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CDS (I) 2025, Elementary Mathematics question paper

100 questions · 100 marks · 120 minutes · +1 per correct answer, -0.33 per wrong one

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  1. 1
    A real number M is squared to give the value N. What is the minimum value of (M+N)(M + N) ?

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  2. 2
    What is the sum of all 3-digit numbers that give a remainder of 5 when they are divided by 50 ?

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  3. 3
    If the average of 64, 69, 72, 75, xx lies between 62 and 76 (excluding 62 and 76), then what is the number of possible integer values of xx ?

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  4. 4
    Let x,y,zx, y, z be variables such that (x+y+z)=k(x + y + z) = k, where kk is a constant. If (x+z−y)×(x−z+y)(x + z - y) \times (x - z + y) is proportional to yzyz, then (y+z−x)(y + z - x) is proportional to :

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  5. 5
    Let pp be the remainder when 7847^{84} is divided by 342 and qq be the remainder when 7847^{84} is divided by 344. What is (p−q)(p - q) equal to ?

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  6. 6
    N is the smallest 5-digit number which when divided by 2,22,23,24,…,2n2, 2^2, 2^3, 2^4, \ldots, 2^n leaves a remainder 1. What is the value of nn ?

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  7. 7
    What is the minimum value of pp for which 1532900+p2266450+p4523900\dfrac{1}{532900} + \dfrac{p^2}{266450} + \dfrac{p^4}{523900} is an integer ?

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  8. 8
    If α\alpha and β\beta are the roots of the equation x+a+b=abxab+ax+bxx + a + b = \dfrac{abx}{ab + ax + bx}, then what is (αβ+α+β)(\alpha\beta + \alpha + \beta) equal to ?

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  9. 9
    Consider a 2-digit number N. Let P be the product of the digits of the number. If P is added to square of the digit in the tens place of N, we get 84. If P is added to the square of the digit in the unit place of N, we get 60. What is the value of P+NP + N ?

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  10. 10
    A mixture of 100 L contains kerosene and turpentine oil in the ratio 3 : 2. What is the minimum quantity of kerosene in litres (whole number) that should be mixed in the mixture so that the resulting mixture has 20% of kerosene ?

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  11. 11
    A lamp is kept on a vertical pole. The height of the top of the lamp above the ground is 532\dfrac{5\sqrt{3}}{2} m. The perpendicular distances of the bottom of the pole from two adjacent walls meeting perpendicularly are 0.7 m and 2.4 m. What is the distance of the top of the lamp from the corner point of the walls on the ground ?

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  12. 12
    C is the centre of a circle of radius 20 cm. AB is a chord of length 32 cm. E is a point on AB such that CE = 13 cm. What is AE×EBAE \times EB equal to ?

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  13. 13
    The inside of a bowl is part of a sphere. When water is put into the bowl to a depth dd, the water surface becomes a circle of radius 2d2d. What is the radius of the sphere ?

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  14. 14
    In a triangle ABC, AB = 2 cm, BC = 4 cm and AC = 3 cm. The bisector of angle A meets BC at D and the bisector of angle B meets AD at E. What is AE : ED equal to ?

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  15. 15
    In a triangle ABC, the bisector of angle A cuts BC at D. If AB + AC = 10 cm and BD : DC = 3 : 1, then what is the length of AC ?

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  16. 16
    In a triangle ABC, AB + BC = 7.1 cm, BC + CA = 12.1 cm and CA + AB = 7.2 cm. What is the area of the triangle ?

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  17. 17
    The adjacent sides of a parallelogram are 10 cm and 8 cm and the angle between them is 150∘150^\circ. What is the area of the parallelogram ?

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  18. 18
    The measure of an angle formed by the bisectors of the angles A and C of the triangle ABC is 130∘130^\circ. What is the measure of the angle B ?

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  19. 19
    What is log⁡102000+log⁡10400+4log⁡1025+5log⁡1020\log_{10} 2000 + \log_{10} 400 + 4 \log_{10} 25 + 5 \log_{10} 20 equal to ?

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  20. 20
    If log⁡10(100001−4x)5−x=1\dfrac{\log_{10}(100001 - 4^x)}{5 - x} = 1, then what is xx equal to ?

