CDS 2010 Maths Question Paper -2

1. Two sides of an acute-angled triangle are 6 cm and 2 cm respectively. Which one of the following represents the correct range of the third side in cm?
   • (4,8)
   • (4, 2√10 )
   • (4 √2,8)
   • (4√2 , 2√10 )
2. What is the last digit in the expansion of 3⁴⁷⁹⁸ ?
   • 1
   • 3
   • 7
   • 9

3. What is equal to?
   • 0
   • 1
   • 5
   • 1/3

4. What are the factors of x² + 4y² + 4y-4xy-2x-8 ?

• (x-2y-4) and (x-2y + 2)
• (x-y + 2) and (x-4y + 4)
• (x-y+2) and (x-4y-4)
• (x + 2y-4) and (x + 2y + 2)

5. What is the HCF of 4x³ + 3x²y-9xy² + 2y³ and x²+xy-2y²?

• x-2y
• x-y
• (x + 2y)(x – y)
• (x-2y)(x-y)

6. What is the sum of the digits of the least number which, when divided by 52, leaves 33 as remainder; when divided by 78, leaves 59 and when divided by 117, leaves 98 as remainder?

• 17
• 18
• 19
• 21

7. If f(x) and g(x) are two polynomials with integral coefficients which vanish at x = 1 / 2, then what is the factor of HCF of f(x) and g(x) ?

• x-1
• x- 2
• 2x-1
• 2x +1

8. If the expression (px³ + x²-2x-q) is divisible by (x – 1) and (x + 1), what are the values of p and q respectively?

• 2, -1
• -2,1
• -2,-1
• 2,1

9. The ratio between the ages of A and B is 2 : 5. After 8 years, their ages will be in the ratio 1:2. What is the difference between their present ages?
   • 20 years
   • 22 years
   • 24 years
   • 25 years

10. A mixture contains milk and water in the ratio 5:1. On Adding 5 litres of water, the ratio of milk and water becomes 5:2. What is the quantity of milk in the original mixture?

• 5 litres
• 25 litres
• 5 litres
• 5 litres

11. What is the smallest positive integer which when divided by 4, 5, 8, 9 leaves remainders 3, 4, 7,8 respectively?
    • 119
    • 319
    • 359
    • 719

12. What is the value of x for which x, x + 1, x+3 are all’ prime numbers?

• 0
• 1
• 2
• 101

13. A person spends 30% of monthly salary on rent, 25% on food, 20% on children’s education and 12% on electricity, and the balance of Rs 1,040 on the remaining items. What is the monthly salary of the person?
    • Rs 8,000
    • Rs 9,000
    • Rs 9,600
    • Rs 10,600
14. If p^x=r^y=m and r^w=p^z=n, then which one of the following is correct?
    • xw = yz
    • xz = yw
    • x + y = w+z
    • x – y = w-z
15. A father and his son start at a point A with speeds of 12 kmph and 18 kmph respectively, and reach another point B. If his son starts 60 minutes after his father at A and reaches B 60 minutes before his father, what is the distance between A and B?
    • 90 km
    • 72 km
    • 36 km
    • None of the above

16. If the equation (a² +b²)x² -2(ac+bd)x + (c² +d²) = 0 has equal roots, then which one of the following is correct?
    • ab = cd
    • ad = bc
    • a² + c² =b² +d²
    • ac = bd

17. Two trains of lengths 100 m and 150 m are traveling in opposite directions! at speeds of 75 kmph and 50 kmph respectively. What is the time taken by them to cross each other?
    • 4 s
    • 2 s
    • 7 s
    • 8 s
18. If (a – b): (a +b) = 1:5, then what is (a² -b²): (a² + b²) equal to?
    • 2:3
    • 3:2
    • 5 : 13
    • 13 : 5
19. If x varies as, the mth power of y, y varies as the nth power of z and x varies as the pth power of z, then which one of the following is correct?

