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CDS (II) 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
    Let p(x)p(x) be a polynomial. When p(x)p(x) is divided by (x−1)(x-1), it leaves 2 as the remainder. When p(x)p(x) is divided by (x−2)(x-2), it leaves 1 as the remainder. What is the remainder when p(x)p(x) is divided by (x−1)(x−2)(x-1)(x-2)?

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  2. 2
    Consider the following in respect of a positive real number xx : I. x+1x>1x+\frac{1}{x}>1 II. (x+1x)2>2\left(x+\frac{1}{x}\right)^2>2 III. (x+1x)4>9\left(x+\frac{1}{x}\right)^4>9 Which of the above are correct?

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  3. 3
    Let pp and qq be natural numbers such that q>pq>p. What is the largest value of pp such that q2−5p−4q^2-5p-4 is negative?

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  4. 4
    Let xx and yy be natural numbers, each less than 20, such that xx, yy, x+yx+y and x−yx-y are prime numbers. How many such combinations of (x, y, x+y, x−y)(x,\ y,\ x+y,\ x-y) are possible?

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  5. 5
    If (x+1)(x+p)(x2+p2)=x4−1(x+1)(x+p)(x^2+p^2)=x^4-1, then what is the value of pp?

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  6. 6
    If (2+3)x+(2−3)x=2(2+\sqrt{3})^x+(2-\sqrt{3})^x=2, then what is (2+3)x−(2−3)x(2+\sqrt{3})^x-(2-\sqrt{3})^x equal to?

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  7. 7
    If 1a+1b=56\frac{1}{a}+\frac{1}{b}=\frac{5}{6} and 1a2+1b2=1336\frac{1}{a^2}+\frac{1}{b^2}=\frac{13}{36}, then what is 1a3+1b3\frac{1}{a^3}+\frac{1}{b^3} equal to?

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  8. 8
    What is the remainder when x6x^6 is divided by x2+1x^2+1?

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  9. 9
    (x+2)(x+2) is a factor of which one of the following?

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  10. 10
    If log⁡102=0.301\log_{10}2=0.301 and log⁡103=0.477\log_{10}3=0.477, then what is the number of digits in the expansion of 606060^{60}?

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  11. 11
    What is the remainder when (1725+1925)(17^{25}+19^{25}) is divided by 18?

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  12. 12
    The HCF of xx and yy is HH. Consider the following statements in respect of the HCF of p=x3+y3x2−xy+y2p=\frac{x^3+y^3}{x^2-xy+y^2} and q=x3−y3x2+xy+y2q=\frac{x^3-y^3}{x^2+xy+y^2} : I. The HCF of pp and qq can be HH. II. The HCF of pp and qq can be 2H2H. Which of the statements given above is/are correct?

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  13. 13
    If x4=x2+1x^4=x^2+1, where x>0x>0, then what is 2x42x^4 equal to?

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  14. 14
    If p+qq+r=r+ss+p\frac{p+q}{q+r}=\frac{r+s}{s+p}; (q+r)≠0(q+r)\ne 0, (s+p)≠0(s+p)\ne 0, then which one of the following is correct?

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  15. 15
    If nn is natural number less than 7, then what is the number of values of nn for which (12n+2)(12n+2) and (8n+1)(8n+1) are relatively prime?

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  16. 16
    What is the HCF of x3+y3+3xy−1x^3+y^3+3xy-1 and (x+y)4−1(x+y)^4-1?

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  17. 17
    Let x=n(n+1)(n+2)x=n(n+1)(n+2), where nn is an even natural number. Which of the following statements is/are correct? I. xx is always divisible by 48. II. x2x^2 is always divisible by 144. Select the answer using the code given below.

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  18. 18
    What is the LCM of x4+x2y2+y4x^4+x^2y^2+y^4, x3y+y4x^3y+y^4 and x4y2−x3y3x^4y^2-x^3y^3?

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  19. 19
    Let XYZXYZ be a 3-digit number. Let DD be the difference between XYZXYZ and ZYXZYX. What is the remainder when DD is divided by 99?

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  20. 20
    Let pp and qq be two natural numbers such that (p+q)p+q(p+q)^{p+q} is divisible by 512. What is the least value of (p+q)(p+q)?

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  21. 21
    What is (a+b)2(c−a)(c+a+b)+(a+b)cc2+bc−a2−ab−(a+2b+c)2(c−a)\frac{(a+b)^2}{(c-a)(c+a+b)}+\frac{(a+b)c}{c^2+bc-a^2-ab}-\frac{(a+2b+c)}{2(c-a)}, a≠ba\ne b, b≠cb\ne c, c≠ac\ne a equal to?