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  21. 21
    If 2sin⁡4α+2cos⁡4α−1=02 \sin^4 \alpha + 2 \cos^4 \alpha - 1 = 0, where 0≤α<π/20 \le \alpha < \pi/2, then what is sin⁡2α+cos⁡2α\sin 2\alpha + \cos 2\alpha equal to ?

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  22. 22
    Consider the following : I. 1−sin⁡6α=cos⁡2α(cos⁡4α−3cos⁡2α+3)1 - \sin^6 \alpha = \cos^2 \alpha (\cos^4 \alpha - 3 \cos^2 \alpha + 3) II. cos⁡8α−sin⁡8α=2sin⁡2α(1−cos⁡4α+sin⁡2αcos⁡2α)\cos^8 \alpha - \sin^8 \alpha = 2 \sin^2 \alpha (1 - \cos^4 \alpha + \sin^2 \alpha \cos^2 \alpha) Which of the above is/are identities ?

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  23. 23
    If p=1cosec⁡θ+cot⁡θp = \dfrac{1}{\operatorname{cosec} \theta + \cot \theta} and q=cosec⁡θq = \operatorname{cosec} \theta, then what is p2−2pqp^2 - 2pq equal to ?

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  24. 24
    Consider the following statements : I. (cosec⁡α−sec⁡α)(\operatorname{cosec} \alpha - \sec \alpha) is always positive in the first quadrant. II. (tan⁡α−cot⁡α)(\tan \alpha - \cot \alpha) is always negative in the first quadrant. Which of the statements given above is/are correct ?

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  25. 25
    A tower subtends an angle 60∘60^\circ at a point A on the same level as the foot of the tower. B is a point vertically above A and AB = h. The angle of depression of the foot of the tower, measured from B is 30∘30^\circ. What is the height of the tower ?

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  26. 26
    What is sin⁡θ1−cot⁡θ+cos⁡θ1−tan⁡θ\dfrac{\sin \theta}{1 - \cot \theta} + \dfrac{\cos \theta}{1 - \tan \theta} (θ≠π/4)(\theta \ne \pi/4) equal to ?

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  27. 27
    The length of an arc of a circle of radius 4 cm is π\pi cm. What is the magnitude of the angle subtended by the arc at the centre ?

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  28. 28
    If cot⁡2θ−33cot⁡θ+6=0\cot^2\theta - 3\sqrt{3}\cot\theta + 6 = 0, where π6≤θ<π2\frac{\pi}{6} \le \theta < \frac{\pi}{2}, then what is a value of sin⁡θ+cos⁡2θ\sin\theta + \cos 2\theta ?

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  29. 29
    Which of the following equations is/are possible ? I. sin⁡2θ=(x+y)24xy\sin^2\theta = \frac{(x+y)^2}{4xy}, where x, y are positive unequal real quantities. II. sin⁡θ+cos⁡θ=x+1x\sin\theta + \cos\theta = x + \frac{1}{x}, where x is a positive real quantity. Select the correct answer using the code given below :

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  30. 30
    If m2(sin⁡θ−1)+n2(sin⁡θ+1)=0m^2(\sin\theta - 1) + n^2(\sin\theta + 1) = 0, where 0<θ<π20 < \theta < \frac{\pi}{2}, then what is (m2+n2)cos⁡θ−(m2−n2)cot⁡θ(m^2 + n^2)\cos\theta - (m^2 - n^2)\cot\theta equal to ?

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  31. 31
    If sin⁡α+cos⁡α=2\sin\alpha + \cos\alpha = \sqrt{2}, where 0<α<π20 < \alpha < \frac{\pi}{2}, then what is sin⁡3α−cos⁡3α\sin^3\alpha - \cos^3\alpha equal to ?

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  32. 32
    What is (1+cot⁡α−cosec⁡α)(1+tan⁡α+sec⁡α)(1 + \cot\alpha - \operatorname{cosec}\alpha)(1 + \tan\alpha + \sec\alpha) equal to ?

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  33. 33
    If tan⁡θ=sin⁡α−cos⁡αsin⁡α+cos⁡α\tan\theta = \frac{\sin\alpha - \cos\alpha}{\sin\alpha + \cos\alpha}, where θ\theta and α\alpha (α≠π4)\left(\alpha \ne \frac{\pi}{4}\right) are acute angles, then what is 2sin⁡θ\sqrt{2}\sin\theta equal to ?

