WI cosec sec – cot tan 25. If tan (3x – 15°) = 1 then what is the value of x. 26. If sin 5 = cos 4, where 5 and 4 are acute angles. Find the value of 27. Simplify : tan2 60° + 4 cos2 45° + 3 (sec2 30° + cos2 90°) 28. Evaluate cos 58 2 sin 32 cos 38cosec 52 3 tan 15 tan 60 tan 75 29. If sin + sin2 = 1 then find the value of cos2 + cos4 30. If sin 2 = cos ( – 36°), 2 and – 36° are acute angles then find the value of . [Class-X – Maths] 44 Page 40 31. Prove that cosec4 32. – cosec2 = cot2 + cot4 . If sin (3x + 2y) = 1 and cos 3x 2y 3 , where 0 (3x + 2y) 90° 2 then find the value of x and y. 33. If sin (A + B) = sin A cos B + cos A sin B then find the value of (a) sin 75° (b) cos 15° cos A cos A cos A, A 45. 1 tan A 1 cot A 34. Prove that 35. Prove that 36. Find the value of sec 1 sec 1 sec 1 2cosec sec 1 T N OI P N WI sin2 5° + sin2 10° + sin2 15° + .... + sin2 85° 37. Prove that tan sec 1 cos . tan sec 1 1 sin 38. If 2 sin 3x 15 sin 2 2x 3 then find the value of 10 tan 2 x 5 . 39. Find the value of sin 60° geometrically. 40. 1 Let p = tan + sec then find the value of p p . 41. Find the value of 2 2 tan cot 90 sec cosec 90 sin 35 sin 55 tan 10 tan 20 tan 30 tan 70 tan 80 45 Page 41 [Class-X – Maths] cos cos m and n show that (m2 + n2) cos2 = n2. cos sin 42. If 43. Prove that cos 1° cos 2° cos 3°.........cos 180° = 0. 44. sin cos sin cos 2 sec Prove that sin cos sin cos . 2 tan 1 45. If A, B, C are the interior angles of a triangle ABC, show that 2 A A B C B C sin cos 2 cos 2 sin 2 1. 2 46. In the given right triangle, if base and perpendicular are represented by ‘Hardwork’ and ‘Success’ and are in the ratio 1 : 1 then find ACB. What mathematical concept has been used? What value is depicted from the problem? T N OI P N WI A B 47. C If x = sin2, y = cos2 Where x represents Honesty and y represents Hardwork. 48. a. What you will get when honesty is added with hardwork? b. Which mathematical concept is used? c. What value is depicted here? If punctuality and regularity are two measurable quantities are numberically equal to A and B respectively such that [Class-X – Maths] 46 Page 42 1 2 1 Cos (A B ) 2 Sin( A – B ) where 0° A + B 90° then find A and B. Which one more value other than punctuality & regularity, would you like to adopt in your life? 1. b 2. a 3. c 4. a 5. d 6. b 7. a 8. b 9. a 11. d 13. a 15. a 17. T N OI 10. b 12. d 14. d 16. cos P 90° 18. 25 12 19. 5 20. 7 21. 5 4 22. 30° 23. 30° 24. 1 25. x = 20. 26. 10° 27. 9 28. 1 29. 1 30. 42° W P N I 47 Page 43 3 5 [Class-X – Maths] 32. 33. x = 20, y = 15 3 1 2 2 , 3 1 2 2 36. 17 2 38. 13 12 40. 2 sec 41. 2 3 46. 45°, Trigonometry, Hardwork and Success 47. (a) 1, (b) Trigonometry, (c) Honesty and Hardwork 48. A = 45°, B = 15° Honesty, Hardwork, Responsibility, Cooperation T N OI P N WI [Class-X – Maths] 48 Page 44 1. The mean for grouped data can be found by : The direct method X (ii) The assumed mean method X a fidi , fi (iii) T N OI P N WI where di = xi –a. The step deviation method X a 2. fixi . fi (i) fiui fi h, where u i xi a . h The mode for the grouped data can be found by using the formula : f1 f 0 mode l h 2 f f f 1 0 2 l = lower limit of the modal class. f1 = frequency of the modal class. f0 = frequency of the preceding class of the modal class. f2 = frequency of the succeeding class of the modal class. h = size of the class interval. Modal class - class interval with highest frequency. 49 Page 45 [Class-X – Maths] 3. The median for the grouped data can be found by using the formula : n 2 Cf median l h f l = lower limit of the median class. n = number of observations. Cf = cumulative frequency of class interval preceding the median class. f = frequency of median class. h = class size. 4. Empirical Formula : Mode = 3 median - 2 mean. 5. Cumulative frequency curve or an Ogive : T N OI (i) Ogive is the graphical representation of the cumulative frequency distribution. (ii) Less than type Ogive : (iii) (iv) P N WI • Construct a cumulative frequency table. • Mark the upper class limit on the x = axis. More than type Ogive : • Construct a frequency table. • Mark the lower class limit on the x-axis. To obtain the median of frequency distribution from the graph : • Locate point of intersection of less than type Ogive and more than type Ogive : Draw a perpendicular from this point on x-axis. • [Class-X – Maths] The point at which it cuts the x-axis gives us the median. 