100 Must Solve Geometry Questions · A triangle of area 80 sq. cms has two of its sides as 12 cms...

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100 Must Solve

Geometry Questions


100 Must Solve Geometry Questions

1. In ∆PQR, QS and SR are angle bisectors and if angle QPR=70⁰. Find the measure of angle

QSR. [125]

2. In quad JKLM, the area of triangular regions JOK, JOM and KOL are 10, 12 and 20

respectively. Find the area of ∆MOL.

A.30 B..24 C.16 D.15

3. a & b are the lengths of the base & height of a right angled triangle whose hypotenuse

is ‘h’. If the value of a & b are positive integers, which of the following cannot be a

value of the square of the

hypotenuse?

A.34 B.26 C..64

D.50

4. Perimeter of an equilateral triangle is 27cm. Find the area of the triangle in mm2.

A..2025√3 B.4050 C.4050√3 D.2025

5. ABCD and OMNP are squares of sides 24 cm and 18 cm respectively. O is the centre of square ABCD, PQ = 4 cm. Find the area of the shaded region.


A.120 B.124 C..144 D.108

6. If tan θ + sin θ = A and tan θ-sin θ= B. Find A2-B2.

A..4√A√B B.2√A√B C.2AB D.√(2AB)

7. All possible obtuse-angled triangles with sides of integer lengths are constructed, such

that two of the sides have lengths 5 and 10. How many such triangles exist?

A.5 B..6 C.7 D.8

8. In a regular hexagon, find the sum of the interior angles and the exterior angles (in

degrees).

A.480 B.1440 C..1080 D.1260

9. Two persons A & B are looking at a bird which is on the top of a wall from opposite

sides of the wall. A is closer to the foot of the wall by 30 m than B. If the angle of

elevation of the bird as observed by A & B are 60° & 30° respectively. The height of the

wall is (in meters) A.30√3 B.45 C.90

D..15√3

10.


Find the measurement of angle PRQ.

A.72⁰ B.120⁰ C..108⁰ D.60⁰

11. If each interior angle of a regular polygon with n sides measures 156°, find the value of

n. A.18 B.17 C.16 D..15

12. When a bullet is shot to hit a target lying on the ground from the top of a tower of

height 20 metre at an angle of depression of 45°, it missed the target & hit a point in

between the line joining the foot of the tower and the target, which is 5 metres away

from the target. What is the tangent of angle of depression at which the bullet should

have been shot so as to hit the target? A.2/3 B..4/5 C.5/6

D.3/4

13.

In ∆ABC, AD is the altitude. P in mid pt of BD and Q in mid pt of AC. BC=18 and AD=24.

Find the length of PQ.

A.10 B.13 C.12 D..15

14. If three angles of a octagon are 80° each and the other remaining five angles measure

k° each, find the value of k?

A.148 B.158 C..168 D.178

15. In a triangle PQR, angle P = 60°. PS is an angle bisector of angle P. Side PQ = 12 cm and

side PR = 20 cms. Calculate the length of


PS. A..7.5√3

B.15√3 C.5√3 D.10√3

16.

In ∆PQR, PX=PY, RY=RZ and angle PQR=90⁰. Find angle YXQ when angle YZQ=125⁰.

A..100⁰ B.105⁰ C.125⁰ D.120⁰

17. There is a regular octagon ABCDEFGH, a frog is at the vertex A. It can jump on to any

vertex except the exact opposite vertex. The frog visits all the vertices exactly once and

then reaches vertex E then how many times did it jump before reaching E?

A. 7 B.8 C.5 D..6

18.

A.28.9 B.24 C..22.9 D.23.5

19. In triangle DEF, points A, B, and C are taken on DE, DF and EF respectively such that

EC=AC and CF = BC. If angle D=40 then what is the angle ACB in degrees?


A..100 B.120 C.90 D.80

20.

XY=XZ. XC∟YZ. ∆ABC is an equilateral triangle. V is the mid pt of XZ. AB=XB. Find the

measurement of angle XVC.

A.115⁰ B..120⁰ C.125⁰ D.110⁰

21.

A.120 B.180 C.140 D..130


22. The length of the common chord of two circles of radii 15 cm and 20 cm, whose

centres are 25 cm apart, is (in cm) [24]

23.

