LV 100 Geometry Questions (Kitabees)

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100 must do Geometry questions Kitabee CAT Preparation 
 
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1. PQRS is a square with an area of 240 units. A, B, C and D are sides on the 
given square such that PA = QB = RC = SD. Also given that PA = 3AQ, then 
what is the area of the shaded region? 
 
 
 
 A. 75 B. 60 C. 55 D. 80 
 
 
2. ABC is a right angled triangle, right angled at B. If AD is the angle bisector of 
angle A and divides BC in two parts such that BD = 2 and CD = 3 units. Then 
what is the length of AD? 
 
 
 A. 4√2 B. 2√6 C. 2√3 D. 2√7 
 
 
 
3. ABCD is a rectangle as shown in the figure below. P is the midpoint of DC, Q 
is the mid-point of AP, R is the midpoint of BQ and S is the midpoint of CR. 
What is the ratio of area of quadrilateral PQRS to the area of rectangle ABCD? 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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 A. 7/16 B. 3/16 C. 7/32 D. 9/32 
 
 
4. PQRS is a trapezium in which PQ || SR. Angle RSP = 1200. Also, RS = PS = 
1/3 QP. What is the measurement of angle PQR? 
 
 
 A. 150 B. 22.50 C. 250 D. 300 
 
 
5. In the figure given below, there is a regular octagon. If the measure of the 
shaded region is 6 units, then what is the area of the octagon? 
 
 
 
 A. 22 B. 24 C. 30 D. 34 
 
 
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100 must do Geometry questions Kitabee CAT Preparation 
 kitabee.in P a g e | 3 
6. PA is the tangent to the circle with centre O while PC bisects the angle APB.
What is the measure of angle ACP?
A. 300 B. 450 C. 600 D. 750
7. In the given figure, ABC is a triangle such that AB = 12, AC = 16 and BC = 20.
Also, M is the mid-point of BC. AMDE is a square and MD intersects the side
AC at the point F. Then, what is the area of the quadrilateral AFDE?
A. 114/7 B. 27/7 C. 125/2 D. 12/3
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100 must do Geometry questions Kitabee CAT Preparation 
 
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8. PQRS is a rectangle and a line L passes through the vertex P. The distance from 
R to L is 4 while the distance from S to L is 12. Also, PS = 2PQ, then what is 
the length of QR? 
 
 
 A. 15 B. 25 C. 20 D. 10 
 
 
9. In the figure given below, ABCD is a rectangle and M is the midpoint of CB. 
DM is perpendicular to the diagonal AC. The, what is the ratio of AB: AD? 
 
 
 A. 2:1 B. √3 : 1 C. 3:2 D. 1: √2 
 
 
 
 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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10. The two circles shown each have a radius of 2 units. There is a square fitted 
snuggly between the horizontal line and the two touching circles. What is the 
area of the square? 
 
 
A. 49/63 B. 64/81 C. 25/36 D. 16/25 
 
 
11. ABCDEF is a regular hexagon and M and N are the mid points of opposite 
edges as shown in the figure. If the area of the hexagon is 120 units. Then what 
is the product of the lengths of AD and MN? 
 
 
 
 
 A.160 B. 200 C. 100 D. 140 
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100 must do Geometry questions Kitabee CAT Preparation 
 
. kitabee.in P a g e | 6 
 
 
 
 
12. In the given figure, the rectangle of length 12 cm encloses an equilateral 
triangle of touching circles as shown. What is the shortest distance between the 
two grey circles? 
 
 
 
A. 4 �3√3− 2� B. 4 �3√3− 1� C. 4 �√3 − 2� D. 4 �√3− 1� 
 
 
13. In the figure shown below, there are two squares with sides 4 and 5 cm while a 
triangle with area 8 cm2 and a shaded parallelogram. What is the area of the 
parallelogram? 
 
