CAT 2013: GEOMETRY (LINES-ANGLES

Transcription

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CAT 2013: GEOMETRY (LINES-ANGLES-CIRCLES)
1. From a point outside a circle of radius 15 cm, two tangents are drawn to the circle. The angle
between the tangents is 74°. Find the area of the triangle formed by the tangents and the chord
joining the points of contact. (sin37° = 0.6)
a) 384 cm2
b) 94 cm2
c) 192 cm2
d) 225 cm2
2. In the figure below (not to scale), PQ is a chord of a circle. O is the centre of
the circle. PQ is extended to R such that QR = OQ. OR is produced to meet the
circle at S. Find
a) 1
b) 1/2
c) 1/3
d) ¼
3. In the figure below (not to scale), O is the centre of the circle. OC is perpendicular
to AB. EC = 8 cm, AB = OC + 12 cm and ED = 3/4 EB. Find
a) 30°
b) 45°
c) 60°
d) 25°
4. In the figure below, the diameter of the circle is 12 cm. PQ and RS are the diameters
of the circle and are perpendicular to each other. PT : TQ = 1 : 3. RU : US = 1 : 3. O is
the centre of the circle. Find XY (in cm) if XZ
a)
b)
c)
d)
5. C1 and C2 are two concentric circles with centre at O. Chord AB of the outer circle C1 touches the
inner circle C2 at C. X is a point on minor arc AB such that XC is perpendicular to AB. If XC = 4 cm
and BC = 8 cm, then find OB.
a) 10cm
b) 12 cm
c) 14 cm
d) 15cm
6. In the figure below, O is the centre of the circle and AB is a diameter of
the circle. DE is a tangent to the circle at C. Find
a) 50°
b) 45°
c) 40°
d) 30°
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7. AB and CD are two chords of a circle with centre O. The
line, AB and CD intersect outside the circle at P. Line
PO cuts BD at E which is its mid point. If OA = 5 cm and
OP = 13 cm, what is the ratio of distances of AB and CD
from O?
A) 5 : 13
b) 1 : 2
c) 5 : 8
d) 1 : 1
8. Two circles touch each other internally at the point C. CD is a diameter of the bigger circle. Chord
AB of the bigger circle is perpendicular to CD and tangent to the smaller
circle.
a) 6 cm
Find the radius of the inner circle.
b) 4 cm
c) 5 cm
d) None of these
9. There are three unit circles intersecting each other as shown in the
figure at points M, N and O, which are the centres of the circles. Find
the total length of arcs MN, NO and OM.
A)
B)
C)
A) 1
B) 0.86
C) √2
10.
4
D) 𝜋
11. Two circles C1 and C2 of the same radius are such that the circle C1 bisects the circumference of
the circle C2. If the radius of the circles is 7 cm, what is the area common to the circles?
A) 10𝜋
B) 15 𝜋
C) 5√5𝜋
D) 12 𝜋
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12. A circle of radius 2 cm touches another circle of radius 10 cm internally. Find the length of a
tangent segment drawn from the centre of the larger circle to the smaller circle.
A) 15√2
B) 15√3
C) 2√15
D) 3√15
13.
a) 75 B) 50
C) 70
D) 65
`
14.
A) 3 cms
B) 9𝜋^2
c) 9𝜋
D) 9√𝜋
15.
A)
b)
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c)
D)
16.
A) 1
B) 2√2-1
D) √2
C) 3√3-2
17.
A) 1
B) 2
C) 3
D) 4
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18.
A)
B)
C)
D)
Directions for questions 19 and 21: Read the information given below and answer the questions
that follow.
19. Find the ratio of lengths of QR to that of PQ
A) 2:3
B) 1:4
C) 3:7
D) 2:5
20. Find the radius of the bigger circle.
A) 16
B) 13
C) 15
D) 9
21. Find the length of QS.
A) 5√11
B) 2√11
D) 4√11
C) 3√11
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22. Two chords of length 12 cm and 16 cm, of a circle of radius 10 cm, are perpendicular to each
other. Find the distance between the midpoints of these two chords.
A) 6
B) 8
C) 10
D) 12
23.
A)
2
B) 1
C) 3
D) None of these
24.
A)
B)
C)
D)
25. Two equal circles with centers O and O’ cut each other at two distinct points C and C’. If neither O
nor O’ falls within the intersection of the circles, what is the smallest range that includes all the
possible values of
∠COC' + ∠CO'C'?
a) Between 0° and 180°
c) Between 0° and 360°
b) Between 0° and 120°
d) Between 0° and 240°
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26. In the figure given below, AF, BF, BE and CE are straight lines.
If the points A, B, C and D are con-cyclic, what is the value of y
– x?