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  22. 22
    If ab=baa^b=b^a, then what is a×(ab)aba(ab)\frac{a\times\left(\frac{a}{b}\right)^{\frac{a}{b}}}{a^{\left(\frac{a}{b}\right)}} equal to?

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  23. 23
    If x=2+21/2+23/2x=2+2^{1/2}+2^{3/2}, then what is x2−4x−10x^2-4x-10 equal to?

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  24. 24
    If p+x+p−xp+x−p−x=p\frac{\sqrt{p+x}+\sqrt{p-x}}{\sqrt{p+x}-\sqrt{p-x}}=p, then what is xx equal to?

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  25. 25
    If (a−b2)x2−(a+b2)x+b=0\left(\frac{a-b}{2}\right)x^2-\left(\frac{a+b}{2}\right)x+b=0, then what are the roots of this equation?

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  26. 26
    If x−1x=2x-\frac{1}{x}=2, x>0x>0; then what is x2−1x2x^2-\frac{1}{x^2} equal to?

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  27. 27
    If (a−b)2+(b−c)2+(c−a)2=6(a-b)^2+(b-c)^2+(c-a)^2=6 and a2+b2+c2=29a^2+b^2+c^2=29, then what is (a+b+c)(a+b+c) equal to?

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  28. 28
    If p=5−25+2p=\frac{\sqrt{5}-2}{\sqrt{5}+2} and q=5+25−2q=\frac{\sqrt{5}+2}{\sqrt{5}-2}, then what is (pq+qp)\left(\frac{p}{q}+\frac{q}{p}\right) equal to?

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  29. 29
    What is the digit at hundreds place of the number (25)10(25)^{10}?

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  30. 30
    A number NN is such that when divided by 4, 6, 7 or 9, it leaves 3 as remainder. What is the smallest 4-digit number that satisfies this property?

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  31. 31
    If 2+2+2+2+…=cosec⁡θ\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\ldots}}}}=\operatorname{cosec}\theta, then what is sin⁡θ\sin\theta equal to?

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  32. 32
    If 8sin⁡θ−cos⁡θ=48\sin\theta - \cos\theta = 4, where 0<θ<π/20 < \theta < \pi/2, then what is cosec θ\mathrm{cosec}\,\theta equal to?

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  33. 33
    If 2tan⁡θ=sec⁡2θ−22\tan\theta = \sec^2\theta - 2, where 0<θ<π/20 < \theta < \pi/2, then what is cot⁡θ\cot\theta equal to?

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  34. 34
    What is (sec⁡θ−tan⁡θ)−1−sin⁡θ1+sin⁡θ(\sec\theta - \tan\theta) - \sqrt{\dfrac{1 - \sin\theta}{1 + \sin\theta}} equal to?

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  35. 35
    If cot⁡θ=7\cot\theta = \sqrt{7}, then what is cosec2θ−sec⁡2θcosec2θ+sec⁡2θ\dfrac{\mathrm{cosec}^2\theta - \sec^2\theta}{\mathrm{cosec}^2\theta + \sec^2\theta} equal to?

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  36. 36
    The difference between the two acute angles in a right-angled triangle is π12\dfrac{\pi}{12} radian. One of the acute angles of the triangle is

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  37. 37
    If α\alpha and β\beta are the roots of the equation log⁡10[998+x2−18x+76]=3\log_{10}\left[998 + \sqrt{x^2 - 18x + 76}\right] = 3 then what is (α−β)2(\alpha - \beta)^2 equal to?

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  38. 38
    If x4+y4=14x2y2x^4 + y^4 = 14x^2y^2, then consider the following : I. log⁡10(x2+y2)=log⁡10x+log⁡10y+2log⁡102\log_{10}(x^2 + y^2) = \log_{10} x + \log_{10} y + 2\log_{10} 2 II. log⁡10(x2−y2)=log⁡10x+log⁡10y+log⁡102+0.5log⁡103\log_{10}(x^2 - y^2) = \log_{10} x + \log_{10} y + \log_{10} 2 + 0.5\log_{10} 3 Which of the above is/are correct?

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  39. 39
    Which of the following is/are the factor(s) of (3x+y)2+(3x+y)(x+5y)−20(x+5y)2(3x + y)^2 + (3x + y)(x + 5y) - 20(x + 5y)^2? I. (4x+13y)(4x + 13y) II. (x+19y)(x + 19y) Select the correct answer using the code given below.