AB = AC = 20. BC = 24. AD ⊥ BC. F is the midpoint of AC and DE = 12. What is the length

of segment DF? [10]

24. Find the sum of A, B, C, D and E.

A.360 B..540 C.450 D.630

25. Each side of a given polygon is parallel to either the X or the Y axis. A corner of such a polygon is said to be convex if the internal angle is 90° or concave if the internal angle is 270°. If the number of convex corners in such a polygon is 25, the number of concave corners must be A..21 B.22 C.23 D.24

26. A triangle of area 80 sq. cms has two of its sides as 12 cms and 16 cms. If the inradius of

the triangle is 4 cms, then find the circumradius.

A.7.5cm B.8cm C..7.2cm D.7.8cm

27. A rectangular courtyard ABCD of side 38 m and 24 m encloses a rectangular lawn EFGH

surrounded by a 4m wide gravel path. Find the area of the gravel path.

A.544 B.600 C.496 D..432


28. In the figure below, the rectangle at the corner measures 10 cm × 20 cm. The corner A of the rectangle is also a point on the circumference of the circle . What is the radius of the circle in cm?

A.10 B.40 C..50 D.20

29. The lengths of the medians of an isosceles triangle are 36, 19.5 and 19.5. Find the area

of the triangle (in sq. units). [180]

30. Radii OA and OB divide the circle into two sectors with central angles 60° and 300°. Find

the ratio of the areas of the largest circles that can be fitted inside each of the two

sectors.

A.1:2 B..4: 9 C.1:5 D.1:3

31. It takes 49 minutes to paint the outer surface of a cube. This cube is cut into 343 smaller

identical cubes. All these 343 smaller cubes are separated into four groups such that in

each group, all the cubes can be glued together to form an individual larger cube. How

much total time (in minutes) will it take to paint the outer surface of all the four new

cubes? A.60 B.57 C.55

D..63

32. A rectangle with perimeter 42 cm is such that it can be divided into squares with each

side equal to 3 cm. Find the maximum possible area of such a rectangle.(in cm2)

A.54 B..108 C.90 D.110


33. A solid right-circular cone has height 56cm and base radius 84cm. The top half of the

cone is cut off and replaced by a solid hemisphere whose circular base exactly fits the

upper circular surface of the remaining portion of the cone. Approximately by what

percentage has the volume of the cube changed? Take ᴨ =22

7

A..25 B.35 C.30 D.27

34. The largest possible square PQRS is inscribed in an equilateral triangle ABC. The largest

possible square is inscribed in ∆APQ. This process is continued infinite times. If the

length of each side of PQRS is 1 units, then find the sum of the perimeters of all the

squares in the figure.

A..4+2√3 B.4-2√3 C.4

D.8

35. Three identical circles with maximum possible area are drawn inside an isosceles

trapezium as shown. Find the perimeter of the trapezium in terms of r, where r is radius

of the circle. (sin22.5=0.38)

36. In the adjoining figure, chord ED is parallel to the diameter AC of the circle. If CBE

= 65°, then what is the value of angleDEC?


A ..25 B.35 C.45 D.55

37.

A.6.5 B.7.2 C..7 D.8

38. In the figure, ACB is a right angled triangle. CD is the altitude. Circles are inscribed

within the triangles ACD, BAD. P and Q are the centers of the circles .The distance PQ is

A .5 B. .5√2 C .7

D.8

39. In a triangle ABC, AB =6, BC = 8 and AC = 10. A perpendicular dropped from B, meets

the side AC at D. A circle of radius BD (with centre B) is drawn. If the circle cuts AB and

BC at P and Q respectively, then AP: QC is equal to

A.1:1 B.3:2 C.2:1 D..3:8

40. A vertical tower OP stands at the centre O of a square ABCD. Let h and b denote the

length of OP and AB respectively. Suppose angle APB = 60° then the relationship

between h and b can be expressed as

A.h2=2b2 B. .2h2=b2 C. 3h2=b2

D. h2=b2


41. ABCDE is a regular pentagon inscribed in a circle with centre 'O'. Another pentagon is drawn by joining the midpoints of sides OA, OB, OC, OD and OE. Find the ratio of the area of the smaller pentagon to the larger pentagon. A..1:4 B.1:3 C.1:2 D.1:√2

42. A cross-block is made from a solid cube of side 12 cm by cutting off cubes of side 4 cm at each of the corners. What is the total surface area of the cross-block? A.1728cm2 B.768cm2 C..864cm2 D.1632cm2

43. A square sheet of paper of length ‘l’ is folded along its diagonal to form a triangle. Some portion of this triangle is cut along the dotted lines, this forms a regular hexagon. O is the mid pt of the base and D is the pt where the paper is cut. What is the area of the

paper cut out? A. 1

1+2√2 l 2 B..