50 Page 46 1. 2. 3. 4. 5. 6. 7. Mean of first 10 natural numbers is (a) 5 (b) 6 (c) 5.5 (d) 6.5 If mean of 4, 6, 8, 10, x, 14, 16 is 10 then the value of ‘x’ is (a) 11 (b) 12 (c) 13 (d) 9 The mean of x, x + 1, x + 2, x + 3, x + 4, x + 5 and x + 6 is (a) x (b) x + 3 (c) x + 4 (d) 3 The median of 2, 3, 2, 5, 6, 9, 10, 12, 16, 18 and 20 is (a) 9 (c) 10 N I W T N I PO (b) 20 (d) 9.5 (b) 3 (d) 2 The median of 2, 3, 6, 0, 1, 4, 8, 2, 5 is (a) 1 (c) 4 Mode of 1, 0, 2, 2, 3, 1, 4, 5, 1, 0 is (a) 5 (b) 0 (c) 1 (d) 2 If the mode of 2, 3, 5, 4, 2, 6, 3, 5, 5, 2 and x is 2 then the value of ‘x’ is (a) 2 (b) 3 (c) 4 (d) 5 51 Page 47 [Class-X – Maths] 8. The modal class of the following distribution is Class Interval Frequency 9. 10. 11. 12. 13. 10–15 15–20 20–25 25–30 30–35 4 7 12 8 2 (a) 30–35 (b) 20–25 (c) 25–30 (d) 15–20 A teacher ask the students to find the average marks obtained by the class students in Maths, the student will find (a) mean (b) median (c) mode (d) sum The empirical relationship between the three measures of central tendency is (a) 3 mean = mode + 2 median (b) 3 median = mode + 2 mean (c) 3 mode = mean + 2 median (d) median = 3 mode – 2 mean T N OI P N WI Class mark of the class 19.5 – 29.5 is (a) 10 (c) 24.5 (b) 49 (d) 25 Measure of central tendency is represented by the abscissa of the point when the point of intersection of ‘less than ogive’ and ‘more than ogive’ is (a) mean (b) median (c) mode (d) None of these The median class of the following distribution is Class Interval : Frequency : 0–10 10–20 20–30 30–40 40–50 50–60 60–70 4 4 8 10 12 8 4 [Class-X – Maths] 52 Page 48 14. 15. 16. 17. 18. 19. 20. (a) 20–30 (b) 40–50 (c) 30–40 (d) 50–60 The mean of 20 numbers is 17, if 3 is added to each number, then the new mean is (a) 20 (b) 21 (c) 22 (d) 24 The mean of 5 numbers is 18. If one number is excluded then their mean is 16, then the excluded number is (a) 23 (b) 24 (c) 25 (d) 26 The mean of first 5 prime numbers is (a) 5.5 (b) 5.6 (c) 5.7 (d) 5 T N OI The sum of deviations of the values 3, 4, 6, 8, 14 from their mean is (a) 0 (c) 2 P N WI (b) 1 (d) 3 If median = 15 and mean = 16, then mode is (a) 10 (b) 11 (c) 12 (d) 13 The mean of 11 observations is 50. If the mean of first six observations is 49 and that of last six observations is 52, then the sixth observation is (a) 56 (b) 55 (c) 54 (d) 53 The mean of the following distribution is 2.6, then the value of ‘x’ is Variable 1 2 3 4 5 Frequency 4 5 x 1 2 53 Page 49 [Class-X – Maths] (a) 24 (b) 3 (c) 8 (d) 13 21. The mean of 40 observations was 160. It was detected on rechecking that the value of 165 was wrongly copied as 125 for computing the mean. Find the correct mean. 22. Find ‘x’ if the median of the observations in ascending order 24, 25, 26, x + 2, x + 3, 30, 31, 34 is 27.5. 23. Find the mean of the following data. x : 10 12 14 16 18 20 f : 3 5 6 4 4 3 24. Variable : 3 Frequency : 6 25. 5 7 T N OI 9 11 13 8 15 p 8 4 Find the value of ‘p’, if mean of the following distribution is 7.5 W P N I From the cumulative frequency table, write the frequency of the class 20–30. Marks Number of Students Less than 10 1 Less than 20 14 Less then 30 36 Less than 40 59 Less than 50 60 [Class-X – Maths] 54 Page 50 26. Following is a cumulative frequency curve for the marks obtained by 40 students. Find the median marks obtained by the student. 