# CDS (II) 2014 Exam: Elementary Mathematics Question Paper & Solution

The UPSC CDS (II) 2014 Elementary Mathematics Question Paper and the Solution is given here. It will help the aspirants in devising the appropriate study plan for the upcoming CDS (II) in November 2017.  The Union Public Service Commission (UPSC) organizes Combined Defense Service (CDS) Examination. Through CDS Exam, candidates get admission to the courses at Indian Military Academy, Indian Naval Academy, Air Force Academy and Officers Training Academy.

UPSC organizes CDS Exam twice in a year. The CDS Exam (I) 2017 was conducted in February 2017. While the notification for the CDS (II) 2017 will be released by the UPSC on 9 August 2017 and the exam will take place on 19 November 2017.

The exam will consist of three papers – Elementary Mathematics, English and General Knowledge. The three papers will be of 100 marks each.

For the benefit of CDS (II) Exam 2017 aspirants, Jagran Josh is providing you the CDS Exam (II) 2014 Elementary Mathematics Question Paper as well as the official answer key as given by the UPSC. It helps the aspirants in understanding the demand as well as the approach of the exam. 1. Pipe A can fill a tank in 3 hours. But there is a leakage also, due to which it takes 3.5 hours for the tank to be filled. How much time will the leakage take in emptying the tank if the tank is filled initially?

(a) 21 hours

(b) 20 hours

(c) 18 hours

(d) 10.5 hours

2. A train takes 10 seconds to cross a pole and 20 seconds to cross a platform of length 200 m. What is the length of the train?

(a) 50 m

(b) 100 m

(c) 150 m

(d) 200 m

3. A, B and C can do a piece of work individually in 8, 12 and 15 days respectively. A and B start working together but A quits after working for 2 days. After this, C joins and works till completion of the work. In how many days will the work be completed?

4. The distance between two points (A and B) is 110 km. X starts running from point A at a speed of 60 km/hr and Y starts running from point Bat a speed of 40 km/hr at the same time. They meet at a point C, somewhere on the line AB. What is the ratio of AC to BC?

(a) 3 : 2

(b) 2 : 3

(c) 3 : 4

(d) 4 : 3

5. A is thrice as efficient as B and hence completes a work in 40 days less than the number of days taken by B. What will be the number of days taken by both of them when working together?

(a) 22.5 days

(b) 15 days

(c) 20 days

(d) 18 days

6. If 10 persons can dig 8 feet trench in 12 days, then how many days will 8 persons take to dig 6 feet trench?

(a) 10 days

(b) 10.25 days

(c) 11 days

(d) 11.25 days

7. The height of a tree varies as the square root of its age (between 5 to 17 years). When the age of the tree is 9 years, its height is 4 feet. What will be the height of the tree at the age of 16 years?

(a) 5 feet 4 inches

(b) 5 feet 5 inches

(c) 4 feet 4 inches

(d) 4 feet 5 inches

8. The ratio of ages of A and B is 2 : 5 and the ratio of ages of B and C is 3 : 4. What is the ratio of ages of A, B and C?

(a) 6: 15 : 20

(b) 8 : 5 : 3

(c) 6 : 5 : 4

(d) 2: 15 : 4

9. When an article is sold at 20% discount, the selling price is Rs. 24. What will be the selling price when the discount is 30%?

(a) Rs. 25

(b) Rs. 23

(c) Rs. 21

(d) Rs. 20

10. A shopkeeper sells his articles at their cost price but uses a faulty balance which reads 1000 g for 800 g. What is his actual profit percentage?

(a) 25%

(b) 20%

(c) 40%

(d) 30%

11. The difference between compound interest and simple interest for 2 years at the rate of 10% over principal amount of Rs. X is Rs. 10. What is the value of X?

(a) Rs. 100

(b) Rs. 1,000

(c) Rs. 500

(d) Rs. 5,000

12. A sum of money becomes 3 times in 5 years at simple interest. In how many years will the same sum become 6 times at the same rate of simple interest?