1

3+2√2 l 2

C. 1

3+√2 l 2 D.

1

1+√2 l 2

44. Let ‘O’ be an incentre and circumcentre of a ∆ABC. Length of OB is 12cm. find the area of the circumcircle (in sq.cm) that is enclosed by the triangle. A.108√3pi-144 B..144pi- 108√3 C.144pi-72√3 D.72√3pi-36

45. Find the approximate sum of the lengths of all diagonals of a regular hexagon having length of each side as 12 cm. A.144cm B.249cm C..197cm D.177cm

46. BC is the diameter of a circle. Chord AD is perpendicular to BC. If angle DAB=30⁰ and

AD=18cm find the distance of chord AB from the centre. A.4.5cm B.6cm C.5cm D..9cm

47. There is a field in the shape of an equilateral triangle with side 18m. There is a circular park with radius 6m with the center at the centroid of the triangular field. The remaining part of the field needs to be covered with stones at the rate of Rs.12 per sq.m. Find the total cost incurred. (pi=3.14) A.325 B..327 C.204 D.287

48. ∆ABC and ∆DFE are congruent right angled triangles with DC : AF=1:3 and A-D-C-F. What is the ratio of areas of ∆BGE to ∆DFE.


A.1:3 B.1:9 C.4:9 D..1:4

49. In ∆ABC let D, E and F be the midpoints of sides BC, CA and AB, respectively. Let the angle bisectors of angle FDE and angle FBD meet at P. If angle BAC=35⁰ and the angle CBA=80⁰, find angle BPD.

A..57.5 B.62.5 C.65 D.67.5

50.

51. There are two concentric circles of radii 12 and 13 respectively centred at point O. A tangent to the inner circle meets the outer circle at points P and Q. Find l(PQ). A.8 B.9 C..10 D.11


52.

OP:OQ=3:4. Find PS:PQ. A..12:25 B.5:12 C.3:4 D.13:20

53. The surface area of solid hemisphere is thrice the curved surface area of a solid cylinder. If the radius of the cylinder is one – fourth of that of the hemisphere, what is the ratio of radius of solid hemisphere to the height of the cylinder? A.2:3 B..1:2 C.2:5 D.1:3

54. In the given fig, ABCDE is a square. AE=BF=CG=HD=0.25AB. Also, AP=PQ=QB=CS=ST=TD. Find the area of hexagon PQRSTU if the side of the square is 12units.

A..54sq.units B.57sq.units C.80sq.units D.65sq.units

55. A sphere is cut into 8 equal parts along 3 mutually perpendicular planes passing through the centre of the sphere. The maximum distance between any two points within each part is 3√2cm. What is the volume of the original sphere? (in cm3 ) A.85.33pi B.49.27pi C..36pi D.27pi

56. Virgil divides his hexagonal farm to be distributed among his three sons. Patrick receives the rectangular region ABDE whose breadth and length are in the ratio 1 : 2. Simon and Wayne receive regions AFE and BCD respectively that are shaped as isosceles right triangles. Find the percentage share of Simon out of the entire farmland.


A.50 B.75 C..25 D.30

57. AB=AC, PB and PC are tangent drawn to the circle from pt P. Find angle BPD when angle BPC=80⁰.

A.37 B.30 C.37.5 D..35

58. A straight wire connects the foot of building A and the top of building B, making an angle of 45° with the ground. The same straight wire is just long enough to connect the top of building B and the top of building A. What will be the ratio of the length of this wire to the length of the wire that connects the top of building A and the bottom of building B? A.1:1 B..√2:√5 C.√2:5 D.2:√5


59. A cone and a sphere have identical radii and identical volume. Find the ratio height of cone and height of sphere. A..2:1 B.1:4 C.2:3 D.3:5

60. A cube of side 12 cm is painted blue on all six surfaces. Then it is cut into smaller cubes of side 1 cm each. How many of the smaller cubes will have blue paint on exactly two faces? A.100 B.112 C.144 D..120

61. How many different types of regular convex polygons having more than 14 sides have the measures (in degrees) of both their interior angles and their central angle (i.e., the angle subtended by one of the side at the centre of the polygon) as integer values (in degrees)? [14]

62.