• P=m+n
• P=m-n
• P=mn
• None of the above

20. If x = (b-c)(a-d), y = (c-a)(b-d), z =(a-b)(c-d), then what is x³ + y³ +z³ equal to?
    • xyz
    • 2 xyz
    • 3 xyz
    • -3 xyz

21. If the 14th term of an arithmetic series is 6 and 6th term is 14, then what is the 95th term?
    • -75
    • 75
    • 80
    • -80

22. What is the solution of the equation
    • 1
    • 2
    • 4
    • None of the above

23. If n is a positive integer, then what is the digit in the unit place of 3²ⁿ⁺¹ +2²ⁿ⁺¹ p
    • 0
    • 3
    • 5
    • 7
24. If k is any even positive integer, then (k² +2k) is ,

• divisible by 24
• divisible by 8 but may not be divisible by 24
• divisible by 4 but may not be divisible by 8
• divisible by 2 but may not be divisible by 4

25. If salary of \X is 20% more than salary of Y, then by how much percentage is salary of Y less than X?
    • 25
    • 20
    • 50/3
    • 65/4
26. What are the roots of the equation 4^x-3.2^x+2 + 32 = 0?

• 1, 2
• 3, 4
• 2, 3
• 1, 3

27. If α and β  are the roots of the equation x² – x-1= 0, then what is the value of ( α⁴ β⁴)?

• 7
• 0
• 2
• None of the above

28. Under what condition do the equations kx-y = 2 and 6x-2y =3 have a unique solution?

• k = 3
• k ≠ 3
• k = 0
• k ≠ 0

29. If sum as well as product of roots of a quadratic equation is 9, then what is the equation?

• x² + 9x -18 = 0
• x² – 18x + 9 = 0
• x² + 9x + 9 = 0
• x² – 9x + 9 = 0

30. Which one of the following is a non-terminating and repeating decimal?

• 13/8
• 3/16
• 3/11
• 137/25

31. If a + b + c = 0, then what is the value , a²/bc+ b²/ca+ c²/ab?
    • – 3
    • 0
    • 1
    • 3
32. If 1 is added to the denominator of a fraction, it becomes 1/2 and if 1is added to the numerator, the fraction becomes 1. What is the fraction?

• 5/9
• 2/3
• 4/7
• 10/11

33. A motorboat takes 2 hours to travel a distance of 9 km down the current and it takes 6 hours to travel the same distance against the current. What is the speed of the boat in still water in kmph?

• 3
• 2
• 5
• 1

34. A can do a piece of work in 24 days. If B is 60% more efficient than A, then how many days does B require to do the same work?

• 12
• 15
• 16
• 18

35. A sum of money lent on simple interest trebles itself in 15 years and 6 months. In how many years will it be doubled?
    • 6 years and 3 months
    • 7 years and 9 months
    • 8 years and 3 months
    • 9 years and 6 months
36. On selling an article for Rs 240, a trader loses 4%. In order to gain 10%, he must sell the article for

• Rs 275
• Rs 280
• Rs 285
• Rs 300

37. In a class, the number of boys is more than the number of girls by 12% of the total strength. What is the ratio of number of boys to that of girls?

• 11 : 14
• 14 : 11
• 28 : 25
• 25 : 28

38. Three numbers are in the ratio 3:2:5 and the sum of their squares is 1862. What are the three numbers?

• 18, 12, 30
• 24, 16, 40
• 15, 10, 25
• 21, 14, 35

39. If one root of the equation ax² +x-3 = 0 is -1, then what is the other root?
    • ¼
    • ½
    • ¾
    • 1
40. What is the value of ½ log₁₀ 25 -21og₁₀ 3 + log₁₀ 18 ?
    • 2
    • 3
    • 1
    • 0
41. The product of a rational number and an irrational number is
    • a natural number
    • an irrational number
    • a composite number
    • a rational number

42. The angle of elevation of the top of a Two poles of heights tower from the bottom of a building is twice that from its top. What is the height of the building if the height of the tower is 75 m and the angle of elevation of the top of the tower from the bottom of the building is 60?