 
 
 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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 A.15 cm2 B. 16 cm2 C. 18cm2 D. 20 cm2 
14. PQRS is a square with sides of length 4. M and N are the midpoints of the 
sides PQ and PS respectively. X is a point on RN such that 3RX = 2XN. Then, 
what is the area of the triangle QMX? 
 
 A.19/7 B. 27/11 C. 20/3 D. 16/5 
 
15. In the given figure, ABCD is a trapezium and X and Y are the mid points of 
AD and BC. If the area of the shaded region is 28 cm2. Then, what is the area of 
ABCD? 
 
 
 
 
 A. 48 B. 56 C. 30 D. 35 
 
 
16. In the given figure, ABCDEF is a regular hexagon with P being the midpoint 
of AB. Q and R are the points of intersection of PD and PE respectively with 
the diagonal CF. What is the ratio of area of triangle PFR and to that of area of 
trapezoid EDQR? 
 
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100 must do Geometry questions Kitabee CAT Preparation 
 
. kitabee.in P a g e | 8 
 
 
 A. ½ B. 1/3 C. ¼ D. 1/√3 
 
17. In the given figure XY is the diameter of the smaller circle and S which is the 
center of the larger circle lies on the smaller circle. If the radius of the larger 
circle is 2 units, then what is the area of the shaded region? 
 
 
 A. 8 B. 10 C. 4 D. 2 
 
18. ABCD is quadrilateral in which AD = DC. Also, angle ADC = angle ABC = 
900. If DM ⊥ AB and DM = 10, then what is the area of the quadrilateral 
ABCD? 
 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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 A. 100 B. 125 C. 81 D. 50 
 
19. There is a circle inscribed inside a hexagon ABCDEF such that all the six sides 
are tangents to the circle. If the lengths of AB, BC, CD, DE and EF are 4,5,6,7 
and 8 units respectively, then what is the length of AF? 
 
 A.9 B. 8 C. 7 D. 6 
 
20. In the given figure PQRS is a square and QRM and SRN are equilateral 
triangles. If SR = 2 units, then what is the length of MN? 
 
 
 
 
 A. 5√2 B. 3√2 C. 2√2 D. √2 
 
 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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21. In the figure shown the bigger triangle is an equilateral triangle with 36 
smaller equilateral triangles with area 2 cm2 each. What is area of triangle 
ABC? 
 
 
 A.22 𝑐𝑐𝑐𝑐2 B. 52 𝑐𝑐𝑐𝑐2 C. 20 𝑐𝑐𝑐𝑐2 D. 72 𝑐𝑐𝑐𝑐2 
 
 
22. ABCD is an isosceles trapezium in which X is the midpoint of AB. If BX = 
4 and angle CXD = 900. Then, what is the perimeter of the trapezium 
ABCD? 
 
 
 
 A. 21 𝑐𝑐𝑐𝑐2 B. 10 𝑐𝑐𝑐𝑐2 C. 24 𝑐𝑐𝑐𝑐2 D. 20 𝑐𝑐𝑐𝑐2 
 
23. As shown in the figure, lines are drawn parallel to the bases and divide 
each of the other two sides of the triangleinto 10 equal segments. What 
percentage of the area of the triangle is grey? 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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 A. 41.75 % B. 42.5% C. 45% D. 46% 
 
24. The side of the square has length of 4 units. The semicircles go through the 
centre of the square and have their centres on the vertices of the square. The 
inner circles have their centres on the sides of the squares and touch the 
semicircles. Then what is the value of the shaded area? 
 
A. 4 𝜋𝜋 (3 − 2√2) 
B. 𝜋𝜋 (3 − 2√2) 
C. 16 𝜋𝜋 (3 − 2√2) 
D. 𝜋𝜋 (3 + 2√2) 
 
 
 
 
 
 
 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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25. Circles are drawn around the vertices of a square. There are two large 
circles and two small ones as shown in the figure. Then what is the ratio of 
the radius of a large circle to the radius of a small circle? 
 