a) 26o
b) 36 o
c) 30 o
d) None of these
27. In a circle the distance between two parallel chords of lengths 6 cm and 8 cm is 7 cm. If the chords
are on opposite sides of the center of the circle, what is the circumference (in cm) of the circle?
a) 8 π
b) 12 π
c) 10 π
d) None of these
28. In the figure given below, AB is a diameter of the circle. If AB || CD, AC ||
BE and ∠BAE = 35°, then the absolute difference between ∠CDB and
∠ABD is
a) 90°
b) 70°
c) 55°
d) 125°
29. A, B, C and D are four points on the circumference of a circle such that DC is a diameter of the
circle. The point of intersection O of AC and BD lies inside the circle. lf ∠ACB = xo, ∠ABD = (X - 10) o
and ∠BDC=(X
+10) o, then the measure of ∠DOC in degrees is
a) 100
b) 90
c) 120 d) Cannot be determined
30. A quadrant of a circle of radius 12 cm is cut out and the remaining part is folded to form a cone.
Find the surface area (in sq.cm.) of the cone.
a) 189 𝜋
b) 81 π
c) 108 π
d) None of these
31. Three circles, each of radius 20, have centres at P, Q and R. Further, AB = 5, CD = 10 and EF = 12.
What is the perimeter of PQR?
32. A certain city has a circular wall around it, and this wall has four gates pointing north, south, east and
west. A house stands outside the city, 3 km north of the north gate, and it can just be seen from a point 9
km east of the south gate. What is the diameter of the wall that surrounds the city?
a. 6 km
b. 9 km
c. 12 km
d. None of these
33.
The length of the common chord of two circles of radii 15 cm and 20 cm, whose centres are 25
cm apart, is
a. 24 cm
b. 25 cm
c. 15 cm
d. 20 cm
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34.
There are two concentric circles such that the area of the outer circle is four times the area of the
inner circle. Let A, B and C be three distinct points on the perimeter of the outer circle such that
AB and AC are tangents to the inner circle. If the area of the outer circle is 12 square centimeters
then the area (in square centimeters) of the triangle ABC would b
b) 9/ 𝜋
c) 9√3/ 𝜋
d) 6√3/ 𝜋
a) 𝜋√12
35. In the figure below, ABCDEF is a regular hexagon and angle AOF is 900 . FO is parallel to ED. What
is the ratio of the area of the triangle AOF to that of the hexagon ABCDEF?
A
F
B
C
O
E
a.
1
12
b.
1
6
D
c.
1
24
d.
1
18
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3 6 . In the figure below, AB is the chord of a circle with center O. AB is extended to C such that BC = OB.
The straight line CO is produced to meet the circle at D. If ACD = y degrees and AOD = x
degrees such that x = ky, then the value of k is
A
D
a. 3
b. 2
B
O
C
c. 1
d. None of the above.
Directions for questions 37 to 39: Answer the questions on the basis of the information given below.
In the adjoining figure I and II, are circles with P and Q respectively, The two circles touch each other and
have common tangent that touches them at points R and S respectively. This common tangent meets the
line joining P and Q at O. The diameters of I and II are in the ratio 4 : 3. It is also known that the length of
PO is 28 cm.
37. What is the ratio of the length of PQ to that of QO?
a. 1 : 4
b. 1 ; 3
c. 3 : 8
38.
39.
d. 3 : 4
What is the radius of the circle II?
a. 2 cm
b. 3 cm
c. 4 cm
d. 5 cm
The length of SO is
a. 8 3 cm
c. 12 3 cm
d. 14 3 cm
b. 10 3 cm
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40. A circle with radius 2 is placed against a right angle. Another smaller circle is
also placed as shown in the adjoining figure. What is the radius of the smaller circle?
a) 3-2√2
b) 4-2√2
c) 7-4√2
d) 6-4√2
41. What is the distance in cm between two parallel chords of lengths 32 cm and 24 cm in a circle of
radius 20 cm?
a. 1 or 7
b. 2 or 14
c. 3 or 21
d. 4 or 28
42. A semi-circle is drawn with AB as its diameter. From C, a point on AB, a line perpendicular to AB is
drawn meeting the circumference of the semi-circle at D. Given that AC = 2 cm and CD = 6 cm,
the area of the semi-circle (in sq. cm) will be:
1. 32
2. 50
3. 40.5
4. 81
5. undeterminable
43.
A)
b)
c)
D)
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CAT 2013: GEOMETRY (LINES-ANGLES

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