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

The UPSC CDS (I) 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 (I) 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. What is the number of integral solutions of the equations HCF (a, b) = 5 and a + b 65?

(a) None

(b) Infinitely many

(c) Less than 65

(d) Exactly one

2. The difference of two consecutive cubes

(a) is odd or even

(b) is never divisible by 2

(c) is always even

(d) Nome of the above

4. The product of four consecutive natural numbers plus one is

(a) a non-square

(b) always sum of two square numbers

(c) a square

(d) None of the above

5. For any integers 'a’ and ‘b’ with HCF (a, b) = 1, what is HCF (a + b, a - b) equal to?

(a) It is always 1

(b) It is always 2

(c) Either 1 or 2

(d) None of the above

6. The expression 2x3 + x2 – 2x – 1 is divisible by

(a) x + 2

(b) 2x + 1

(c) x – 2

(d) 2x – 1

7. A positive number, when increased by 10, equals 200 times its reciprocal. What is that number?

(a) 100

(b) 10

(c) 20

(d) 200

8. x3 + 6x2 + 11x + 6 is divisible by

(a) (x + 1) only

(b) (x + 2) only

(c) (x + 3) only

(d) All of the above

9. The present age of Ravi’s father is four times Ravi’s present age. Five years back he was seven times as old as Ravi was at that time. What is the present age of Ravi's father?

(a) 84 years

(b) 70 years

(c) 40 years

(d) 35 years

10. The average age of male employees in a firm is 52 years and that of female employees is 42 years. The mean age of all employees is 50 years. The percentage of male and female employees are respectively

(a) 80% and 20%

(b) 20% and 80%

(c) 50% and 50%

(d) 52% and 48%

11. 15 men complete a work in 16 days. If 24 men are employed, then the time required to complete that work will be

(a) 7 days

(b) 8 days

(c) 10 days

(d) 12 days

12. A train takes 9 seconds to cross a pole. If the speed of the train is 48 km/hr, the length of the train is

(a) 150m

(b) 120m

(c) 90m

(d) 80m

13. Ravi’s brother is 3 years elder to him. His father was 28 years of age when his sister was born while his mother was 26 years of age when he was born. If his sister was 4 years of age when his brother was born, the ages of Ravi's rather and mother respectively when his brother was born were

(a) 32 years and 23 years

(b) 32 years and 29 years

(c) 35 years and 29 years

(d) 35 years and 33 years

14. In solving a problem, one student makes a mistake in the coefficient of the first degree term and obtains - 9 and - 1 for the roots. Another student makes a mistake in the constant term of the equation and obtains 8 and 2 for the roots. The correct equation was

(a) x2 + 10x + 9 = 0

(b) x2 — 10x + 16 = 0

(c) x2 — 10x + 9 = 0

(d) None of the above

15. If m and n are the roots of the equation ax2+ bx + c = 0, then the equation whose roots are (m2-+1)/m and (n2 + 1)/n is

(a) acx2 + (ab + bc) x + b2 + (a-c)2 = 0

(b) acx2 + (ab - bc) x + b2 + (a-c)2 = 0

(c) acx2  + (ab - bc) x + b2 - (a-c)2 = 0

(c) acx+ (ab + bc) x + b2 - (a-c)2=0

16. The value of x2 - 4x + 11 can never be less than

(a) 7

(b) 8

(c) 11

(d) 22

18. If 3x4 - 2x3 + 3x2 - 2x + 3 a is divided by (3x + 2), then the remainder is

(a) 0

(b) 185/27

(c) 181/25

(d) 3/4

19. What should be added to the expression x(x + a) (x + 2a) (x + 3a) so that the sum may be a perfect square?

(a) 9a2

(b) 4a2

(c) a4

(d) None of the above

20. If the roots of the equation Ax2 + Bx + C = 0 are -1 and 1, then which one not the following is correct?

(a) A and C are both zero

(b) A and B are both positive

(c) A and C are both negative

(d) A and C are of opposite sign

Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) neither 1 nor 2

23. A water pipe is cut into two pieces. The longer piece is 70% of the length of the pipe. By how much percentage is the longer piece longer than the shorter piece?

(a) 140%

(b) 400%

(c) 40%

(d) None of the above

24. The sum of two positive numbers x and y is 25 times their difference. If the product of the numbers is 84, then what is the sum of the two numbers?