 
 A. 2/9 B. 1 + √2 C. 1 D. 2.5 
 
 
26. In triangle PQR, PS is the median. The measure of angle PRQ is 300 and 
the angle PSQ has measure of 450. What is the measure of angle QPS? 
 
 
 
 A. 450 B. 300 C. 250 D. 200 
 
 
27. In the given figure, the square has a side length of 2 units. What is the 
radius of the smaller circle? 
 
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100 must do Geometry questions Kitabee CAT Preparation 
 
. kitabee.in P a g e | 13 
 
 
 
A. 2+2√2.
√2−1
 𝐵𝐵. 2.
√2+1
 𝐶𝐶. 2√2.
√2+1
 𝐷𝐷. 2√2−2
√2+1
 
 
28. ABC is a triangle with area 2 units. Sides ABC are extended to both sides to 
points P, Q, R, S, T and U such that PA = AB = BS, TC = CA = AQ and UC 
= BC = RB. What is the area of the hexagon PQRSTU? 
 
 
 A. 26 B. 16 C. 21 D. 30 
 
29. ABCD is a rectangle in which AM is the angle bisector of angle BAD and 
BD is a diagonal. If the distances between M and the sides BC and CD are 
respectively 8 and 1. Then, what is the length of CD? 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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 A. 8 + 2√2 B 11 - √2 C. 10 D. 8 + 3√2 E. 11 + 1/√2 
 
 
30. The two regular hexagons are equal to each other as shown in the figure. 
What fraction of the parallelogram’s area is shaded? 
 
 
 
 A. 1/3 B. 1/2 C. ¼ D. 1/5 E. 1/6 
 
31. In the figure shown PQR is an isosceles triangle with PQ = PR. Also, MN is 
perpendicular to PQ. If angle QMR = 1200 and angle PQM = 500, then what 
is the measure of angle MQR? 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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 A. 50 B. 100 C. 150 D. 200 E. 250 
 
 
32. Triangle PQR is a right angled triangle at P and has PQ =2 and PR = 2√3. 
Altitude PL intersects median RM at F what is the length of PF? 
 
 
 A. √3
2
 B. 3√3
7
 C. 4√3
7
 D. 5√3
9
 
 
33. ABC is a triangle inside which a circle is inscribed as shown in the figure. 
Also, segment ED is tangent to the circle. If AC = 5, AB = 6 and BC = 3, 
then, what is the perimeter of the triangle ADE? 
 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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 A. 7 B. 4 C. 9 D. 6 E. 8 
 
34. The square ABCD has a side of length 2 and M is the midpoint of AB. 
What is the area of the shaded region? 
 
 
 A. 1/4 B. 1 C. 1/2 D. 1/3 
 
 
35. Square ABCD has an area of 4. E is the Midpoint of AB. Similarly, F, G, H 
and I are the mid points of DE, CF, DG and CH respectively. The area of 
∆𝐼𝐼𝐷𝐷𝐶𝐶 𝑖𝑖𝑖𝑖 
 
 
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100 must do Geometry questions Kitabee CAT Preparation 
 
. kitabee.in P a g e | 17 
 
 
 A. ¼ B. 1/8 C. 1/16 D. 1/32 
 
36. AD is the diameter of the circle and AB and AC are two chords drawn. If 
the measure of angle BAC = 60 and BE is perpendicular to AC then what is 
the length of chord BD given that AB = 18 and EC = 2 cm? 
 
 
 A. 6 √3 B. 6 √2 C. √3 D. 4 √3 
 
 
37. The regular hexagon shown has a side length of 3 units. The sectors of 
circles each have a radius of 3 and centres in the vertices of the hexagon. 
What is the area of the shaded region? 
 