A..(2-√3):1 B. (2+√3):1 C. (2-√3):2 D. (2+√3):2

63. In the adjoining diagram, ABCD is a parallelogram with m∠BCD = 135°. AB is diameter of the circle and ℓ(AB) = 20 cm. What is the area common between the circle and the parallelogram (in cm2)?

A.25(pi-2) B.25pi C..25(pi+2) D.25pi/2


64.

[30]

65. Two identical rectangles having length = 28 units and breadth = 8 units are inscribed in a circle of area ‘a’ as shown in the figure given below. The area of the shaded region is (a − b). The value of 'b' is:

A.374 B.404 C..384 D.354

66. For what value of k, is the line (3x – y + 1) + k(x + 3y + 2) = 0 perpendicular to x = 4? A..-3 B.3 C.-2 D.1/3

67. A farmer has 198 m of barbed wire fencing. With this he encloses a field of area X sq. m. What can be said about the maximum possible value of X? A. 1600 < X < 1650 B. 1870 < X < 2200 C. 2500 < X < 3000 D.. 3000 < X < 3500


68. ABCDEF is a regular hexagon and EFGHI is a regular pentagon such that the pentagon lies outside the hexagon. Side CD and side HI are extended to intersect at point P. Find m∠DPI (in degrees).

[96]

69. All the six faces of a cube of side 4 cm are initially painted using white colour. The cube is then cut into 64 identical cubes having side 1 cm each. The faces of all these small cubes that do not have white colour on them are then painted using green colour. What is the sum of the total surface areas (in cm2) of the 27 small cubes that have been painted using green colour? A.384 B..288 C.96 D.108

70.

A.

283

5224 B.

283

653 C..

283

5877 D.

653

5877

71.

A.60m B..60√3m C.120m D.60/√3m

72. ABCD is a square as shown below. The area of the trapezium ABED is four times the area of the triangle DEC. What is the ratio of area of trapezium AECD to the area of the

square? A..7:10 B.3:4 C.3:7 D.4:7

73. The curved surface area of a solid cone is three-fifths of the total surface area of the cone. If the volume of the cone is given by c𝜋h3, where 'h' is the height of the cone and 'c' is a constant, find the value of 'c'. A.1/5 B..4/15 C.1/15 D.1/9


74. In the figure given below, PQR is an equilateral triangle. PR and PQ are tangents to the circle and TR = 4 cm.

If the area of the triangle is 36√3cm2 then find the area of the circle in cm2. A.16pi/3 B.48pi/3 C..64pi/3 D.49pi/3

75. Frederickson park is a circular garden of radius 100 m. Four poles A, B, C, D are fixed at four different points in clockwise direction (in that order) on the circumference of the garden. Angle ADC = 45°. Marcus completed 10 round trips along the circumference of the garden taking path ABC (i.e. path A-B-C-B-A in one round trip). Markief similarly walked along the circumference of the garden and completed ‘n’ round trips taking path ADC (i.e. path A-D-C-D-A in one round trip). If Markief covered 60% the distance covered by Marcu, then what is the value of ‘n’? A..2 B.3 C.4 D.5

76. The perimeter of an equilateral triangle is same as that of an isosceles right angled triangle. If the area of the isosceles right angled triangle is 18 cm2, then what is the length (approximate) of side of the equilateral triangle? A.6.2cm B.6.4cm C.6.6cm D..6.8cm

77.