(a) 26

(b) 24

(c) 22

(d) 20

25. If n is a whole number greater than 1, then n2 (n2 — 1) is always divisible by

(a) 12

(b) 24

(c) 48

(d) 60

26. If the roots of the equation (a2 - bc) x2 + 2 (b2 - ac) x + (c2 - ab) = 0 are equal, where b ≠ 0, then which one of the following is correct?

(a) a + b + c = abc

(b) a2 + b2 + c2= 0

(c) a3- b3 + c3= 0

(d) a3- b3 + c3 = 3abc

27. What is the remainder when 41000 is divided by 7?

(a) 1

(b) 2

(c) 4

(d) None of the above

28. A, B and C can do a piece of work individually in 8, 12 and 15 days respectively. A and 13 start working, but A quits after working for 2 days. After this, C joins B till the completion of work. In how many days will the work be completed?

29. If x + y = 5, y + z = 10 and z + x = 15, then which one of the following is correct?

(a) z > x > y

(b) z > y > y

(c) x > y > z

(d) x > z > y

30. On a 20% discount sale, an article costs Rs. 596. What was the original price of the article?

(a) Rs. 720

(b) Rs. 735

(c) Rs. 745

(d) Rs. 775

31. What is the HCF of 8 (x5 - x3 + x) and 28 (x6 + 1)?

(a) 4(x4- x2 + 1)

(b) 2(44 -x2+1)

(c) (x4- x2 +1)

(d) None of the above

32. Two chairs and one table cost 700 and 1 chair and 2 tables cost 800. If the cost of m tables and m chairs is 30,000, then what is m equal to?

(a) 60

(b) 55

(c) 50

(d) 45

33. Consider the following statements:

1. No integer of the form 4k + 3, where k is an integer, can be expressed as the sum of two squares.

2. Square of an odd integer can be expressed in the form 8k + 1, where k is an integer.

Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

34. For any integer n, what is HCF (22n + 7, 33n + 10) equal to?

(a) n

(b) 1

(c) 11

(d) None of the above

35. In a fire range, 4 shooters are firing at their respective targets. The first, the second, the third and the fourth shooter hit the target once every 5s, 6s, 7s and 8s respectively. If all of then hit their target at 9:00 am, when will they hit their target together again?

(a) 9:04 am

(b) 9:08 am

(c) 9:14 am

(d) None of the above

(a) A

(b) Complement of A

(c) ϕ

(d) E

37. If A:B = 2:3, B:C = 5:7 and C:D = 3:10, than what is A:D equal to?

(a) 1:7

(b) 2:7

(c) 1:57

(d) 5:1

38. If the average of A and B is 30, the average of C and D is 20, then which of the following is/are correct?

1. The average of B and C must be greater them 25.

2. The average of A and D must be less than 25.

Select the correct answer using the code given below?

(a) 1 only

(b) 2 only

(c) Either 1 or 2

(d) Neither 1 nor 2

39. Out of 105 students taking an examination for English and Mathematics, 80 students pass n English, 75 students pass in Mathematics and 10 students fail in both the subjects. How many students pass in only one subject?

(a) 26

(b) 30

(c) 35

(d) 45

40. At what point of time after 3 ‘o’clock. do the hour-hand and the minute-hand of a clock occur at right angles for the first times?

41. If 64 identical small spheres are made out or a big sphere of diameter 8 cm, what is the surface area of each small sphere?

(a) π cm2

(b) 2π cm2

(c) 4π cm2

(d) 8π cm2

42. AB and CD are two diameters of a circle of radius r and they are mutually perpendicular. What is the ratio of the area of the circle to the area of the triangle ACD?

(a) π/2

(b) π

(c) π/4

(d) 2π

43. A circle of radius 10 cm has an equilateral triangle inscribed in it. The length of the perpendicular drawn from the centre to any side of the triangle is

(a) 25 √3 cm

(b) 5 √3 cm

(c) 10 √3 cm

(d) None of the above

44. The dimensions of a field are 15 in by 12 m. A pit 8 m long, 2.5 m wide in 2 m deep is dug in one corner of the field and the earth removed is evenly spread over the remaining area of the field. The level of the field is raised by

(a) 15 cm

(b) 20 cm

(c) 25 cm

(d) 200/9 cm

45. The sides of a right-angled triangle are equal to three consecutive numbers expressed in centimeters. What can be the area of such a triangle?