(a) 10 years

(b) 12 years

(c) 12.5 years

(d) 10.5 years

13. A man buys 200 oranges for Rs. 1,000. How many oranges for Rs. 100 can he sell so that his profit percentage is 25%?

(a) 10

(b) 14

(c) 16

(d) 20

14. If m% of m + n% of n = 2% of (m x n), then what percentage of m is n?

(a) 50%

(b) 75%

(c) 100%

(d) Cannot be determined due to insufficient data

15. If the side of a cube is increased by 100%, then by what percentage is the surface area of the cube increased?

(a) 150%

(b) 200%

(c) 300%

(d) 400%

16. How many pairs of positive integers m and it satisfy the equation

1/m + 4/n = 1/12

where it is an odd integer less than 60?

(a) 7

(b) 5

(c) 4

(d) 3

17. The sides of a triangle are in the ratio 1/2 : 1/3 : 1/4. If its perimeter is 52 cm, then what is the length of the smallest side?

(a) 9 cm

(b) 10 cm

(c) 11 cm

(d) 12 cm

18. The diameter of a metallic sphere is 6 cm. The sphere is melted and drawn into a wire of uniform circular cross-section. If the length of the wire is 36 m, then what is its radius equal to?

(a) 0.1 cm

(b) 0.01 cm

(c) 0.001 cm

(d) 1.0 cm

19. Consider all those two-digit positive integers less than SO, which when divided by 4 yield unity as remainder. What is their sum?

(a) 310

(b) 314

(c) 318

(d) 323

20. If every side of an equilateral triangle is doubled, then the area of new triangle becomes k times the area of the old one. What is k equal to?

(a) √3

(b) 2

(c) 4

(d) 8

21. If an = 3 – 4n, then what is a1 + a2 + a3 + … + an equal to?

[1 + 2 + 3 + 4 +… + n = n(n + 1)/2]

(a) -n(4n - 3)

(b) -r(2n - 1)

(c) -n2

(d) -n(2n + 1)

22. A train travels at a speed of 40 km/hr and another train at a speed of 20 m/s. What is the ratio of speed of the first train to that of the second train?

(a) 2 : 1

(b) 5 : 9

(c) 5 : 3

(d) 9 : 5

23. (x + y) : (x – y) = 3: 5 and xy = positive imply that

(a) x and y are both positive

(b) x and y are both negative

(c) one of them is positive and one of them is negative

(d) no real solutions for x and y exist

24. How many pairs of X and Y are possible in the number 763X4Y2, if the number is divisible by 9?

(a) 8

(b) 9

(c) 10

(d) 11

25. What is the remainder when 41012 is divided by 7?

(a) 1

(b) 2

(c) 3

(d) 4

26. What is the highest common factor of 2x3 + x2 – x – 2 and 3x3 – 2x2 + x – 2?

(a) x – 1

(b) x + 1

(c) 2x + 1

(d) 2x – 1

27. What is the remainder when (1235 x 4523 x 2451) is divided by 12?

(a) 1

(b) 3

(c) 5

(d) 7

28. What is the remainder when (1723 + 2323 + 2923) is divided by 23?

(a) 0

(b) 1

(c) 2

(d) 3

29. p, q and r are prime numbers such that p < q < r < 13. In how many cases would (p + q + r) also be a prime number?

(a) 1

(b) 2

(c) 3

(d) None of the above

30. The LCM of two numbers is 90 times their HCF. The sum of LCM and HCF is 1456. If one of the numbers is 160; then what is the other number?

(a) 120

(b) 136

(c) 144

(d) 184

31. The LCM of two integers is 1237. What is their HCF?

(a) 37

(b) 19

(c) 1

(d) Cannot be determined

(a) a + b

(b) a – b

(c) 1

(d) 0

33. A man rides one-third of the distance from A to B at the rate of x km/hr and the remainder at the rate of 2y km/hr. If he had travelled at a uniform rate of 6z km/hr, he could have ridden from A to B and back again in the same time. Which one of the following is correct?