27. The following ‘more than ogive’ shows the weight of 40 students of a class. What is the lower limit of the median class. T N OI P N WI 55 Page 51 [Class-X – Maths] 28. The mean of the following frequency distribution is 62.8. Find the values of x and y. Class Interval : 0–20 20–40 40–60 60–80 80–100 100–120 Total 5 x 10 y 7 8 50 Frequency : 29. The following frequency distribution gives the daily wages of a worker of a factory. Find mean daily wage of a worker. Daily Wage (in ) More than 300 0 More than 250 12 More than 200 21 More than 150 44 More than 100 53 T N I More than 50 O P IN More than 0 30. W No. of students 60 10-20 20-30 30-40 40-50 50-60 60-70 70-80 Total 12 20 x 65 y 25 18 230 Find the mean, median and mode of the following : Class Interval : Frequency : 32. 59 The median distance of the following data is 46. Find x & y. Distance (m) 31. Number of Workers 0–10 10–20 20–30 30–40 40–50 50–60 60–70 6 8 10 15 5 4 2 The following frequency distribution shows the marks obtained by 100 students in a school. Find the mode. [Class-X – Maths] 56 Page 52 33. Marks Number of Students Less than 10 10 Less than 20 15 Less than 30 30 Less than 40 50 Less than 50 72 Less than 60 85 Less than 70 90 Less than 80 95 Less than 90 100 Draw ‘less than’ and ‘more than’ ogives for the following distribution Marks : 20–30 30–40 40–50 50–60 60–70 70–80 80–90 8 12 24 6 10 15 25 No. of Students : T N OI Also find median from graph. 34. Class Interval : Frequency : 35. P N WI The mode of the following distribution is 65. Find the values of x and y, if sum of the frequencies is 50. 0–20 20–40 40–60 60–80 80–100 100–120 120–140 6 8 x 12 6 y 3 Following is the table people violating traffic rules under different age groups: Age (In yrs) No. of persons violating traffic rules 18-20 20-22 22-24 24-26 26-28 28-30 63 50 35 27 16 9 Find mean. Why should we obey traffic rules. What values are depicted here? 36. The following table shows the ages of patients in a private hospital after Parliamentary Elections held in November for free treatment for those who cast their votes. 57 Page 53 [Class-X – Maths] Age (In yrs) 18-28 28-38 38-48 48-58 58-68 68 onwards Total No. of Patients 6 11 21 23 14 5 80 Find the mode of the above data. What values are depicted here? 37. Following table shows distance covered by 50 students of a school in a shot-put competition. Distance (M) No. of St. 0-20 20-40 40-60 60-80 80-100 6 11 17 12 4 Construct cumulative frequency distribution table. Find median. What values are depicted here? 38. The number of persons living in an old age home of a city are as follows: Age (In yrs) No. of old Persons 50-55 55-60 10 12 60-65 65-70 T N OI 17 13 70-75 75-80 16 22 Find the mean and median. What steps should be taken to improve the condition of old people in our society. What values are shown in this question? 39. P N WI During the last two months the quantity of PNG used in a locality of 30 houses was recorded as follows: Quantity (M3) 38-40 40-42 42-44 44-46 46-48 48-50 50-52 No. of families 3 2 4 5 14 4 3 Find mean, median & mode of the above data. 40. The haemoglobin level of 35 students of a class is as follows: Hb Level No. of Students Less then 8 Less then 10 Less then 12 Less then 14 Less then 16 3 7 13 23 35 Construct "less than type" of give. What value is depicted in the question? [Class-X – Maths] 58 Page 54 1. c 2. b 3. b 4. a 5. b 6. c 7. a 8. b 9. a 10. b 11. c 12. b 13. c 14. a 15. d 16. b 17. a 18. d 19. a 20. c 21. 161 22. x = 25 23. 14.8 24. p = 3 25. 22 27. 147.5 T N I PO 26. 40 28. x = 8, y = 12 30. x = 34, y = 46 33. 60 35. (a) 22.1 years (b) discipline (c) knowledge & obeying traffic rules 29. 182.50 31. Mean = 30, 32. 41.81 34. x = 10, y = 5 36. mode – 49.81 Social responsibility 37. m edian = 49.41m , (A w areness for physical fitness), (Im portance of sports), (H ealthy com petition) 38. (a) (m ean) = 66.88, N I W Median = 30.67, mode = 33.33 (m edian) = 67.3 (b) (generation gap) (developing m oral social values), (motivation) (c) (moral values) 39. 45.8, 46.5, 47.9 40. (Awarness for fitness), (Moral values). 59 Page 55 [Class-X – Maths]