geometry and mensuration

1. In the following figure AB||DF, if ABC = 63 and CDE = 37,

then find BCD:a) 110 b) 90 c) 100 d) 95

2. In the following figure, AB||CD||EF||GH||IJ, if AI is 100cm, AC = 2CE = 3EG = 4GI, find the value of

a) 40cm b) 36 cm c) 44cm d) 50cm

3. Find how many points strictly lie inside the region made by the line x+y = 21, and the co-ordinate axis:a) 210 b) 231 c) 190 d) 189

4. A circle has 8 lines, not more than 2 are concurrent, what is the maximum regions that can be formed inside the circle?a) 9 b) 20 c) 36 d) 37

5. If (10, 15, x) are length of sides of a triangle, what will be the range pf the perimeter of the same? (integral values)a) 31 to 49 b) 30 to 48 c) 32 to 50 d) 33 to 47

6. How many scalene triangles with perimeter 7cm will have integral side lengths?a) 2 b) 3 c) 4 d) 5

7. How many isosceles triangles can be formed with one of their side lengths 6cm?a) 20 b) 11 c) 30 d) infinitively many

8. In the above problem, how many triangles can be formed with the equal sides being 6cm?a) 10 b) 11 c) 12 d) 9

9. If (7,24, x) are sides of an obtuse angled triangle, how many values can x assume?a) 11 b) 12 c) 10 d) 9

10. ABC is a right-angled triangle, right angled at B, BD AC, AB : BC = 3:4, then calculate BD : BCa) 5 : 12 b) 3:4 c) 3:5 d) Cannot be determined

For further details CALL/SMS on 9920935456 or email to [email protected]

11. In the above problem, find the ratio of AD : DC a) 3 : 4 b) 4:3 c) 9 : 16 d) Cannot be determined

12. In the following figure, ABC is a right angled triangle, AD = AE and CE = CF as shown. What is the measure of DEF?a) 35 b) 40 c) 45 d) 50

13. ABC is right angled at B, AC = 20cm, if AD and CE are the medians drawn from A and C respectively, find AD2 + CE2

a) 400 b) 500 c) 300 d) Cannot be determined14. Which of the following may ir may not be inside the triangle?

a) In-centre b) Orthocentre c) Circumcentre d) Centriod15. If x, y and z are length of sides of a triangle satisfying x2 + y2 + z2 = xy + yz + xz, then the triangle must

bea) Isosceles b) Equilateral c) Right-angled d) Scalene

16. In triangle ABC, AB = 3cm, BC = 4cm, ABC = 120, find length of BC :a) √30 b) √34 c) √37 d) √40

17. In triangle ABC, A = 60. AD, BE and CF are the angle bisectors of the interior angles A, B and C respectively which meet at O , BF and CF are the bisectors of the exterior angles B and C respectively, find the ratio of BOC and BFC :a) 2 : 1 b) 1 : 2 c) 2 : 3 d) 3 : 2

18. In triangle ABC, AD, BE and CF are the perpendicular bisectors of BC, AC and AB respectively which are concurrent at P, find the ratio of BAC and BPC:a) 1 : 2 b) 2 : 3 c) 2 : 1 d) 3 : 2

19. In triangle ABC, DE||FG||BC, AD = DF = FB, find AE : AC :a) 1 : 2 b) 1 : 3 c) 2 : 3 d) None of these

20. In the above figure, find the ratio of areas of triangles ADE, AFG and ABCa) 1 : 4 : 9 b) 1 : 3 : 6 c) 1 : 3 : 5 d) None of these

21. Points D, E and F divide the sides of triangle ABC (AB, BC and AC respectively) in the ratio of 3: 2, 3: 1 and 2 : 3 respectively. Find the ratio of areas of triangles ABC and DEF.a) 121 : 100 b) 100 : 81 c) 100 : 69 d) 100 : 84

22. ABC is an equilateral triangle with side 8cm, find the sum of squares of the medians of ABC :


a) 40 b) 52 c) 48 d) 6023. In the figure below, ABD = CDB = PQD = 90. AB : CD = 3 : 1, then find CD : PQ (CAT 2003)

a) 1 : 0.69 b) 1:0.5 c) 1 : 0.75 d) 1 : 0.66

24. In ABC, D is a point on AB such that, BD = 9cm, BC = 12cm and CD = 6cm. If BCD = BAC, find the ratio of perimeters of triangles ADC to BDC? (CAT-05)a) 7 : 9 b) 8 : 9 c) 6 : 9 d) 5 : 9