[10]

78. An ant named ‘Luke’ and a fly named ‘Manny’ are standing at the center of a right circular cone having diameter of the base equal to 10 cm and height 12 cm. They notice a sugar particle inside the cone at the tip of the cone. Luke first walks to his left in a straight line and reaches a point on the circumference of the base of the cone. He then walks along the circumference of the base of the cone and reaches diametrically opposite point of the base of the cone. Luke then walks up-to the tip of the cone along curved surface of the cone taking shortest possible distance. On the other


hand, Manny directly flies vertically upwards to the tip of the cone. If both Luke and Manny start at the same time and reach the tip of the cone at the same time, what is the ratio of Luke’s walking speed to Manny’s flying speed? A..(5pi+18)/12 B. (5pi+13)/12 C. (5pi+15)/12 D. (5pi-13)/12

79. The area of a rhombus is 54 cm2. What is the length of side of the square (in cm) whose area is thrice the area of the rectangle having length equal to the longer diagonal of the rhombus and breadth equal to the shorter diagonal of the rhombus? A.12 B.. 18 C.20 D.16

80. Let us assume that a spherical object becomes a cube. What is the ratio of the minimum distances that an insect has to travel along the surface to get to a point diagonally opposite on the cube to the distance that it would travel on the spherical one. Assume that the length of a side of the new object is the radius of the earlier one. A.(1+√2):pi B.pi:√3 C.22:7√5 D..7√5:22

81. In the given diagram, the line segment AB, which is 2 cm long, is a tangent to the inner of the two concentric circles with centre O and intersects the outer circle at B. What is the area of the circular region between the circles? A.pi B.2pi C.3pi D..4pi

82. The sum of the radii of a frustum-shaped bucket is 12 units and their product is 32 units. Some water is filled in the bucket. The ratio of the volume of the water to the volume of the bucket is 1 : 3 and the ratio of the height of the water level to that of the bucket is 2 : 3. Find the diameter of the surface of the water level. A.1/3 B.3/4 C..1/2 D.5/6

83. In a regular hexagon PQRSTU, find the ratio of the area of ∆ PRT to that of the hexagon PQRSTU. 4.6 B..9.2 C.7.8 D.6.8

84. Find the sum of areas of regular polygons having four, five, six and eight number of sides in sq.units. These polygons are inscribed in a circle of radius 10 units. [sin72° = 0.951]. A.600 B.766.35 C..980.39 D.832

85. In figure, segment AB is divided into three parts, BD: DQ: QAsegment AB is divided into three parts, BD: DQ: CR and segment PR is divided into three parts, PD: DC: CR=1:2:3. If PR=AB and angle A=11.5, find angle DQR and QXC (take sin11.5=0.2).


A.30,120 B..30,161.5 C.25,115 D.25,150

86. 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..pi units B.2pi units C.pi/2 units D.2pi/3units

87. Let x and y denote the height and the side respectively of a regular square pyramid. Suppose the angle between two slant edges at the vertex is 60°, what is the relation between x and y? A.x2=2y2 B. x2=3y2 C.. 2x2=y2 D. x2=y2

88. ∆PRQ is an isosceles triangle whose area is 8 sq.units. A circle is drawn such that that PQ is tangent to the circle at s. R is the centre of the circle. The area of the shaded region is

equal to: A.0.86 B.√2 C.2√2 D..1.716

89. In the figure given below, AB is the chord of a circle with centre 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.2 B..3 C.1 D.4

90. The perimeter of a parallelogram, a hexagon and a rectangle are equal. The areas covered by parallelogram, hexagon and rectangle are p, h, r respectively. If the


parallelogram and rectangle have equal bases, but different heights, then: A..h>r>p B.r>h>p C.p>r>h D.p>h>r

91. A father and his son are waiting at a bus stop in the evening. There is a lamp post behind them. The lamp post, the father and his son stand on the same straight line. The father observes that the shadows of his head and his son’s head are incident at the same point on the ground. If the heights of the lamp post , the father and his son are 6 metres, 1.8 metres and 0.9 metres respectively, and the father is standing 2.1 metres away from the post, then how far (in meters) is the son standing from his father? A. 0.9 B. 0.75 C. 0.6 D.. 0.45

92. Ray and Roy are two boys sitting on a see-saw. Intially, when Ray is sitting at ground level and Roy is up in the air, the angle made by the see-saw with respect to the ground is 60o. Ray moves upwards in such a way that at a certain moment when Ray is still at a lower height than Roy, the extended line from the see-saw makes an angle of 45o with respect to the ground. Find the height above the ground that Ray has reached at that moment if initially Roy was 6.92 feet in the air. A.0.814 B.0.732 C..0.636 D.0.312

93. P, Q, S and R are points on the circumference of a circle of radius r, such that PQR is an equilateral triangle and PS is a diameter of the circle. What is the perimeter of the quadrilateral PQSR?