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  34. 34
    For how many values of α\alpha does the expression (sin⁡α+2)(sin⁡α+4)(sin⁡α−2)(sin⁡α−4)(\sin\alpha + 2)(\sin\alpha + 4)(\sin\alpha - 2)(\sin\alpha - 4) become zero ?

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  35. 35
    What is the value of x, where 0≤x<30∘0 \le x < 30^\circ, satisfying tan⁡3xtan⁡6x=1\tan 3x \tan 6x = 1 ?

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  36. 36
    What is (a−b)2(b−c)(c−a)+(b−c)2(c−a)(a−b)+(c−a)2(a−b)(b−c)−3\frac{(a-b)^2}{(b-c)(c-a)} + \frac{(b-c)^2}{(c-a)(a-b)} + \frac{(c-a)^2}{(a-b)(b-c)} - 3 equal to, where a≠b≠ca \ne b \ne c ?

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  37. 37
    Given that 100×99×98×…×3×2×1100n\frac{100 \times 99 \times 98 \times \ldots \times 3 \times 2 \times 1}{100^n} is an integer. What is the largest value of n for which this is true ?

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  38. 38
    A man starting from a place P went x metre (x>120 m)(x > 120\text{ m}) East before turning South. He went 40 m straight before turning to West. He went 60 m to reach a place Q. From Q he went 200 m North and reached a place R. If PR = 200 m, then what is x equal to ?

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  39. 39
    If x2+y2+z2=3x^2 + y^2 + z^2 = 3, where x, y and z are integers, then how many values can (xy+yz+zx)(xy + yz + zx) have ?

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  40. 40
    If x, y, z are real numbers such that x+y+z=10x + y + z = 10 and xy+yz+zx=18xy + yz + zx = 18, then what is the value of x3+y3+z3−3xyzx^3 + y^3 + z^3 - 3xyz ?

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  41. 41
    What is 17−415+8−215\sqrt{17 - 4\sqrt{15}} + \sqrt{8 - 2\sqrt{15}} equal to ?

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  42. 42
    What is the maximum value of the sum of the numbers 36, 33, 30, 27, 24, ... ?

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  43. 43
    There are two natural numbers m and n (m>n)(m > n). When m is divided by 12, it leaves a remainder 4. When n is divided by 12, it leaves a remainder 6. Which of the following statements is/are correct ? I. The remainder when (m+n)(m + n) is divided by 12 is 10. II. The remainder when (m−n)(m - n) is divided by 12 is 10. Select the correct answer using the code given below :

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  44. 44
    If (x+y):(y+z):(z+x)=3:5:6(x + y) : (y + z) : (z + x) = 3 : 5 : 6 and x+y+z=14x + y + z = 14, then what is x2+y2+z2x^2 + y^2 + z^2 equal to ?

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  45. 45
    The ratio of sum of two numbers to their difference is 5:15 : 1. What is the ratio of the sum of their squares to the difference of their squares ?

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  46. 46
    Travelling at 3/5th3/5^{\text{th}} of his usual speed, a man is late by 20 minutes. What is the usual time if he travels with his usual speed ?

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  47. 47
    What is the remainder when 2p−12^p - 1 is divided by p, where p>5p > 5 is a prime number ?

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  48. 48
    What is the number of factors of 243−163−8324^3 - 16^3 - 8^3 ?

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  49. 49
    What is the least number of complete years in which a sum of money put out at 20% compound interest (compounded annually) will be more than doubled ?

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  50. 50
    A train of certain length takes time t to pass completely through a station of length x. The same train with same speed takes time 2t to pass completely through another station of length y. What is the time taken by the train to pass completely through a station of length (x+y)(x + y) ?

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  51. Consider the following for the next two (02) items that follow : A frustum of a right cone has a top of diameter 2k, bottom of diameter 2.5k and height k.
    51
    What is the whole surface area of the frustum ?

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  52. 52
    What is the volume of the frustum ?