25. Triangle ABC is right angled at B, AB = 10cm, AC = 26cm, AD bisects A, find the length of BD?a) 10 b) 7 c) 20/3 d) 26/3

26. In triangle ABC, AB = BC, D is a point on AC such that AB = BD. If DCB = 25, find BAC?a) 55 b) 60 c) 65 d) 70

27. In triangle ABC, AD is the median. BE bisects AD and meets AC at point F. If AC = 30cm, find AF?a) 20cm b) 15cm c) 10cm d) 12cm

28. In triangle ABC, the lengths of sides of AB, BC and AC are 12cm, 18cm and 20cm respectively. D is a point on AC such that BD = AB. Find AD : DCa) 1 : 1 b) 2 : 3 c) 10 : 9 d) 11 : 9

29. Two runners P and Q start from A, P along AB in the anticlockwise direction and Q along AC in the clockwise direction. They meet at a point D. What is the ratio of the speeds of P and Q in that order?AB = 10 cm, AC = 15 cm and DE = DF as shown in the figure

a)2 : 3 b) 3 : 4 c) 3 : 2 d) 1 : 2

30. A trapezium ABCD is such that AB || CD. Diagonals AC and CD meet at point O. If AO : OC is 1 : 5 and DO = 15cm, find OB :a) 5cm b) 6cm c) 10cm d) 7cm


31. In the above question find the ratio of areas of triangle AOB and COD :a) 1 : 16 b) 9 : 25 d) 7 : 16 d) 1: 25

32. ABCD is an isosceles trapezoid with AB = 10cm, CD = 6cm and the distance between these parallel lines is 8cm. Find the perimeter of ABCD :a) 16 + 2√11 b) 16 + 8√15 c) 16 + 4√17 d) 16 + 4√13

33. A circle is inscribed in an isosceles trapezoid with lengths of parallel sides as 75cm and 108cm, find the diameter of the circle inscribed :a) 90 b) 85 c) 100 d) 92

34. 4 equilateral triangles of side 2cm are joined together to form a parallelogram as shown in the figure, find the length of diagonal AC :

a) √14 b) √18 c) 6.5 d) 2√735. ABCD is a parallelogram with sides 15 and 20, if one diagonal is of length 16, find the other :

a) 18 b) 30 c) √997 d) None of these36. ABCD is a parallelogram, AE and AF are two cevians drawn on the side CD such that CD is trisected.

Find the ratio of areas of triangles AEF and ACB.a) 1 : 2 b) 2 : 3 c) 1: 3 d) 1 : 4

37. If in a parallelogram PQRS, the angular bisector of QPS meets QR at ‘O’ and if OR = 24 cm and OS = OP = 9 cm then what is the length of PQ?a) 3 b) 4 c) 5 d) 6

38. In a rectangle PQRS, X is a point inside the rectangle, then which of the following is definitely true?a) XP + XR = XQ + XSb) XP2 + XQ2 = XR2 + XS2

c) XP2 + XR2 = XQ2 + XS2

d) XP + XS = XR + XQ39. P is a point inside the rectangle ABCD such that AP = 4, BP = 3, PD = 5, Find PC

a) 3 b) 4√3 c) 5 d) 3√2 40. ABCD is a rectangle, F and E are points on AB and CD such that AFCE is a rhombus, AB = 16 units

and BC = 12 units. Calculate EF :a) √175 b) 13 c) √193 d) 14

41. ABCD and ABE is an equilateral triangle. Find DECa) 15 b) 30 c) 45 d) 60


42. An isosceles right angled triangle is inscribed in a square such that its hypotenuse becomes the mid-segment of the square. Find the ratio of area of the triangle to the square:a) 1: 3 b) 1 : 3√2 c) 1 : 4 d) 1 : 2

43. PQRS is a cyclic quadrilateral in which the angular bisector of P meets the angular bisectors of S and Q at A and B respectively and the angular bisector of R meet the angular bisectors of S and Q at C and D respectively. Then the quadrilateral ABDC is aa) Square b) Rectangle c) Cyclic Quadrilateral d) Rhombus

44. ABCD is a rhombus whose side is 5 cm in which diagonals AC and BD are in the ratio 4 : 3. If E is a point on AC such that AE : AC = 3 : 4 then what is the length BE?a) √13 b) 85/3 c) √13/3 d) 13