(a) z = x + y

(b) 3z = x + y

(c) 1/z = 1/x + 1/y

(d) 1/2z = 1/x + 1/y

(a) a + b

(b) 2ab

(c) a3 + b3

(d) a4 + b4

35. Consider the following statements:

1. (a – b – c) is one of the factors of 3abc + b3 + c3 – a3

2. (b + c – 1) is one of the factors of 3bc + b3 + c3 – 1.

Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

36. 710 – 510 is divisible by

(a) 10

(b) 7

(c) 5

(d) 11

37. Consider the following statements in respect of four spheres A, B, C and D having respective radii 6, 8, 10 and 12 cm:

1. The surface area of sphere C is equal to the sum of surface areas of spheres A and B.

2. The volume of sphere D is equal to the sum of volumes of spheres A, B and C.

Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

38. What is the number of divisors of 360?

(a) 12

(b) 18

(c) 24

(d) None of the above

39. The multiplication of a three-digit number XY5 with digit Z yields X215. What is X + Y + Z equal to?

(a) 13

(b) 15

(c) 17

(d) 18

40. If the equation x2 + 2(1 + k)x + k2 = 0 has equal roots, then what is the value of k?

(a) 1/2

(b) – 1/2

(c) 1

(d) – 1

41. If in and n are the roots of the equation x2 + ax + b = 0, and m2 and n2 are the roots of the equation x2 - d = 0, then which of the following is/are correct?

1. 2b – a2 = c

2. b2 = d

Select the correct answer using the code given below.

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

42. If N2 – 33, N2 – 31 and N2 – 29 are prime numbers, then what is the number of possible values of N, where N is an integer?

(a) 1

(b) 2

(c) 6

(d) None of the above

43. There are 48 cricket balls, 72 hockey balls and 84 tennis balls, and they have to be arranged in several rows in such a way that every row contains the same number of balls of one type. What is the minimum number of rows required for this to happen?

(a) 12

(b) 16

(c) 17

(d) 19

44. The HCF of two natural numbers in and it is 24 and their product is 552. Flow many sets of values of Wand n are possible?

(a) 1

(b) 2

(c) 4

(d) No set of m and n is possible satisfying the given conditions

45. If in and n(m > n) are the roots of the equation

7(x +2a)2 + 3a2 = 5a(7x + 23a)

where a > 0, then what is 3m – n equal to?

(a) 12a

(b) 14a

(c) 15a

(d) 18a

46. A person selling an article for Rs. 96 finds that his loss percent is one-fourth of the amount of rupees that he paid for the article. What can be the cost price?

(a) Rs. 160 only

(b) Rs. 240 only

(c) Rs. 160 or Rs. 240

(d) Neither Rs. 160 nor Rs. 240

47. If (x + k) is the common factor of x2 + ax + b and x2 + cx + d, then what is k equal to?

(a) (d – b) / (c – a)

(b) (d – b) / (a – c)

(c) (d + b) / (c + a)

(d) (d – b) / (c + a)

48. What is the remainder when x5 – 5x2 + 125 is divided by x + 5?

(a) 0

(b) 125

(c) – 3125

(d) 3125

49. What is the lowest common multiple of ab(x2 + 1) + x(a2 + b2) and ab(x2 – 1) + x(a2 – b2)?

(a) (a2x2 – b2)(a + bx)

(b) (a2x2 – b2)(a + bx)2

(c) (a2x2 – b2)(a – bx)

(d) (a2x2 – b2)(a – bx)2

50. A certain number of two digits is three times the sum of its digits. If 45 is added to the number, the digits will be reversed. What is the sum of the squares of the two digits of the number?

(a) 41

(b) 45

(c) 53

(d) 64

51. If from the top of a post a string twice the length of the post is stretched tight to a point on the ground, then what angle will the string make with the post?

(a) π/6

(b) π/4

(c) 5π/12

(d) π/3

52. The price of a commodity increased by 5% from 2010 to 2011, 8% from 2011 to 2012 and 77% from 2012 to 2013. What is the average price increase (approximate) from 2010 to 2013?