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  53. Consider the following for the next two (02) items that follow : ABC is a triangle right-angled at B. The perimeter of the triangle is 24 cm and the difference between the sum of the perpendicular sides and the hypotenuse is 4 cm.
    53
    What is the area of the triangle ABC ?

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  54. 54
    A circle is inscribed in the triangle. What is its radius ?

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  55. Consider the following for the next two (02) items that follow : A circle M of radius 8 cm touches externally with another circle N of radius 16 cm. Let P, Q be the points where the common tangent touches the circles M and N respectively.
    55
    What is the length of the common tangent PQ ?

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  56. 56
    If U, V are the centres of the circles M and N respectively, then what is the area of the quadrilateral formed by the points P, Q, V and U ?

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  57. Consider the following for the next two (02) items that follow : The perimeter of a triangle ABC is 105 cm. The altitudes AD, BE and CF are in the ratio 3:5:63 : 5 : 6.
    57
    What is AB : BC : CA equal to ?

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  58. 58
    What is the approximate area of the triangle ABC ?

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  59. A pot is made from a hollow sphere of inner radius 20 cm by cutting its upper portion horizontally. The height of the pot is 30 cm.
    59
    What is the inner radius of the circular opening of the pot so formed ?

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  60. 60
    What is the angle made by the line joining the centre of the sphere and any point on the rim of the circular opening with a vertical line passing through the centre ?

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  61. A hall is of length ll, breadth b and height h. The maximum distance between any two points (say P and Q) inside the hall is 14 m, whereas the maximum distance between two points (say P and R) on the floor is 656\sqrt{5} m.
    61
    What is h equal to ?

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  62. 62
    If α\alpha is the angle between the line joining P and Q, and the line joining P and R, then what is cos⁡α\cos\alpha equal to ?

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  63. The sides of an open box are 0.5 cm thick and bottom is 1 cm thick. The internal length, breadth and depth are respectively 14 cm, 10 cm and 8 cm. It is completely filled with water.
    63
    If the material weighs 2000 kg per cubic metre, then what is the weight of the material used in the construction of the box ?

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  64. 64
    If water weighs 1000 kg per cubic metre, then what is the weight of the box with water ?

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  65. ABC is a triangle right-angled at A. Further, AB = 8 cm, BC = 10 cm. D is the point on BC such that AD is perpendicular to BC.
    65
    What is AD equal to ?

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  66. 66
    What is ratio of area of triangle ADC to area of triangle ADB ?

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  67. The annual rainfall at a place is 40 cm. The weight of water is 1 metric tonne per cubic meter.
    67
    What is the volume of rainfall in cubic meter per hectare ?

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  68. 68
    What is the weight of water (in metric tonnes) of annual rainfall falling there on a hectare of land ?

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  69. The angle at the vertex of a conical body is 120∘120^\circ.
    69
    What is the ratio of the radius of the conical body to its slant height ?

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  70. 70
    If the sum of slant height, height and radius is (9+33)(9 + 3\sqrt{3}) cm, then what is the volume of the cone ?

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  71. 71
    A person sells article X for Rs. 34,500 and makes a profit of 15%. He sells article Y at a loss of 10%. He neither loses nor gains on the whole because of these two transactions. What is the selling price of article Y ?

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  72. 72
    100 quintals is what percent of 10 metric tonnes ?

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  73. 73
    A circle is inscribed in an equilateral triangle. The radius of the circle is 2 cm. What is the area of the triangle ?

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  74. 74
    The sides of a triangle are k, 1.5k and 2.25k. What is the sum of the squares of its medians ?

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  75. 75
    If 2s=a+b+c2s = a + b + c, then what is s (s−a) (s−b) (s−c)[1s−a+1s−b+1s−c−1s]s\,(s - a)\,(s - b)\,(s - c)\left[\frac{1}{s - a} + \frac{1}{s - b} + \frac{1}{s - c} - \frac{1}{s}\right] equal to ?

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  76. 76
    How much will Rs. 10,000 amount to in one year's time at 4% rate of interest per annum if the interest is compounded once in every three months ? (take approximate value)

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  77. 77
    If p=0.09‾p = 0.0\overline{9}, then what is the value of 70p2+43p−570p^2 + 43p - 5 ?

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  78. 78
    What is the remainder when 21012^{101} is divided by 101 ?