45. The quadrilateral ABCD in the following figure is formed by the angular bisectors of a parallelogram PQRS where PQ = 4√3 and QR = 4 and QPS = 600. Then what is the ratio AB: BC?

a) 1 : √3 b) √3 : 1 c) 4 : √3 d) √3 : 4

46. There aretwo circles (I and II) with centers P and Q respectively. The two circles touch each other and have a 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. (CAT 2004)Find the ratio of length PQ to QOa) 1 : 4 b) 1 : 3 c) 3 : 4 d) 3 : 8

47. In the above question what will be the radius of circle II?a) 2cm b) 3cm c) 4cm d) 5cm

48. Length of SO isa) 8√3 b) 10√3 c) 12√3 d) 14√3

49. What is the distance in cm between two parallel chords of length 32 cm and 24 cm in a circle of radius 20 cm? (CAT 2005)a) 1 or 7 b) 2 or 14 c) 3 or 21 d) 4 or 28

50. chord ED is parallel to the diameter AC of the circle. If CBE = 65°, then what is the value of angle DEC? (CAT 2004)a) 35 b) 55 c) 45 d) 25

51. In the figure given below (not drawn to scale), A, B and C are three points on a circle with centre O. The chord BA is extended to a point T such that CT becomes a tangent to the circle at point C. If ∠ATC = 30° and ∠ACT = 50°, then the angle ∠BOA is : (CAT 2003)


a) 100 b) 150 c) 80 d) Cannot be determined

52. AB,AC, PQ are tangents of a circle as shown in the figure and AB = 5 cm, then find the perimeter of APQ

a) 5 b) 10 c) 15 d) 20

53. In the figure ABCD is a Rhombus. D is the centre of the circle what is the angle subtended by chord AC on circumference of the circle i.e., APC

a) 45 b) 50 c) 60 d) 75

54. Two gears are circular and kept tangential, if their centers are fixed and their radii are 30cm and 40cm, how many revolutions will the larger one make if the smaller one makes 4 revolutionsa) 3 b) 4 c) 5 d) None of these


55. Two circles are externally tangent with a common external tangent. If theradii of the circles are 9 and 16, what is the distance between points oftangency?a) 20 b) 22 c) 25 d) 24

56. Three circles are in a row touching each other such that all three of them have two common tangents. The radii of the largest and the smallest circle are 9 and 4 respectively. Line segment AB passes through the centers of the circles and lies on the two outer circles What is the length of AB?a) 19 b) 38 c) 26 + 3√32 d) None of these

57. Two circles touch each other externally and also touch a bigger circle of diameter 10 cm internally, as shown in the figure. A triangle is formed by joining the centers of the three circles. The perimeter of the triangle, in cm, is a) 5 b) 10 c) 15 d) 20

58. A regular polygon with n sides has interior angles measuring 178. What is the value of n?a) 120 b) 180 c) 150 d) 144

59. What is the value of θ in the following regular nonagon ABCDEFGHI?

a) 10b) 20 c) 30 d) 40

60. Consider a regular polygon of p sides. The number of values of p for which the polygon will have angles whose values in degrees can be expressed in integers?a) 24 b) 23 c) 22 d) 21


MENSURATION

1. Find the area of the region bounded by |x| + |y| = 10.a) 100 b) 150 c) 150√2 d) 200

2. A right triangle with legs measuring 12 cm and 16 cm is inscribed in a circle. What is the circumference of the circle in centimeters?a) 14π b) 25π c) 20π d) None of these

3. A circle is inscribed inside a square. The square is inscribed insideanother circle. If the area of the small circle is 4π, what is the area of the large circle, in square centimeters?a) 4√2π b) 8π c) 6π d) 8√2π

4. A circle is inscribed in a triangle with sides measuring 4 cm, 6 cm, and 8 cm.What is the area of the circle in square centimeters?a) 7π/6 b) 3π/2 c) 5π/3 d) 7π/4

5. An equilateral triangle E1 has area 81√3 sq. cm. A second triangle, E2, is drawn with vertices on the midpoints of the sides of E1. The midpoints of the sides of E2 are the vertices of triangle E3, and so on. What is the sum of the perimeters, in centimeters, of all the triangles, E1, E2, E3… ∞?a) 100 b) 50√2 c) 108 d) 110

6. You have 6 bars of lengths 10, 20, 30, 40, 50 and 60 cm. The number of non-congruent triangles that can be formed by choosing three of the sticks to make the sides isa) 6 b) 7 c) 8 d) 9