(a) 26%

(b) 32%

(c) 24%

(d) 30%

53. A railroad curve is to be laid on a circle. What radius (approximate) should be used if the track is to change direction by 25° in a distance of 120 m?

(a) 300 m

(b) 280 m

(c) 275 m

(d) 264 m

54. If 0 < θ < π/4, then what is √1 – 2 sin θ cos θ equal to?

(a) cosθ – sinθ

(b) sinθ – cosθ)

(c) ±(cosθ – sinθ)

(d) cosθ sinθ

55. If tan θ + cot θ = 2, then what is sin θ + cos θ equal to?

(a) 1/2

(b) 1/√3

(c) √2

(d) 1

56. What is sec x/cot x + tan x equal to?

(a) sin x

(b) cos x

(c) tan x

(d) cot x

57. From a certain point on a straight road, a person observes a tower in the west direction at a distance of 200 m. He walks some distance 'along the road and finds that the same tower is 300 m south of him. What is the shortest distance of the tower from the road?

(a) 300/√13 m

(b) 500/√13 m

(c) 600/√13 m

(d) 900/√13 m

58. What is sin x - cos x + 1/sin x + cos x - 1 equal to?

(a) sin x – 1/cos x

(b) sin x + 1/cos x

(c) sin x – 1/cos x +1

(d) sin x + 1/cos x + 1

59. What is (sin2 x – cos2 x)(1 – sin2 x cos2 x) equal to?

(a) sin4 x – cos4 x

(b) sin6 x – cos6 x

(c) cos8 x – sin8 x

(d) sin8 x – cos8 x

60. What is (sin x cos y + cos x sin y) (sin x cos y – cos x sin y) equal to?

(a) cos2 x – cos2 y

(b) cos2 x – sin2 y

(c) sin2 x – cos2 y

(d) sin2 x – sin2 y

61. What is (1 + cot x – cosec x ) (1 + tan x + sec x) equal to?

(a) 1

(b) 2

(c) sin x

(d) cos x

62. What is (cosec x – sin x)(sec x – cos x)(tan x + cot x ) equal to?

(a) sin x + cos x

(b) sin x – cos x

(c) 2

(d) 1

63. Consider the following statements:

1. sin 1° > sin 1

2. cos 1° < cos 1

Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

64. If sin x + cosec x = 2, then what is sing x + cosec9 x equal to?

(a) 2

(b) 18

(c) 512

(d) 1024

65. If sin x + cos x = p and sin3 x + cos3 x = q, then what is p3 – 3p equal to?

(a) 0

(b) – 2q

(c) 2q

(d) 4q

66. What is the number of pairs of perpendicular planes in a cuboid?

(a) 4

(b) 8

(c) 12

(d) None of the above

67. How many equilateral triangles can be formed by joining any three vertices of a cube?

(a) 0

(b) 4

(c) 8

(d) None of the above

For the next two (2) items that follow:

ABCD is a trapezium in which AB is parallel to CD. Let M be the midpoint of BC.

68. Consider the following statements:

1. 'Area of triangle ADM + Area of triangle DCM' is equal to three-fourth of the area of trapezium ABCD, if AB = CD.

2. 'Area of triangle DCM + Area of triangle ABM' is always greater than half of the area of trapezium ABCD.

Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

69. Consider the following statements:

1. 'Area of triangle ADM - Area of triangle ABM' is always equal to area of triangle DCM, if AB = CD.

2. Half of area of triangle ABM is equal to one-eighth of area of trapezium ABCD, if AB = CD.

Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

70. ABCD is a parallelogram. P and R are the midpoints of DC and BC respectively. The line PR intersects the diagonal AC at Q. The distance CQ will be equal to

(a) AC/4

(b) BD/3

(c) BD/4

(d) AC/3

71. Consider the following statements in respect of an equilateral triangle:

1. The altitudes are congruent.

2. The three medians are congruent.

3. The centroid bisects the altitude.

Which of the above statements are correct?