A.. 2r(1+ 3) B. 2r(2 + 3) C.r(1+ 5) D. 2r + 3

94.

A.9:5 B.7:5 C.9:7 D..11:5

95. An equilateral triangle BPC is drawn inside a square ABCD. What is the value of the angle APD in degrees?

A. 75 B.90 C.. 150 D.120

96. 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.15.5 B.14 C.49 D.. none of these

97.

Two circles with centres P and Q cut each other at two distinct points A and B. The circles have the same radii and neither P nor Q falls within the intersection of the circles. What is the smallest range that includes all possible values of the angle AQP in degrees? A.Between 0 and 45 B.Between 0 and 90 C. Between 0 and D.. Between 0 and 60 98. A and B are travelling at the speed of 3 km/hr and 2 km/hr respectively. A travels

towards North and B travels towards East from the same starting point. B travels for 4 hours towards East, then for 4 hours towards North. At this point B receives a call from A who has stopped after travelling for 4 hours towards North from the starting point. B decides to meet A and travels for 2 hours towards West. At this point at what angle should B turn so as to traverse the minimum distance to reach A? A..45 B.60 C.30 D.75

99. In a triangle ABC, the lengths of the sides AB and AC equal 17.5 cm and 9 cm respectively. Let D be a point on the line segment BC such that AD is perpendicular to BC. If AD = 3 cm, then what is the radius (in cm) of the circle circumscribing the triangle ABC? A.18.25 B.27.5 C.30 D..26.25

100.

Which of the following equation represents the graph? A.x2+y2=9 B..|x|+|y|=3 C.|x|=|y|=3 D.|x+y|


Solutions

1. [125]

Angle QPR=70, therefore angle PQR+ angle PRQ=110. Now, as QS and SR are angle

bisectors, angle SQR + angle SRQ=55. Which gives angle QSR=125.

2. B.24

Let JO=A, KO=B, LO=C and MO=D. We get the areas as JOK=0.5(A.B)=10,

KOL=0.5(B.C)=20, JOM=0.5(D.A)=12 and MOL=0.5(C.D).from these we can find out the

values of C and D and thus obtain area of MOL.

3. C.64

26=52 +12, 34=32+52, 50=52 +52 and 64 cannot be expressed as sum of square of

integers

4. A.2025√3 Perimeter=27cm => side=9cm=>area =√3

4xsidexside

5. C.144


6. A.4√A√B

7. B.6

8. C.1080

9. D.15√3

tan30=h/(x+30)=1/√3 and tan 60=h/x=√3. Solving these equation we get h=15√3

10. C.108⁰

∆ARB and ∆RPQ are similar =>angle PRQ =angle ARB=180-(ang RAB + ang RBA). Let

angle RAC=x, angle PRQ=180-2x and (ang RAB + ang RBA)=180-3x =>x=36⁰

11. D.15

Exterior angle= 180-Interior angle =24⁰


Also, Exterior angle=360/n

Hence, n=360/24=15

12. B.4/5

13. D.15

If we drop a perpendicular from Q to BC to a pt say E, then QE will half of AD (12) as

∆ACD and ∆QCE are similar. Therefore, ED is half of DC. Which gives PE=0.5BC=9 as P is

mid pt of BD. Thus, PQ can be obtained using Pythagoras theorem.

14. C.168

Sum of interior angles of an octagon is 1080. If 3 are angles 80⁰ and the rest angles are

equal (k). Then k=(1080-240)/5

15. A.7.5√3

16. A..100⁰

RY=RZ so angles RYZ=RZY=180-125=55. Angle R=70=>angle P=20. Therefore angle PXY=

angle PYX=(180-20)/2=80 =>angle YXQ=100


17. D.6

A frog couldn’t jump on the vertices E. Therefore, there are 6 other vertices where it jumped.

Thus, there are 6 jumps before reaching E.

18. C.22.9

19. A.100

20. B.120⁰

angle ABX=180-60=120 =>angle BAX = angle BXA=30, since XY=XZ, angle BXA= angle


BXV=30 =>angle YXZ =60 also XYZ = XZY =60= angle VCZ as ∆CVZ and ∆XYZ are similar.