 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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A. (6 𝜋𝜋 + 9 √3) 
B. 3 (6 𝜋𝜋 − 9 √3) 
C. (6 𝜋𝜋 + 3 √3) 
D. 3 (6 𝜋𝜋 + 10) 
 
38. The two semicircles are drawn as shown in the figure. Chord CD is of 
length 8 units is parallel to the diameter AB of the bigger semicircle and 
touches the smaller semicircle. The, what is the area of the shaded region? 
 
 
 A. 10 𝜋𝜋 B. 8 𝜋𝜋 C. 12 𝜋𝜋 D. 16 𝜋𝜋 
 
39. In the diagram, PQRS is a rectangle. Point T is outside the rectangle so 
that ∆ PTQ is an isosceles right angled triangle with hypotenuse PQ. If PQ = 4 
QR= 3 then what is the area of ∆𝑃𝑃𝑃𝑃𝑃𝑃? 
 
 
 
 A. 5 B. 6 C. 7 D. 8 
 
 
40. A square PQRS with sides of length 20 cm is rolled without slipping along a 
line. Initially P and Q are on the line and the first roll is around point Q. The 
rolling stops when P returns to the line. What is the length of the curve that P 
has travelled? 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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A. 20 𝜋𝜋 ( 1 + √2) 
B. 20 𝜋𝜋 (√2) 
C. 10√2 𝜋𝜋(1 + √2) 
D. 𝜋𝜋 ( 1 + √2) 
 
41. The coloured small squares have the same area. The side of a small square is 
equal to 1. What is the side of the outer square? 
 
 
 A. 2√2 B. √2 C. 2 + 2√2 D. 2√2 + 4 
 
 
42. As shown in the figure, there are two squares with side 2 units each with a 
common vertex. The edge of one of them lies along the diagonal of the other. 
What is the area of the shaded region? 
 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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 A. 4-2√2 B. 1-2√2C. 4-√2 D. None of these 
 
43. Two congruent equilateral triangles with perimeters 24 cm are overlapped with 
their sides parallel as shown in the figure. What is the perimeter of the resulting 
hexagon? 
 
 
 A. 11 cm B. 16 cm C. 12 cm D. 14 cm 
 
 
 
 
 
 
 
 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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44. A square with a total area of 125 cm2 is divided into five parts as shown- four 
squares and an L shaped figure each having equal area. Then, what is the length 
of the shortest side of the L-shaped figure? 
 
 
 
 A. 1cm B. 1.2 cm C. 5√5-10 D. 3(√5 -1) cm 
 
 
45. A rectangle in the figure is divided into six squares. The length of the smallest 
square is 1 cm. Then, find the length of the side of the largest square? 
 
 
 
 A. 4 cm B. 7 cm C. 6cm D. 5cm 
 
 
 
 
 
 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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46. In the given figure the ratio of radii of the sector and the incircle is 3:1. What is 
the ratio of their areas? 
 
 A. 3:2 B. 4:3 C. 5:3 D. 6:5 
 
47. In the figure shown, if the area of the green portion is 1200 cm2, then what is 
the area of the blue portion? 
 
 
 A. 1600 B.1200 C.1400 D. 1000 
 
48. In the figure shown, there is a rectangular parallelopiped. If the lengths of PQ, 
QR and PR are 10, 12 and 16 respectively. Then what is the length of the 
diagonal PA? 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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 A. 10 B. 5√10 C. 5√5 D. 5√15 
 
49. ABCD is a rectangle. The numbers shown represent areas. What is the value of 
x where x represents area of GHID? 
 
 
 A. 8 B. 11 C. 10 D. 12 
 
 
50. ABCDEF is a regular hexagon and P is the midpoint of side AB. What is the 
ratio of area of ∆PAD to area of hexagon ABCDEF? 
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100 must do Geometry questions Kitabee CAT Preparation 
 
. kitabee.in P a g e | 24 
 
 
 A. 
1
5
 B. 
1
4
 C. 1
3
 D. 
1
6
 
 
51. Three equal squares are placed together as shown in the figure. If the lines AE 
and CH intersect at point P, then what is the measure of angle CPE? 
 