(a) 1 and 2 only

(b) 2 and 3 only

(c) 1 and 3 only

(d) 1, 2 and 3

72. Consider the following:

ABC and DER are triangles in a plane such that AB is parallel to DE, BC is parallel to EF and CA is parallel to FD.

Statement-I: If angle ABC is a right angle, then angle DEF is also a right angle.

Statement-II: Triangles of the type. ABC and DEF are always congruent.

Which one of the following is correct in respect of the above statements?

(a) Statement—I and Statement—II are correct and Statement—II is the correct explanation of Statement—I

(b) Statement—I and Statement—II are correct and Statement—II is not the correct explanation of Statement—I

(c) Statement—I is correct and Statement—II is incorrect

(d) Statement—I is incorrect and Statement—II is correct

73. Let the incircle to a triangle ABC touch BC, AC and AB respectively at the points X, Y and Z. Statement—I: If AB > BC, then AB + AZ < BC + XC

Statement—II: AZ = AY

Which one of the following is correct in respect of the above statements?

(a) Statement—I and Statement—II are correct and Statement—II is the correct explanation of Statement—I

(b) Statement—I and Statement—II are correct and Statement—II is not the correct explanation of Statement—I

(c) Statement—I is correct and Statement—II is incorrect

(d) Statement—I is incorrect and Statement—II is correct

74. Let ABC be a triangle in which ∠ACB = 600 and AC = x < BC. Let the circle with centre at C and radius x meet BC at D. Let CF be the perpendicular drawn from C meeting AD at F.

Statement—I: Triangle ACD is isosceles but not equilateral.

Statement—II: DF = x/2

Which one of the following is correct in respect of the above statements?

(a) Statement—I and Statement—II are correct and Statement—II is the correct explanation of Statement—I

(b) Statement—I and Statement—II are correct and Statement—II is not the correct explanation of Statement—I

(c) Statement—I is correct and Statement—II is incorrect

(d) Statement—I is incorrect and Statement—II is correct

75. Bisectors of two adjacent angles A and B of a quadrilateral ABCD intersect each other at a point P. Which one of the following is correct?

(a) 2∠APB = ∠C + ∠D

(b) ∠APS = ∠C + ∠D

(c) ∠APB = 180° - (∠A + ∠B)

(d) ∠APB = 180° - (∠C + ∠D)

76. In a triangle ABC, AD is the median through A and E is the midpoint of AD, and BE produced meets AC at F. Then AF is equal to

(a) AC/5

(b) AC/4

(c) AC/3

(d) AC/2

77. Three straight lines are drawn through the three vertices of a triangle ABC, the line through each vertex being parallel to the opposite side. The triangle DEF is bounded by these parallel lines. Consider the following statements in respect of the triangle DEF:

1. Each side of triangle DEF is double the side of triangle ABC to which it is parallel.

2. Area of triangle DEF is four times the area of triangle ABC.

Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

78. Let ABCD be a parallelogram. Let X and Y be the midpoints of the sides BC and AD respectively. Let M and N be the midpoints of the sides AB and CD respectively. Consider the following statements:

1. The straight line MX cannot be parallel to SW.

2. The straight lines AC, BD, XY and MN meet at a point.

Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

79. The chord of an arc of a circle is of length x, the height of the arc is y and the radius of the circle is a z. Which one of the following is correct?

(a) y(2z – y) = x2

(b) y(2z – y) = 4x2

(c) 2y (2z – y) = x2

(d) 4y(2z – y) = x2

80. In a triangle ABC, ∠B = 2 ∠C = 2 ∠A. What is the ratio of AC to AB?

(a) √2 : 1

(b) 1 : 1

(c) √3 : 1

(d) 1 : 4

82. What is the maximum distance between two points of a cube of side 2 cm?

(a) √3 cm

(b) 2√3 cm

(c) 4√3 cm

(d) 2√2 cm

82. The areas of the three adjacent faces of a cuboidal box are x, 4x and 9x square unit. What is the volume of the box?

(a) 6x2 cubic unit

(b) 6x3/2 cubic unit

(c) 3x3/2 cubic unit

(d) 2x3/2 cubic unit

83. A cylinder circumscribes a sphere. What is the ratio of volume of the sphere to that of the cylinder?

(a) 2 : 3

(b) 1 : 2

(c) 3 : 4

(d) 3 : 2

84. If for a triangle the radius of the circumcircle is double the radius of the inscribed circle, then which one of the following is correct?