• 25 m
• 5 m
• 50 m
• 60 m

43. If cos θ ≥ 1/2 in the first quadrant, then which one of the following is correct?

44. What is the value of cos 1°cos 2° cos 3°……. cos 90°?

• 1/2
• 0
• 1
• 2

45. Two poles of height 6m and 11m stand vertically upright on a plane ground. If the distance between their feet is 12 m, what is the distance between their tops?
    • 11 m
    • 12 m
    • 13 m
    • 14 m

46. If sin θ+cos θ= 1, then what is the value of sinθ  cos θ?

• 2
• 0
• 1
• ½

47. What is equal to?
    • secθ -tanθ
    • secθ +tanθ
    • cosecθ +cotθ
    • cosecθ -cotθ
48. The shadow of a tower is 15 m when the Sun’s altitude is 30°. What is the length of the shadow when the Sun’s altitude is 60° ?
    • 3 m
    • 4 m
    • 5 m
    • 6 m
49. The arithmetic mean of 100 numbers was computed as 89.05. It was later found that two numbers 92 and 83 have been misread as 192 and 33 respectively. What is the correct arithmetic mean of the numbers?
    • 55
    • 55
    • 55
    • Cannot be calculated from the given data
50. In a triangle, if sum of two angles is equal to the third angle (considering the interior angles only), then the triangle is

• right-angled
• acute-angled
• equilateral
• obtuse-angled

51. The side BC of the triangle ABC is extended to D. If ∠ACD = 120°, ∠ABC = 2/3 ∠CAB, then what is ∠BAC ?

• 60°
• 45°
• 30°
• 72°

52. Let PAB be a secant to a circle intersecting at points A and B, and PC is a tangent. Which one of the following is correct?
    • The area of rectangle with PA, PB as sides is equal to the area of square with PC as side
    • The area of rectangle with PA, PC as sides is equal to the area of square with PB as side
    • The area of rectangle with PC, PB as sides is equal to the area of square with PA as side
    • The perimeter of rectangle with PA, PB as sides is equal to the perimeter of square with PC as side

53. The line segments AB and CD intersect at O. OF is the internal bisector of obtuse angle BOC and OE is the internal bisector of acute angle AOC. If ∠BOC = 130°, what is the measure of ∠FOE?
    • 90°
    • 110°
    • 115°
    • 120°
54. In the figure given above, if ∠BAD = 60°, ∠ADC= 105°, then what is ∠DPC equal to?
    
    • 40°
    • 45°
    • 50°
    • 60°
55. In the figure given above, ABCD is a quadrilateral with AB parallel to DC and AD parallel to BC. ADC is a right angle. If the perimeter of the triangle ABE is 6 units, what is the area of the quadrilateral?
    

• 2 √3 square units
• 4 square units
• 3 square units
• 4√3 square units

56. In the figure given above, PQ is parallel to RS. What is the angle between the lines PQ and LM ?
    
    • 175°
    • 177″
    • 179″
    • 180°
57. In the figure’ given above, L is any point on the; bisector of the acute angle ABC and the line ML is- parallel to EC. Which one of the following is correct?
    

• The triangle BML is equilateral
• The triangle BML is isosceles but right-angled
• The triangle BML is isosceles but not right-angled
• The triangle BML is not Isosceles

58. In the figure given above, PQ is a diameter of the circle whose centre is at O. If ∠ROS = 44°, then what is the value of   ∠RTS ?
    
    • 46°
    • 64°
    • 69°
    • None of the above
59. In the figure given above, O is the centre of the circle. AC and BD intersect at P. If ∠AOB= 100°, what is   ∠APB ?
    

• 77°
• 80°
• 85°
• 90°

60. In the figure given above, ABCD is a parallelogram. P is a point on BC such that PB: PC = 1:2. DP produced meets AB produced at Q. If the area of the triangle BPQ is 20 square units, what is the area of the triangle DCP?
    
    • 20 square units
    • 30 square units
    • 40 square units
    • None of the above
61. In the figure given above, AB is parallel to CD What is ∠XOY?
    
    • 80°
    • 90°
    • 95°
    • 100°
62. What is the radius of the circle inscribed in |a triangle having side lengths 35 cm, 44 cm and 75 cm?

• 3 cm
• 4 cm
• 5 cm
• 6 cm

63. What is the maximum length of a rod that can be placed inside a box having the shape of a cuboid of length 30 cm, breadth 24 cm and height 18 cm?
    • 30 cm
    • 30 √2 cm
    • 24 cm
    • 18 √5 cm
64. The volume of a sphere is 8 times that of another sphere, What is the ratio of their surface areas?