 
 
 A. 300 B. 450 C. 600 D. 500 
 
 
52. If a and b are the perpendicular sides of a right angled triangle. If d1 is the 
diameter of the incircle and d2 is the diameter of the circumcircle of this 
triangle, then what is the value of d1 + d2 ? 
 A. (a+b) B. 2(a+b) C. ½ (a+b) D. √𝑎𝑎𝑎𝑎 
 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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53. In the given figure PQRS is a cyclic quadrilateral and the diagonal QS bisects 
the angle PQR. Also, PR = QR. If angle QSR = 800 and angle PRQ = 200, then 
what is the measure of angle QPS? 
 
 
 A. 100 B. 120 C.60 D.110 
 
 
 
54. In ∆ABC, ED is the bisector of ∠BDA and DF is the bisector of ∠ADC. If 
AE = 3, BE = 7, BD = 3DC and AC = 32, then what is the length of FC? 
 
 
 A.14 B.12 C.11 D.10 E.16 
 
 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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55. ABCD is a pyramid such that all plane angles with vertex D are equal to 900. If 
the area of lateral surfaces DAB, DAC and DBC are 4, 6 and 8 respectively. 
Then what is the volume of ABCD? 
 
 
 
 A. 10 B. 5 C. 8 D. 6 
 
56. In the figure shown, rhombus PQRS is similar to rhombus STQU. If the area of 
PQRS is 48 and angle QPS = 600, then what is the area of STQU? 
 
 
 A.10 B.15 C.18 D.16 
 
 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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57. If the lengths of two altitudes of a triangle are 10 and 11 cm, then which of the 
following cannot be the length of the third altitude? 
 
 A. 5 cm B. 6 cm C. 7cm D. 10 cm 
 
 
58. In the figure shown the inner triangle is equilateral with side length 2 units and 
circles drawn around its vertices. What is the distance between the two points of 
intersection of the bigger circles? 
 
 
 A. 2√15 B. 3√15 C.3√10 D.4√20 
 
 
59. If the lengths of the arcs AP and BP are 20 and 16 respectively, then what is 
the value of angle AXP? 
 
 
 A. 300 B. 240 C. 180 D. 100 
 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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60. In the given figure a square and an equilateral triangle are drawn. If the area of 
the square is 144 units, then what is the perimeter of the equilateral triangle? 
 
 
A) 12(3 + √3 ) B) 4(1 + √3 ) C) 4(1 + √2 ) D) 4(2 + √3 ) 
 
 
 
61. A, B, C, D, and E are five points on a circle. Also, a tangent is drawn to the 
circle at the point A. All the five angles marked with X are equal. Then, what is 
the measure of angle ABD? 
 
 
 A. 660 B. 70.50 C. 720 D. 750 
 
 
62. ABC is a right angled triangle right angled at A. The bisectors of the acute 
angles intersect at the point M. If the distance from M to the hypotenuse is 2√2, 
then what is the distance from M to A? 
 A. 4 B. 3 C. 8 D. 2√3 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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63. In an isosceles triangle ABC, AB = AC = 20 and BC = 16. H is a point on BC 
such that CH = 4. What is the length of AH? 
 
 A. 4√2 B. 4√22 C. 8√11 D. 2√5 
 
 
64. PQRS is a square with a side length of 2 units. Two equilateral triangles PQE 
and PQF are drawn such that E is on the interior of PQRS and F is on the 
exterior of PQRS. Then, what is the area of triangle RFE? 
 
 A. √5 B. √7 C. √3 D. √5 
 
 
65. We have a square ABCD with side length 1 unit such that isosceles triangle 
ABE where AE = BE is drawn on the exterior of the square. If the lengths CD, 
AC and CE are in increasing geometric progression, then what is the area of 
triangle ABE? 
 