7. The area of ∆ABC is 80 square units. If BD = 15 units and DC = 10 units, what is the area of ∆ABD?a) 48 b) 32 c) 40 d) 50

8. If the base of a triangle is increased by 20% and the height decreased by 20%, what is the change in the area?a) No change b) 10% fall c) 4% fall d) 4% rise

9. In the following figure, CD, DE, EF and FG are medians drawn to triangles ABC, ADC, ADE and AFE respectively. Find the ratio of areas of triangles AFG to ABC

a) 1 : 8 b) 1 : 15 c) 1: 16 d) 2 : 15

10. In triangle ABC, sides AB, AC, and BC are extended till Q, P and R such that AC = AP, BC = CR, and AB = BQ, as shown in the adjacent figure. It is known that the area of triangle ABC is 30 cm2. Find the area of triangle PQR.


a) 200cm2 b) 180cm2 c) 210cm2 d) 225cm2

11. ABCD is a square with side length 10. A circle is drawn through Aand D so that it is tangent to BC.a) 5cm b) 6cm c) 5.5cm d) 6.25cm

12. Five lines parallel to the base of a triangle divide each of the other sides into five equal segments and the area into five distinct parts. If the area of the largest of these parts is 33, then what is the area of the original triangle?a) 100 b) 75 c) 121 d) 108

13. Square ABCD is inscribed inside a circle. Another square is inscribed between square ABCD and the circle such that its two vertices are on the circle and one side lies along AB, find the ratio of lengths of smaller square to the larger one:a) 1 : 5 b) 2 : 5 c) 3: 7 d) 4 : 11

14. Four points A, B, C, and D lie on a straight line in the X-Y plane, such that AB = BC = CD, and the length of AB is 1 metre. An ant at A wants to reach a sugar particle at D. But there are insect repellents kept at points B and C. The ant would not go within one metre of any insect repellent. The minimum distance in metres the ant must traverse to reach the sugar particlea) π b) 2π c) π + 1 d) π + 2

15. A cow is tied with a 50 m rope to a corner of a 20 m by 30 m rectangular field. The field is completely fenced and the cow can graze on the outside only. What area of the land can the cow graze?a) 2000π b) 2200π c) 2500π d) 3000π

16. Rectangular tiles each of size 80 cm by 25 cm must be laid horizontally on a rectangular floor of size 120 cm by 140 cm, such that the tiles do not overlap. A tile can be placed in any orientation so long as its edges are parallel to the edges of the floor. No tile should overshoot any edge of the floor. The maximum number of tiles that can be accommodated on the floor isa) 8 b) 9 c) 7 d) 6

17. A punching machine is used to punch a circular hole of diameter two units from a square sheet of aluminum of width 2 units, as shown below. The hole is punched such that the circular hole touches one corner P of the square sheet and the diameter of the hole originating at P is in line with a diagonal of the square.


a) π/4 b) (π-1)/2 c) (π – 1)/4 d) (π – 2)/4

18. In the following image OB = 7 cm. Find the area of shaded region

a) 77 b) 88 c) 2π d) 154

19. A boy wishes to find the depth of a lake. He finds out that there is a lily whose stem is straight and stretches to the bottom of the lake. He measures that the lily is 1meter above the surface of lake. The Boy gradually bends the lily till its top completely submerges in water. He finds that it takes a distance of 4 meter form its initial position to submerge the lily find the depth of lake :a) 14 b) 15 c) 7.5 d) 8.5

20. Three horses are grazing within a semi-circular field. In the diagram given below, AB is the diameter of the semi-circular field with centre at O. Horses are tied up at P, R and S such that PO and RO are the radii of semi-circles with centres at P and R respectively, and S is the centre of the circle touching the two semi-circles with diameters AO and OB. The horses tied at P and R can graze within the respective semi-circles and the horse tied at S can graze within the circle centred at S. The percentage of the area of the semi-circle with diameter AB that cannot be grazed by the horses is nearest to

a) 20 b) 30 c) 28 d) 40


21. A man has rectangle field of land which is used to graze his cattle and sheep. He confines the cattle to graze a triangular portion of the field formed by ΔCEF. F is midpoint of BE. The Rest of the portion of the field will be grazed by sheep. The grass per unit area in ΔCEF is 4 times the grass grazed per unit area of the field. Find the ratio of amount of grass grazed in ΔCEF to that of rest of the field.