(a) 6 cm2

(b) 8 cm2

(c) 10 cm2

(d) 12 cm2

46. How many circular pates or diameter d can be taken out of a square plate of side 2d with minimum loss of material?

(a) 8

(b) 6

(c) 4

(d) 2

47. The volume of a hollow cube is 216 x3. The surface area of the largest sphere which can be enclosed in it is

(a) 18πx2

(b) 27πx2

(c) 36πx2

(d) 72πx2

48. What is the diameter of the largest circle lying on the surface of a sphere of surface area 616 square cm?

(a) 14cm

(b) 10.5 cm

(c) 7cm

(d) 3.5 cm

49. What is the total area of three equilateral triangles inscribed in a semicircle of radius 2 cm?

(a) 12cm2

(b) 3√3/4 cm2

(c) 9√3/4 cm2

(d) 3√3 cm2

50. In a trapezium, the two non-parallel sides are equal in length, each being of 5 units. The parallel sides are at a distance of 3 units apart. If the smaller side of the parallel sides is of length 2 units, then the sum of the diagonals of the trapezium is

(a) 10√5 units

(b) 6√5 units

(c) 5√5 units

(d) 3√5 units

51. The shadow of a tower standing on a level plane is found to be 50 m longer when the Sun's elevation is 30° than when it is 60°. What is the height of the tower?

(a) 25 m

(b) 25√3 m

(c) 25/√3 m

(d) 30 m

52. The angle of elevation of the top of a tower 30 ni high from the foot of another tower in the same plane is 60° and the angle of elevation of the top of the second tower from the foot of the first tower is 30°. The distance between the two towers is m times the height of the shorter tower. What is m equal to?

(a) √2

(b) √3

(c) 1/2

(d) 1/3

53. If cos x + sec x = 2, then what is cosn x + secn x equal to, where n is a positive integer?

(a) 2

(b) 2n – 2

(c) 2n – 1

(d) 2n

54. What is sin 25° sin 35° sec 65° sec 55° equal to?

(a) – 1

(b) 0

(c) 1/2

(d) 1

55. The value of cos 25° – sin 25° is

(a) positive but less than 1

(b) positive but greater than 1

(c) negative

(d) 0

56. In a right-angled triangle ABC, right-angled at B, if cos A = 4/5, then what is sin C equal 5 to?

(a) 3/5

(b) 4/5

(c) 3/4

(d) 2/5

57. If α and β are complementary angles, then what is √cos α cosec β – cos α sin β equal to?

(a) sec β

(b) cos α

(c) sin α

(d) - tan β

58. If 2 cot θ = 3, then what is 2 cos θ – sin θ /2 cos θ + sin θ equal to?

(a) 2/3

(b) 1/3

(c) 1/2

(d) 3/4

59. If sin θ cos θ = 1/2, then what is sin6 θ + cos6 θ equal to?

(a) 1

(b) 2

(c) 3

(d) 1/4

60. If sec θ + tan θ = 2, what is the value of sec θ?

(a) 3/2

(b) √2

(c) 5/2

(d) 5/4

61. What is cosec (75° + θ) – sec (15° – θ) – tan (55° + θ) + cot (35° – θ) equal to?

(a) – 1

(b) 0

(c) 1

(d) 3/2

62. If sin θ + 2 cos θ = 1, where 0 < θ < π/2, then what is 2 sin θ – cos θ equal to?

(a) – 1

(b) 1/2

(c) 2

(d) 1

63. If tan 8θ = cot 2θ, where 0 < 8θ < π/2, then what is the value of tan 5θ?

(a) 1/ √3

(b) 1

(c) √3

(d) 0

64. A square is inscribed in a circle of diameter 2a and another square is circumscribing the circle. The difference between the areas of the outer and inner squares is

(a) a2

(b) 2a2

(c) 3a2

(d) 4a2

65. The area of a sector of a circle of radius 36 cm is 72π cm2. The length of the corresponding arc of the sector is

(a) π cm

(b) 2π cm

(c) 3π cm

(d) 4π cm

66. A cone of radius r cm and height h cm is divided into two parts by drawing a plane through the middle point of its height and parallel to the base. What is the ratio of the volume of the original cone to the volume of the smaller cone?