 A. √15
4
− 1
2
 B. √15
2
− 1C. √15
4
 D. √15
2
 
 
 
66. A circle is circumscribed about a triangle PQR such that PQ = 6, QR = 8 and 
PR = 9. Point M is chosen on this circle such that chord PM intersects QR at its 
midpoint N. What is the length of NM? 
 
 A. 4� 2
75
 B. 16�1
2
 C. �2
5
 D. 16� 2
85
 
 
 
 
 
 
 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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67. There is a sector of 600 of a circle with radius 12 units inside which a square is 
inscribed such that two of its consecutive vertices lie on the arc of the sector. 
What is the area of the square? 
 
 A. 144(2 − √3) B. 24(2 − √3) C. 40(2− √3) D. 54(2 − √3) 
 
68. A figure is formed such that it contains a square ABCD with side length 2 units 
and four equilateral triangles drawn on the exterior of ABCD such that each 
triangle shares one distinct side with the square. What is the area of the smallest 
square which just contains this figure? 
 
 A. 2 + 2√3 B. 1 + √3 C. 4 + 2√3 D. 8 + 4√3 
 
 
69. A right triangle DEF is such that DE = 12, EF = 16 and DF = 20 units. P, Q and 
R are on sides DE, EF and DF such that EP = 4, EQ = 8 and FR = 6. What is the 
area of triangle PQR? 
 
 A. 65 B. 101
2
 C. 104
5
 D. 200 
 
 
70. S1 is a square with side length 20 units. S2 is formed by connecting the 
midpoints of sides of S1 and this process is continued such that Sn is formed by 
connecting the midpoints of the sides of SH-1. What is the value of ∑ [𝑆𝑆𝐻𝐻]∞𝐻𝐻=1 
where [H] denotes the area of the region H. 
 
 A. 800 B. 200 C. 135.9 D. 1000 
 
71. A point P is inside a square ABCD such that AP = 4, BP = 6 and CP = 5. What 
is the length of DP? 
 
 A. √2 B. √3 C. √5 D. √6 
 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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72. A trapezium with bases of length 12 inches and 18 inches and height of 8 
inches is revolved around the longer base to form a three dimensional figure. 
Then, what is the volume of the figure so formed? 
 
 A. 386𝜋𝜋 B. 768 𝜋𝜋 C. 748 𝜋𝜋 D. 400𝜋𝜋 
 
73. PQRS is a quadrilateral in which PQ = 5, QR = 4 and RS = 10. Then, how 
many integer values can PS take? 
 
 A.17 B. 18 C. 19 D. infinitely many 
 
 
74. A1 is a solid figure formed when an equilateral triangle is rotated 3600 around 
one of its sides. A2 is the solid figure formed when the same triangle is rotated 
3600 around one of its altitudes. What is the ratio of the volume of A1 to the 
volume of A2? 
 
 A. √3
6
 B. √3
3
 C. √3 D. 2√3 
 
 
75. It is given that the sum of the volumes of a regular hexagonal prism and a 
regular hexagonal pyramid is 196. If the bases have the same areas and the 
height of the pyramid is 3 times that of the prism, then, what is the volume of 
the pyramid? 
 
 A. 98 B. 49 C. 92 D. 47 
 
76. There is a rectangular prism with a square base kept inside a cylinder as shown 
in the figure. The vertices of the prism lie on the circles of the cylinder’s bases. 
If the radius of the cylinder is 2 units while the total surface area is 20π. Then, 
what is the surface area of the inscribed prism? 
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100 must do Geometry questions Kitabee CAT Preparation 
 
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A. 16√2 +24 B. 24√2 +16 C. 12√2 + 8 D. 8√2 + 12 
 
77. A. right circular cone has a slant height of 5. The measure of the vertex angle 
i.e. the top angle of the triangular cross section is given by the function A(t) = t2 
+7t +60 where A(t) is in degrees. Assuming that the slant height remains 
constant, what is the difference in areas of the cone’s base at t=5 and t=3? 
 