a) 1 : 2 b) 4 : 7 c) 4 : 1 c) 3 : 7

22. In a square three circles are formed along the diagonal as shown in the figure given below. The radius of the middle circle is double as the radius of the other two. Find the length of a side of square AB. It is given that the radius of the small circle is a.

a) 12√2a b) 4√2a c) 14√2a d) None of these23. In the figure below (not drawn to scale), rectangle ABCD is inscribed in the circle with center at O. The

length of side AB is greater than that of side BC. The ratio of the area of the circle to the area of the

rectangle ABCD is π : √3. The line segment DE intersects AB at E such that ∠ODC = ∠ADE. What is the ratio AE : AD?

a) 2 : √2 b) 1: √3 c) 1 : 2 d) None of these

24. 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 bea) 6√3/n b) 9√3/n c) 9/n d) √11/n


25. There are two concentric circles with radii 2√3 cm and 4√3 cm. Two tangents (not parallel to each other) are drawn to the smaller circle which cut the bigger circle at A and B and touch the small circle at C, AB passes through the center O. Then find the perimeter of trapezium ABCD.

a) 10 b) 10√3 c) 12 + 10√3 d) 4 + 8√3

26. Find the area of cyclic quadrilateral, with lengths of sides 8,9,10 and 11.a) 12√55 b) √7900 c) 12√44 d) 10√65

27. ABCDEF is a regular hexagon and ∠AOF = 90 , FO is parallel to ED. AO FO, What is the ratio of the area of the triangle AOF to that of the hexagon ABCDEF?a) 1/6 b) 1/12 c) 1/15 d) 1/20

28. The length, breadth and height of a cuboid are in the ratio 1 : 2 : 3, if the volume is 7986cm3, calculate the surface area :a) 1331 b) 2662 c) 1210 d) 3993

29. Three cubes of volumes 216cm3, 512cm3 and 1000cm3 are melted and recasted to form a new cube. Find the diagonal of the cube?a) 12√2 b) 12√3 c) 10 d) 10√2

30. If a square cardboard of side 6 cm is folded to make a cubic box, then what is the volume of the box?a) 8 b) 10 c) 12 d) 16

31. If the radius of a cylinder is increased by 20%, by what percentage should the height be decreased to keep the volume constant?a) 20% b) 30% c) 30 5/9 % d) 33 1/3 %

32. Let A and B be two solid spheres such that the surface area of B is 300% higher than the surface area of A. The volume of A is found to be k% lower than the volume of B. The value of k must be :a) 85.5b) 87.5 c) 90 d) 92.5

33. A sphere is carved out of a cone with height 15cm and radius of base circle 12cm. What is the maximum volume of the sphere?a) 800 b) 750 c) 805 d) 900


34. The volume of the solid generated by the revolution of an isosceles right angled triangle about its hypotenuse of length 3x is:a) 3πx2/2 b) 9πx2/4 c) 7πx2/3 d) None of these

35. A solid metallic cylinder of base radius 3 cm and height 5 cm is melted to make ‘n’ solid cones of height 1 cm and base radius 1 mm. Find the value of ‘n’a) 150 b) 135 c) 100 d) 120

36. A cylinder with height and radius in a ratio of 2: 1 is full of soft drink. It is tilted so as to allow the soft drink to flow off till the point where the level of soft drink just touches the lowest point of the upper mouth and the highest point of the base. If 2.1l is retained, what is the capacity of the cylinder?a) 3.6 b) 4.2 c) 6.3 d) 4

37. One day Sumit planned to make lemon tea and used a portion of thespherical lemon as shown in the figure. Find out the volume of the remaining lemon. Radius of the lemon is 10cm, central angle of the cut is 60.

a) 10000π/3 b) 10000π/9 c) 1000π d) 729π

38. A cone is cut by two planes parallel to the base such that the height is trisected, find the ratio of volumes of the three solids formed :a) 1 : 8 : 27 b) 1 : 7 : 17 c) 1 : 7 : 19 d) 2 : 9 : 21

39. There are two closed hollow cylinders with radii 2 cm and 2.5 cm respectively and of same height 3.5 cm. The two cylinders, taken together, can hold 170 gm of sand. If the two cylinders are melt and a hollow sphere is formed, then approximately how much sand can be filled in the sphere?a) 300gm b) 250gm c) 270gm d) 296gm


geometry and mensuration

Documents

points d

triangles abc

sides of triangle abc

exterior angles b

medians of abc

rightangled triangle

equilateral triangle

right angled triangle