(a) The triangle is right angled

(b) The triangle is isosceles

(c) The triangle is equilateral

(d) None of the above

For the next two (2) items that follow:

A toy is in the form of a cone mounted on the hemisphere with the same radius. The diameter of the base of the conical portion is 12 cm and its height is 8 cm.

85. What is the total surface area of the toy?

(a) 132 π cm2

(b) 112 π cm2

(c) 96 π cm2

(d) 66 π cm2

86. What is the volume of the toy?

(a) 180 π cm3

(b) 240 π cm3

(c) 300 π cm3

(d) 320 π cm3

For the next three (3) items that follow:

A right triangle having hypotenuse 25 cm and legs in the ratio 3 : 4 is made to revolve about its hypotenuse. (π = 3.14)

87. What is the volume of the double cone so formed?

(a) 3124 cm3

(b) 3424 cm3

(c) 3768 cm3

(d) 3924 cm3

88. What is the surface area of the double cone so formed?

(a) 1101.2 cm2

(b) 1111.4 cm2

(c) 1310.4 cm2

(d) 1318.8 cm2

89. Consider the following

1. The volume of the cone generated when the triangle is made to revolve about its longer leg is same as the volume of the cone generated when the triangle is made to revolve about its shorter leg.

2. The sum of the volume of the cone generated when the triangle is made to revolve about its longer leg and the volume of the cone generated when the triangle is made to revolve about its shorter leg is equal to the volume of the double cone generated when the triangle is made to revolve about its hypotenuse.

Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

For the next three (3) items that follow:

A piece of land is in the form of a parallelogram and the perimeter of the land is 86 m. The length of one side exceeds the other by 13 m and one of the diagonals is 41 m.

90. What is the area of the parallelogram?

(a) 63 m2

(b) 96 m2

(c) 126 m2

(d) 252 m2

91. What is the shorter height of the parallelogram?

(a) 9.0 m

(b) 7.5 m

(c) 5.5 m

(d) 4.5 m

92. Consider the following statements:

1. The difference between the diagonals of the parallelogram is more than 20 m.

2. The difference between the heights of the parallelogram is more than 10 m.

Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

93. If one of the roots of the equation px2 + qx + r = 0 is three times the other, then which one of the following relations is correct?

(a) 3q2 = 16pr

(b) q2 = 24 pr

(c) p = q + r

(d) p + q + r = 1

94. If the radius of a circle is increased by 6%, then its area will increase by

(a) 6%

(b) 9%

(c) 12.36%

(d) 16.64%

95. The class which has maximum frequency is known as

(a) median class

(b) mean class

(c) modal class

(d) None of the above

96. Consider the following statements related to cumulative frequency polygon of a frequency distribution, the frequencies being cumulated from the lower end of the range:

1. The cumulative frequency polygon gives an equivalent representation of frequency distribution table.

2. The cumulative frequency polygon is a closed polygon with one horizontal and one vertical side. The other sides have non-negative slope.

Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

97. Consider the following data:

1. Number of complaints lodged due to road accidents in a State within a year for 5 consecutive years

2. Budgetary allocation of the total available funds to the various items of expenditure

Which of the above data is/are suitable for representation of a pie diagram?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

98. When we take class intervals on the x-axis and corresponding frequencies on the y-axis, and draw rectangles with the areas proportional to the frequencies of the respective class intervals, the graph so obtained is called

(a) bar diagram

(b) frequency curve

(c) ogive

(d) None of the above

(a) 0

(b) – 1

(c) 1

(d) 2

100. Ten observations 6, 14, 15, 17, x + 1, 2x – 13, 30, 32, 34, 43 are written in ascending order. The median of the data is 24. What is the value of x?

(a) 15

(b) 18

(c) 20

(d) 24