 A.125𝜋𝜋(3−2�2)
24
 B. 25𝜋𝜋/4 C. 3/2 D. 125𝜋𝜋(3−�2)
24
 
 
 
78. The base of the triangular piece of paper ABC is 12 cm long. The paper is 
folded down over the base of the paper. The area of the triangle that projects 
below the base 16% that of the area of the triangle ABC. The length of DE in 
cm, is 
 
 
 
 
 
 
 
 
 A. 9.6 B. 8.4 C. 7.2 D. 4.8 
 
 
 
 
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79. There is a regular hexagon ABCDEF of side length 2. Diagonals AC, BD, CE, 
DF, EA and FB are drawn to form another hexagon in the middle. Then, what is 
the area of the smaller hexagon? 
 
 
 
 
 A. 1/√3 B. √3 C. 2√3 D. 2/√3/9 
 
80. A regular dodecagon is inscribed in a circle of radius 4 units. What is its area? 
 
 A. 48 B. 24 C. 28 D. 44 
 
 
81. Three circles of radius 4 units each are stacked together so that each is 
externally tangent to the other two. A smaller circle is placed in the middle so 
that it touches all the three larger circles. What is the perimeter of the smaller 
circle? 
 
 A. √3+12 
3
 B. 4√3−12 
5
 C. 8√3−12 
3
 D. 4 
 
 
82. What is the length of the angle bisector to the hypotenuse drawn in a right 
triangle with lengths 7, 24 and 25? 
 A. 84∗2√2
31
 B. 24∗2√2
31
 C. 44√2
31
 D. 24∗2√2
31
 
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83. PQR is a triangle in which point T is on PQ and S is a point on PR. If PT:TQ = 
4:5 and PS : SR = 2:3 and point U is the intersection of SQ and TR. Then, what 
is the ratio of lengths of QU to US? 
 
 A. 25:12 B. 25:3 C. 3:4 D. 4:3 
 
84. In the given figure chords CE and DE are equal to each other while O is the 
centre of the circle. If the measure of arc AB is 600, then what is the ratio of 
area of triangle CED to the area of triangle AOB? 
 
 
 
A. √3/4 B. 3√3/2 C. 4/√3 D. √3/3 
 
85. A right triangle with sides AB = 9, BC = 12 and AC = 15 is drawn. BF and BE 
are the trisectors of angle ABC. What is the length of BE? 
 
 
 
 A. 12
4+3√3
 B. 72
11+3√3
 C. 72
4+3√3
 D. 72
3√3
 
 
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86. If the slant height of a regular right hexagonal pyramid is 6 while its height is 
10. What is the volume of the pyramid? 
 
 A. 72√3 B. 96√3 C. 192√3 D. 160√3 
 
 
87. In the given figure, there is a cube of side length 8 units. M is the midpoint of 
AB and N is the midpoint of N. The face opposite to ABCD is a square PQRS. 
What is the area of the triangle MNQ? 
 
 
 A. 2√5 B. 4√10 C. 16√5 D. None of these 
 
88. There is right circular cone of diameter 12 and height 6 units. Water is being 
poured into this cone. When the cone is 1/9th filled then what is the distance of 
water’s surface from the top of thecone? 
 
 
 
A. √143 B. √193 C. 2√243 D. √493 
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89. In triangle ABC, BM and AN are medians. If BM = 15, AN = 18 and AB = 21. 
Then what is the perimeter of triangle OMN 
 
 
 
 
 A. 21.5 B. 24 C. 20.5 D. 25.5 
 
 
 
 
 
90. There is an equilateral triangle of perimeter 3√3 inscribed inside another 
equilateral triangle as given in the figure. AB = 1, then what is the area of the 
larger equilateral triangle? 
 
 A. 3√3
2
 B. 9√3
4
 C. 3√3 D. 3/2 
 
 
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91. In the given figure, the regular octagon formed has a perimeter of 32 units. 
What is the area of the quadrilateral QOPH? 
 