• 8:1
• 4:1
• 2:1
• 4:3

65. A rectangular area of 6 square metres is to be painted on a 3 m x 4 m board leaving a border of uniform width on all sides. What should be the width of the border?

• 25 m
• 5 m
• 1 m
• 3 m

66. A solid cylinder of height 9 m has its curved surface area equal to one-third of the total surface area. What is the radius of the base?

• 9 m
• 18 m
• 27 m
• 30 m

67. A figure is formed by revolving a rectangular sheet of dimensions 7 cm x 4 cm about its length. What is the volume of the figure thus formed?
    • 352 cm³
    • 296 cm³
    • 176 cm³
    • 616 cm³
68. The diagonals of the three faces of a cuboid are x, y, z respectively. What is the volume of the cuboid?
    
    • None of the, above

69. Half of a large cylindrical tank open at the top is filled with water and identical heavy spherical balls are to be dropped into the tank without spilling water out:. If the radius and the height of the tank are equal and each is four times the radius of a ball, what is the maximum number of balls that can be dropped?

• 12
• 24
• 36
• 48

70. The diameters of two circles are 18 cm and 8 cm The distance between their centres is 13 cm. What is the number of common tangents?
    • 1
    • 2
    • 3
    • None of the above

71. If the height of a cone is increased by 50%, then what is the percentage increase in the volume of the cone?
    • 100/3
    • 40
    • 50
    • 200/3
72. If three cubic biscuits having edges 0-3 m, 0-4 m and 0-5 m respectively are melted and formed into a single cubic biscuit, then what is the total surface area of the cubic biscuit?
    • 08 m²
    • 56 m²
    • 84 m²
    • 16 m²
73. A wheel of a bicycle has inner diameter 50 cm and thickness 10 cm. What is the speed of the bicycle if it makes 10 revolutions in 5 seconds?

• 5 m/s
• 4 m/s
• 3 m/s
• 2 m/s

74. A wire of length 36 cm is bent in the form of a semicircle. What is the radius of the semicircle?

• 9 cm
• 8 cm
• 7 cm
• 6 cm

75. In order to fix an electric pole along a roadside, a pit with dimensions 50 cm x 50 cm is dug with the help of a spade. The pit is prepared by removing earth by 250 strokes of spade. If one stroke of spade removes 500 cm3 of earth, then what is the depth of the pit?

• 2 m
  1. m
• 75 m
• 5 m

76. In the figure given above, what is the sum of the angles formed around A, B, C except the angles of the triangle ABC?
    
    • 360°
    • 720°
    • 900°
    • 1000°
77. In the figure given above, A and B are end points of diameter of a circle with centre at P, and C is a point on the circumference of the circle such that ∠ABC = 35°, then what is ∠PCA?
    

• 25°
• 30°
• 35°
• 55°

78. In the figure given above, ABC is an equilateral triangle of side length 30 cm. XY is parallel to BC, XP is parallel to AC and YQ is parallel to AB. If (XY + XP + YQ) is 40 cm, then what is PQ equal to?
    

• 5 cm
• 12 cm
• 15 cm
• None of the above

79. In the figure given above, AS is a line of length 2a, with M as mid- point. Semicircles are drawn on one side with AM, MB and AB as diameters. A circle with centre O and radius r is drawn such that this circle touches all the three semicircles. What is the value of r ?
    

• 2a/3
• a/2
• a/3
• a/ 4

80. The mean weight of 150 students in a certain class is 60 kg. The mean weight of the boys from the class is 70 kg, while that of girls is 55 kg. What is the number of girls in the class?

• 105
• 100
• 95
• 60

81. In an examination, 40% of the candidates wrote their answers in Hindi and the others in English. The average marks of the candidates written in Hindi is 74 and the average marks of the candidates written in English is 77. What is the average marks of all the candidates?
    • 5
    • 8
    • 0
    • 8
82. Consider the following frequency distribution :

Class	0-10	0-20	0-30	0-40	0-50
Frequency	3	8 83.      1	14	20	25

What is \ the above frequency distribution known as?