Now by exterior angle theorem we get angle XVC=120

21. D.130


22. [24]

23. [10]

Since, F is the mid pt of AC, FC=0.5AC.∆ABC and ∆FCD are similar. Therefore,

DF=0.5AB=10

24. B.540

Consider angle A, A=180-Exterior angle. Let ABCDE be a regular pentagon, then exterior

angle=72⁰. Doing this for all the angles or we can consider all the angles equal as it is a

regular polygon. Thus, 540.


25. A.21

The number of convex corners is always less than concave corners by 4. Hence 25 – 4 =

21.

26. C.7.2cm

27. D..432

28. C.50

29. [180]

The median on the base of the isosceles triangle will be perpendicular to the base and

the median is divided by the centroid into a ratio of 2:1. Thus, by using Pythagoras

theorem on the right angled triangle formed at the base by the lower part of the

medians the area is calculated.

30. B.4:9


31. D.63

Taking the cube root of 343 we get the side of the cube as 7 units. Therefore, the

surface area of this will be 294sq.units. It takes 49 minutes to paint the outer surface of

a cube, so it takes 10sec to paint 1sq.unit.This cube is cut into 343 smaller identical

cubes, only possible combination of 4 cubes having an integer as side is 1,1,5 and 6.

Now, calculate the total surface areas of these cubes and then divide it by the rate of

painting to obtain the answer.

32. B.108

Let the length and breadth of the rectangle be ‘l’ cm and ‘b’ cm respectively.

Therefore, 2(l + b) = 42

Therefore, l + b = 21

Now, l and b have to be multiples of three for the rectangle to be divided into squares

with side 3 cm.

Therefore, all possible pairs of l and b (in no particular order) are (18, 3), (15, 6) and (12,

9).

Therefore, the maximum possible area is 12 × 9 = 108 cm2.

33. A.25

Volume of the original cone= 413952.

Volume of the cut out portion of the cube =51744.

Volume of the hemispherical portion =155232.

The change in volume =155232-51744=103488.

103488 is 25% of 413952.


34. A.4+2√3

35. [19.176r]

AD=2r as it equal to the diameter of the circle. Now, perimeter=AC+BE+2AB, as angle

B=45⁰ AB=√2BD and BE=AC+2BD. Perimeter = AC+AC+2BD+2√2BD=2AC+2BD(1+√2).

Now for BD: AD=BD, as angle B=45⁰, BD=2r.

Now for AC: AC=(4r)+2OF; AO=R, angle OAF=(135/2)-45=22.5⁰ . OF=AO.sin(22.5)=0.38r.

therefore, Perimeter=2(4r+0.76r)+4r(1+√2)=19.176r


36. A.25


37. C.7


38. B.5√2


39. D.3:8


40. B.2h2=b2

41. A.1:4


42. C.864cm2

43. B.1

3+2√2l 2

Thus, using the value sides DE and DC the area of ∆EDC is obtained, 4 times this is the

total area of the paper cut out.


44. B.144pi- 108√3

As the incentre and the circumcentre are located at a common pt the triangle is an

equilateral triangle. Hence, the radius of the incircle is 6cm and the side of the triangle is

12√3cm.

area of the triangle=(12√3).2√3/4 =108√3 sq.cm

area of thr circumcircle=pi(12)2 =144pi sq.cm

Thus , B

45. C.197cm

A hexagon has 9 diagonals out of which 3 are longer than the rest. These 3 diagonals are

twice the length of the side of the hexagon as a hexagon is made up of 6 equilateral

triangles. The remaining diagonals can be calculated as:

Drop a perpendicular from B to AC at P. Now, the ∆APB is made up of

angles 30, 60 and 90. Using this we get the length of AP=6√3. Therefore, AC=12√3cm.

Thus, the sum of diagonals=3(24)+6(12√3)≈197cm.


46. D.9cm

Therefore, AE=OF=9cm.

47. B.327

Area of triangle – area of circle=area to be covered with stones

(√3

4 x18x18) –(3.14x6x6)=27.25

Cost incurred=27.25x12=327

48. D.1:4

AC=DF as ∆ABC and ∆DFE are congruent right angled triangles. AD+DC=DC+CF.