(a) 4 : 1

(b) 8 : 1

(c) 2 : 1

(d) 6 : 1

67. Which one of the following is a Pythagorean triple in which one side differs from the hypotenuse by two units?

(a) (2n + 1, 4n, 2n2 + 2n)

(b) (2n, 4n, n2 + 1)

(c) (2n2, 2n, 2n + 1)

(d) (2n, n2 – 1, n2 + 1)

where n is a positive real number.

68. A cube has each edge 2 cm and a cuboid is 1 cm. long, 2 cm wide and 3 cm high. The paint in a certain container is sufficient to paint an area equal to 54 cm2.

Which one of the following is correct?

(a) Both cube and cuboid can be painted

(b) Only cube can be painted

(c) Only cuboid can be painted

(d) Neither cube nor cuboid can be painted

69. ABC is a triangle right-angled at A where AB = 6 cm and AC 8 cm. Semicircles are drawn (outside the triangle) on AB, AC and BC as diameters which enclose areas x, y and z square units respectively. What is x + y – z equal to?

(a) 48 cm2

(b) 32 cm2

(c) 0

(d) None of the above

70. The three sides of a triangle are 15, 25 and x units. Which one of the following is correct?

(a) 10 < x < 40

(b) 10 ≤ x ≤ 40

(c) 10 ≤ x < 40

(d) 10 < x ≤ 40

71. Two light rods AB = a + b, CD = a – b are symmetrically lying on a horizontal plane. There are kept intact by two strings AC and BD. The perpendicular distance between the rods is ‘a’. The length of AC is given by

(a) a

(b) b

(c) √a2 – b2

(d) √a2 + b2

72. Let PQRS be a rectangle such that PQ = √3 QR. What is PRS equal to?

(a) 60°

(b) 45°

(c) 30°

(d) 15°

73. The area of a rectangle lies between 40 cm2 and 45 cm2. If one of the sides is 5 cm, then its diagonal lies between

(a) 8 cm and 10 cm

(b)  9 cm and 11 cm

(c) 10 cm and 12 cm

(d) 11 cm and 13 cm

74. A cylinder is surmounted by a cone at one end, a hemisphere at the other end. The common radius is 3.5 cm, the height of the cylinder is 6.5 cm and the total height of the structure is 12.8 cm. The volume V of the structure lies between

(a) 370 cm3 and 380 cm3

(b) 380 cm3 and 390 cm3

(c) 390 cm3 and 400 cm3

(d) None of the above

75. In a triangle ABC, AB = BC = CA. The ratio of the radius of the circumcircle to that of the incircle is

(a) 2 : 1

(b) 3 : 1

(c) 3 : 2

(d) None of the above

76. If sin (A + B) = 1, where 0 < B < 45°, then what is cos (A – B) equal to?

(a) sin 2B

(b) sin B

(c) cos 2B

(d) cos B

77. For a plot of land of 100 m x 80 m, the level is to be raised by spreading the earth from a stack of a rectangular base 10 m x 8 m with vertical section being a trapezium of height 2 m. The top of the stack is 8 m x 6 m. By how many centimeters can the level be raised?

(a) 3 cm

(b) 2.5 cm

(c) 2 cm

(d) 1.5 cm

78. AB is a straight line; C and D are points on the same side of AB such that AC is perpendicular to AB and DB is perpendicular to AB. Let AD and BC meet at E.

What is AE/AD + BE/BE equal to?

(a) 2

(b) 1.5

(c) 1

(d) None of the above

79. Consider an equilateral triangle of a side of unit length. A new equilateral triangle is formed by joining the mid-points of the first one, then a third equilateral triangle is formed by joining the mid-points of the second. The process is continued. The sum perimeter of all triangles thus formed is

(a) 2 units

(b) 3 units

(c) 6 units

(d) Infinity

80. Let ABCD be a parallelogram. Let P, Q, R, S be the mid-points of sides AB, BC, CD, DA respectively. Consider the following statements:

1. Area of triangle APS < Area of triangle DSR if BD < AC.

2. Area of triangle ABC = 4 (Area of triangle BPQ).

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

81. AB and CD are two chords of a circle meeting externally at P. Then which of the following is/are correct?

1. PA x PD = PC x PB.

2. Triangles PAC and PDB are similar.

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

82. Consider the following statements:

1. Let ABCD be a parallelogram which is not a rectangle.

Then 2 (AB2 + BC2)  AC2 + BD2

2. If ABCD is a rhombus with AB = 4 cm, then AC2 + BD2 = n3 for some positive integer n.

Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

83. ABC is triangle right-angled at B. If AB= 6 cm and BC = 8 cm, then what is the length of the circum-radius of the triangle ABC?