• Cumulative distribution in more than type
• Cumulative distribution in less than type
• Continuous frequency distribution
• None of the above

83. Consider the following statements in respect of histogram :
84. Histogram is an equivalent graphical representation of the frequency distribution.
85. Histogram is suitable for continuous random variables, where the total frequency of an interval is evenly distributed over the interval.

Which of the statements given above is/are correct?

• 1only
• 2 only
• Both 1 and 2
• Neither 1 nor 2

84. What is the median of the values 11, 7, 6, 9, 12, 15, 19 ?

• 9
• 11
• 12
• 15

85. Let (x₁, y₁), (x₂, y₂),…, (x_n, y_n) are n pairs of positive numbers. The arithmetic mean and geometric mean of any set of positive numbers (c₁,c₂,….., c_n) are denoted by M(c_i),G(c_i) respectively.

Consider the following :

1. M(x_i+y_i) = M(x_i)+M(y_i)
2. G(x_i y_i) = G(x_i)G(y_i)

Which of the above is/are correct?

• 1only
• 2 only
• Both 1 and 2
• Neither 1 nor 2

86. A circle and a square have the same perimeter. Which one of the following is correct?

• The area of the circle is equal to that of square
• The area of the circle is larger than that of square
• The area of the circle is less than that of square
• No conclusion can be drawn

87. If (x⁴ + x^-4) = 322, what is one of the values of (x-x^-1 )?

• 18
• 16
• 8
• 4

88. Consider the following statements :
89. If two triangles are equiangular, then they are similar.
90. If two triangles have equal area, then they are similar.

Which of the statements given above is/Eire correct?

• 1 only
• 2 only
• Both 1 and 2
• Neither 1 nor 2

89. If two corresponding sides of two similar triangles; are in the ratio 9 : 4, then what is the ratio of their areas?

• 9:4
• 3:2
• 81: 16
• 27 : 8

90. ABCD is a quadrilateral, the sides of which touch a circle. Which one of the following is correct?

• AB+ AD = CB+ CD
• AB: CD = AD: BC
• AB + CD = AD + BC
• AB: AD = CB: CD

91. By selling 8 dozen pencils, a shopkeeper gains the selling price of 1 dozen pencils. What is the gain?

• 12 ½ %
• 13 1/7 %
• 14 2/7 %
• 87 ½ %

92. The ratio of A to B is x: 8 and the ratio of B to C is 12: z. If the ratio of A to C is 2 : 1, then what is the ratio of x:z?

• 2:3
• 3 :2
• 4:3
• 3:4

93. If α and β are the roots of the equation x² + px + q = 0, then – α^-1, β^-1 are the roots of which one of the following equations?

• qx² – px + 1 = 0
• qx² + px + 1 = 0
• x₂ + px -q = 0
• x²-px + q = 0

94. For a set of positive numbers, consider the following statements :
95. If each number is reduced by 2, then the geometric mean of the set may not always exist.
96. If each number is increased by 2, then the geometric mean of the set is increased by 2.

Which of the above statements is/are correct? ,

• 1 only
• 2 only
• Both 1 and 2
• Neither 1 nor 2

For the next three (3) items which follow :

Let C be a right-circular cone. It is given that the two ends of a| frustum of C are of radii 3 cm and 6 cm, and the height of the frustum is 9 cm.

95. What is the slant height of the given frustum? i

• 3√10 cm
• 6√10 cm
• 12 cm
• 15 cm

96. What is the height of the cone?

• 9 cm
• 12 cm
• 5 cm
• 18 cm

97. What is the total surface area of the given frustum?
    

For the next three (3) items which follow :

Let ABCD be a quadrilateral. Let the diagonals AC and BD meet at O. Let the perpendicular drawn from A to CD meet CD at E. Further, AO: OC = BO: OD, AB = 30 cm, CD = 40 cm and the area of the quadrilateral ABCD is 1050 cm².

98. What is BE equal to?

• 30 cm
• 30√2 cm
• 30 √3cm
• None of the above
  99. What is the area of the triangle ADC equal to?
• 300 cm²
• 450 cm²
• 600 cm²
• None of the above

100. What is ∠AEB equal to?

• 30°
• 45°
• 60°
• None of the above