Therefore, AD=CF=DC. Thus, C is the mid pt of DF. Therefore, ∆DCG≈∆DFE similarly

∆DCG≈∆ABC. Now, angle BGE=angle DGC, BG=GC and DG=GE. So ∆BGE and ∆DGC are

congruent. Now areas of DGC:DFE=1:4 property of similar triangles. Since, area ∆BGC=

area ∆DGC, hence BGE:DFE=1:4.

49. A.57.5

Angles A, C and B are 35, 65 and 80 respectively. E and D are mid pts of AC and BC


respectively. So, ED||AB, hence angle EDC= angle B and angle FDB= angle C. Therefore,

angle FDE=180-(C+B) =35, using angles of ∆PBD we can find angle BPD.

50. [30]

51. C.10

Using Pythagoras theorem on ∆RQO we can find the length of RQ and PQ=2RQ.

52. A.12:25

Consider area of ∆OPQ, area ∆OPQ=0.5xOPxOQ=0.5xSPxPQ. If OP:OQ=3X:4X, Then we

get SP=12

5x.

53. B.1:2

3 x 2pi x R.h=3pi x r2


Since, R=1

4r we get, r:h=1:2

54. A.54sq.units

55. C.36pi


56. D.25

57. D.35

58. B. ..√2:√5

AB’=√2X and BA”=√5X


59. A.2:1

60. D.120

Only the cubes that are formed from edges, except the corner cubes, will have exactly two painted faces. Cubes on one edge = 10. Total edges = 12

Therefore, number of cubes with exactly two painted faces = 10 × 12 = 120

61. [14]

62. A. (2-√3):1


63. C..25(pi+2)

64. [30]

65. C.384

Area of the shaded region = (a – b), where ‘b’ is the area covered by the rectangles. Area covered by the rectangles = (28 × 8) + (28 – 8)(8) = 384

66. A.-3


67. D. 3000 < X < 3500 The max possible area is a circle. The wire forms the circumference of the circle. Thus, we get the radius of circle as 31.5m and area as 3118.5m.

68. [96]

69. B..288 Total surface area of the 64 small cubes = 64 × 6 × 12 = 384 sq. cm. Out of this, the total surface area of the cubes that have painted using white colour= 6 × 42 = 96 sq. cm. Total surface area of the 27 small cubes that have been painted using blue colour=384-96= 288 sq. cm

Therefore, the required answer is 288.


70. C.283

5877

71. C.60√3m Let PQ be the lighthouse and A & B be the two positions for the ship. Because the angle of elevation remains the same along the path AB, AB is apart of a circle with centre Q and radius AQ = BQ.


72. A.7/10

73. B.4/15


74. C.64pi/3


75. A.2

76. D.6.8


77. [10]

78. A. (5pi+18)/12

79. B.18 Given that the area of the rhombus is 54 cm2. Let‘x’ and ‘y’ be the lengths of the diagonals of the rhombus. Therefore, xy = 2(54) = 108 cm2 Let the side of the square be ‘a’. Therefore, a2 = 3(108) = 324 Therefore, a = 18 cm

80. D..7√5:22 On the sphere to travel a diagonally opposite point, the insect will travel half the circumference of the sphere, i.e. piR, where R is the radius of the sphere. This is a


constant.

81. D.4pi

82. B.9.2 Let R be the larger radius and r be the smaller radius. R+r=12 and Rr=32. Solving the equations we get R=8 and r=4. Now, let the height of the vessel be h, Volume =pih(R2+r2+Rr )/3=112pih/3. The volume of water=1/3 volume of bucket and the height


of water level is 2/3 of the height of the bucket.

83. C.1/2


84. C.980.39


85. B.30,161.5

86. A.pi units

87. C.2x2=y2 In the pyramid the distance of the base of the slant edge to the centre of the base is equal to the height of the pyramid i.e. x. By Pythagoras theorem we get y=√2.


88. D.1.716S


89. B.3


90. A.h>r>p

91. D.. 0.45



92. C.0.636


93. A. 2r(1+ 3)


94. D.11:5


95. C.150

96. D. none of these


97. D.Between 0 and 60

98. A.45


99. D.26.25

100. B..|x|+|y|=3

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100 Must Solve Geometry Questions · A triangle of area 80 sq. cms has two of its sides as 12 cms...

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