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  79. 79
    If p (≠0)p\,(\ne 0) and q (≠0)q\,(\ne 0) are the roots of the equation x2+px+q=0x^2 + px + q = 0, then what is p2+q2p^2 + q^2 equal to ?

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  80. 80
    The equations x2+px+q=0x^2 + px + q = 0 and x2+qx+p=0x^2 + qx + p = 0 (p≠q)(p \ne q) have a common root. What is the value of (p+q)(p + q) ?

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  81. 81
    If x2−5x+4x^2 - 5x + 4 is a factor of x4−px2+qx^4 - px^2 + q, then what are the values of p and q respectively ?

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  82. 82
    If two quadratic equations px2+px+4=0px^2 + px + 4 = 0 and x2+qx+q=0x^2 + qx + q = 0 have a common root 2, then what is p+qp + q equal to ?

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  83. 83
    What is the HCF of the polynomials x8+x4+1x^8 + x^4 + 1 and x4+x2+1x^4 + x^2 + 1 ?

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  84. 84
    An arc AB of a circle subtends an angle x radian at the centre O. If the area of the sector AOB is equal to half of the square of length of arc AB, then what is x equal to ?

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  85. 85
    Consider the following statements in respect of prime numbers p and q : I. Their LCM is always an odd number. II. Sum of their LCM and HCF is always an even number. Which of the statements given above is/are correct ?

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  86. Consider the following for the next five (05) items that follow : The frequency distribution of 205 observations on X is given below :
    X3567
    Frequencyff + 2f - 3f + 6
    86
    What is the value of f ?

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  87. 87
    What is the median of the frequency distribution ?

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  88. 88
    What is the mode of the frequency distribution ?

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  89. 89
    What is the most appropriate graphical representation of the given frequency distribution of X ?

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  90. 90
    What is the mean of the frequency distribution ?

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  91. Consider the following for the next two (02) items that follow : The frequency distribution of marks obtained by students in an English examination is given below :
    Marks obtainedNumber of Students
    Below 4050
    Below 50125
    Below 60210
    Below 70315
    Below 80350
    91
    What is the number of students who scored between 60 and 70 marks ?

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  92. 92
    What is the number of students who scored more than 50 marks ?

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  93. Consider the following for the next three (03) items that follow : Circulation figures (in thousands) of different newspapers (A, B, C, D, E) for five years are given below :
    YearABCDE
    2019201015820
    20201212181212
    20212414171415
    2022261016159
    20232216141611
    93
    In which of the given years was the circulation of the newspaper D close to its average circulation over all the years ?

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  94. 94
    In which of the years from 2019 to 2022 was the circulation of the newspaper D close to the average circulation of all the newspapers in that year ?

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  95. 95
    How many cases are there in which average of the circulation for an individual newspaper was more than the average of the circulation of all the newspapers ?

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  96. Consider the following for the next three (03) items that follow : Study the following Pie Charts : Chart-I : Number of houses : 1,20,000 Chart-II : Population : 21,00,000 Chart-III : Number of families : 3,60,000
    96
    On an average, how many persons live in every pucca house ?

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  97. 97
    If 5000 more Kutcha houses are built, then what will be approximate change in angle for Kutcha houses in Pie Chart-I ?

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  98. 98
    If 300 families from the “Houseless” category shift into Kutcha houses, what will be the average number of families in every Kutcha house ?

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  99. 99
    Marks obtained by 60 students in a test (Maximum Marks-100) are given below : 60, 82, 63, 40, 88, 80, 24, 60, 09, 58, 44, 74 57, 60, 63, 30, 31, 64, 00, 62, 37, 54, 10, 99 35, 83, 58, 33, 92, 43, 66, 41, 78, 72, 32, 33 65, 11, 60, 17, 53, 93, 57, 03, 29, 31, 82, 50 63, 15, 41, 55, 72, 42, 32, 23, 30, 48, 67, 34 What is the percentage (approximate) of students getting marks higher than or equal to 60 but less than 80 ?

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  100. 100
    The arithmetic mean of 100 observations was 50. Later on, it was found that two observations were misread as 82 and 6 instead of 182 and 56. What is the value of the corrected mean ?

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