(a) 10 cm

(b) 7 cm

(c) 6 cm

(d) 5 cm

84. If AD is the internal angular bisector of triangle ABC with AB = 3 cm and AC= 1 cm, then what is BD : BC equal to?

(a) 1 : 3

(b) 1 : 4

(c) 2 : 3

(d) 3 : 4

85. The sides of a triangle are in geometric progression with common ratio r < 1. If the triangle is a right-angled triangle, then r2 is given by

86. In a triangle ABC, AD is perpendicular to BC and BE is perpendicular to AC. Which one of the following is correct?

(a) CE x CB = CA x CD

(b) CE x CA = CD x CB

(c) AD x BD = AE x BE

(d) AB x AC = AD x BE

87. ABCD is a parallelogram. E is a point on BC such that BE : EC = m : n. If AE and DB intersect in F, then what is the ratio of the area of triangle FEB to the area of triangle AFD?

(a) m/n

(b) (m/n)2

(c) (n/m)2

(d) [m/(m + n)]2

88. If triangles ABC and DEF are similar such that 2AB = DE and BC = 8 cm, then what is EF equal to?

(a) 16 cm

(b) 12 cm

(c) 10 cm

(d) 8 cm

89. A rectangle of maximum area is drawn inside a circle of diameter 5 cm. What is the maximum area of such a rectangle?

(a) 25 cm2

(b) 12.5 cm2

(c) 12 cm2

(d) None of the above

90. A drainage tile is a cylindrical shell 21 cm long. The inside and outside diameters are 4.5 cm and 5.1 cm respectively. What is the volume of the clay required for the tile?

(a) 6.96π cubic cm

(b) 6.76π cubic cm

(c) 5.76π cubic cm

(d) None of the above

91. The diameter of the base of a cone is 6 cm, altitude is 4 cm. What is the approximate curved surface area of the cone?

(a) 45 square cm

(b) 47 square cm

(c) 49 square cm

(d) 51 square cm

92. What is the area of the larger segment of a circle formed by a chord of length 5 cm subtending an angle of 900 at the centre?

93. Consider the following in respect of the variate which takes values 2, 2, 2, 7, 7, 2, 7, 7 :

1. The median is equal to mean.

2. The mode is both 2 and 7.

Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

94. Consider the following statements pertaining to a frequency polygon of a frequency distribution of a continuous variable with seven class intervals of equal width:

1. The original frequency distribution can be reconstructed from the frequency polygon.

2. The frequency polygon touches the x-axis in its extreme right and extreme left.

Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

95. The mean of the following distribution is 18.

 Class interval Frequency 11 — 13 3 13 — 15 9 15 — 17 6 ‎17 — 19 13 19 — 21 f 21 — 23 5 23 — 25 4

What is the value of f?

(a) 8

(b) 9

(c) 10

(d) 11

96. Which of the following items of information is a good example of statistical data?

(a) A table of logarithms of numbers

(b) A list of names of 120 students of a class

(c) A list of annual incomes of the members of a club

(d) Holiday list of the offices of Government of India in the year 2013

97. The following table gives ‘less than’ type frequency distribution of income per day:

What is the modal class?

(a) 250 – 500

(b) 500 – 750

(c) 750 – 1000

(d) None of the above

98. Consider the following:

1. The arithmetic mean of two unequal positive numbers is always greater than their geometric mean.

2. The geometric mean of two unequal positive numbers is always greater than their harmonic mean.

Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

99. Consider the following statements in respect of a discrete set of numbers:

1. The arithmetic mean uses all the data and is always uniquely defined.

2. The median uses only one or two numbers from the data and may not be unique.

Which of the above statements is/are correct?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

100. The geometric mean of (x1, x2, x3, …xn) is X and the geometric mean of (y1, y2, y3, … yn) is Y. Which of the following is/are correct?

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