 
 
 A. 8 + 4√8 B. 4√8 C. 6 + 4√8 D. 2 + 4√8 
 
92. In the diagram, right angled triangles AED and BFC are constructed inside 
rectangle ABCD so that F lies on DE. If AE = 21, ED = 72 and BF = 45 what is 
the length of AB? 
 
 
 
 A. 50 B. 48 C. 52 D. 54 E. 56 
 
 
93. In the figure given below, ABC is a right triangle right angled at A and angle 
DAC = 600. Also, AD ⊥ BC, ED ⊥ AB, EF ⊥ BC. Also, BC = 80 units. Find the 
value of BF + BE + BD + AB. 
 
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 A. 15 B. 35 C. 75 D. 80 
 
 
 
 
94. In triangle PQR, PQ = 8, PR = 10 and QR = 9 units. Line segments AC and BD 
are drawn parallel to side PR with A and B on PQ and C and D on side QR with 
BD closer to PR such that the triangle is divided into three equal areas. What is 
the length of BD? 
 
 A. 10√6
2
 B. 5√6
3
 C. 13√6
3
 D. 10√6
3
 
 
95. In triangle ABC, DE and DF are parallel to AC and BC respectively. If BE is 5 
and EC is 3 units, then what is the ratio of the area of triangle AFD to the area 
of the parallelogram DFCE? 
 
 
 A. 3/10 B. 7/15 C. 5/8 D. 3/2 
 
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96. DEF is a right angled triangle right angled at E. Inside it a square PQRS is 
constructed as shown in the figure. If EP = 48 and EQ = 36, then what is the 
length of the segment DF? 
 
 
A. 188 B.185 C. 187 D. 180 
 
97. In an equilateral triangle, six perpendiculars are drawn from the midpoints of 
the sides as shown in the figure. What fraction of the area of the initial triangle 
does the resulting hexagon cover? 
 
 
 A. 1/3 B. 2/5 C. 4/9 D. ½ 
 
 
98. There is a rectangle ABCD in which S, T, U and V are the midpoints of sides 
AB, BC, CD and DA respectively. If M is the midpoint of the segment UV, 
then what is ratio of area of triangle MST to area of the rectangle ABCD. 
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 A. 2 B.4 C. 1 D. 6 
 
 
99. There are four semicircles drawn of radius 2 units as shown in the figure and 
their centres are at the midpoints of the sides of a square. What is the radius of 
the smaller circle that touches all the four semicircles? 
 
 A. 3√2− 2 B. 2√2− 2 C. √2 − 2 D. 2√2 
 
100. In the figure shown below, it has two squares of sides 4 and 2 cm 
respectively. What is the area of the shaded region? 
 
 
 A.8 B.4 C. 6 D.2 
 
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ANSWER KEY: 
1. A 11. A 21. A 31. A 41. C 51. B 61. C 71. C 81. C 91. A
2. B 12. D 22. C 32. C 42. A 52. A 62. A 72. B 82. A 92. A
3. C 13.B 23. C 33. E 43. B 53. B 63. B 73. A 83. A 93. C
4. D 14. D 24. C 34. D 44. C 54. A 64. C 74. D 84. C 94.D
5. B 15.B 25. B 35. B 45. B 55. C 65. A 75. A 85. C 95. A
6. B 16.A 26. B 36. A 46. A 56. D 66. D 76. B 86. C 96. B
7. C 17.D 27. D 37. B 47. A 57. A 67. A 77. B 87. C 97. D
8. C 18. A 28. A 38. B 48. B 58. A 68. D 78. B 88. C 98. A
9. D 19. D 29. A 39. C 49.B 59. D 69. C 79. C 89. A 99. B
10. D 20.C 30. B 40. C 50. D 60. A 70. A 80. A 90. B 100. B
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