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  40. 40
    What is xx−y+yy−z+zz−xx+yx−y+y+zy−z+z+xz−x+3\cfrac{\frac{x}{x-y} + \frac{y}{y-z} + \frac{z}{z-x}}{\frac{x+y}{x-y} + \frac{y+z}{y-z} + \frac{z+x}{z-x} + 3} equal to?

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  41. Let psin⁡2α+qcos⁡2α=mp\sin^2\alpha + q\cos^2\alpha = m, qsin⁡2β+pcos⁡2β=nq\sin^2\beta + p\cos^2\beta = n; p≠m,np \ne m, n and q≠m,nq \ne m, n.
    41
    What is (tan⁡αtan⁡β)2\left(\dfrac{\tan\alpha}{\tan\beta}\right)^2 equal to?

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  42. 42
    If α\alpha and β\beta are complementary angles, then which one of the following is correct?

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  43. Let cosec θ−sin⁡θ=p\mathrm{cosec}\,\theta - \sin\theta = p and sec⁡θ−cos⁡θ=q\sec\theta - \cos\theta = q.
    43
    What is (psin⁡θ+qcos⁡θ)(p\sin\theta + q\cos\theta) equal to?

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  44. 44
    What is p2q2(p2+q2+3)p^2q^2(p^2 + q^2 + 3) equal to?

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  45. Let sin⁡αsin⁡β=423\dfrac{\sin\alpha}{\sin\beta} = \dfrac{4\sqrt{2}}{3} and cos⁡αcos⁡β=239\dfrac{\cos\alpha}{\cos\beta} = \dfrac{2\sqrt{3}}{9}.
    45
    What is tan⁡2α\tan^2\alpha equal to?

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  46. 46
    What is tan⁡2β\tan^2\beta equal to?

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  47. Let 1+sin⁡θcos⁡θ=p+p2+1\dfrac{1 + \sin\theta}{\cos\theta} = p + \sqrt{p^2 + 1}.
    47
    What is sec⁡θ\sec\theta equal to?

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  48. 48
    What is tan⁡θ\tan\theta equal to?

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  49. Let sin⁡θ+cos⁡θ=p\sin\theta + \cos\theta = p and sec⁡θ+cosec θ=q\sec\theta + \mathrm{cosec}\,\theta = q, where p≠1p \ne 1.
    49
    What is the relation between pp and qq?

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  50. 50
    What is tan⁡θ+cot⁡θ\tan\theta + \cot\theta equal to?

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  51. 51
    Two trains XX and YY are travelling in the same direction at 100 km/hr and 60 km/hr respectively. Train XX crosses a man in train YY in 9 seconds. What is the length of train XX?

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  52. 52
    Two persons XX and YY leave place PP for place QQ at 7:00 a.m. and 7:10 a.m. respectively along the same path. XX walks at a speed of 4.8 km/hr and YY walks at a speed of 6 km/hr. How many kilometres from place PP will XX meet YY?

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  53. 53
    There are two employees XX and YY. XX's salary is first increased by 12% and then decreased by 10%, and YY's salary is first increased by 10% and then decreased by 12%. If their salaries at present are equal, then what was the ratio of initial salary of XX to initial salary of YY?

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  54. 54
    In a village consisting of pp persons, x%x\% can read and write. Of the males, only y%y\% can read and write. Of the females, only z%z\% can read and write. If x,y>zx, y > z, then what is the number of males in the village?

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  55. 55
    XX and YY are two alloys of copper (Cu) and zinc (Zn). Alloy XX is prepared by mixing Cu and Zn in the ratio 5:45 : 4, and alloy YY is prepared by mixing Cu and Zn in the ratio 5:135 : 13 respectively. If equal quantities of alloys XX and YY are melted to form a third alloy ZZ, then what is the ratio of Cu to Zn in ZZ ?

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  56. 56
    Two students XX and YY appeared in a test. The score of XX is 20 more than that of YY. If the score of XX is 75%75\% of the sum of the scores of XX and YY, then what is the ratio of score of XX to score of YY ?

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  57. 57
    If one root of the equation 2x2−5px+2p2=02x^2 - 5px + 2p^2 = 0 exceeds the other by 4, then what is the value of pp ?

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  58. 58
    An amount of Rs. 10,000 is borrowed at 10%10\% per annum on compound interest for 3 years, compounded annually, and paid back in 3 equal annual installments during these years. What is the amount of each installment (approximately)?

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  59. 59
    What is the solution of the inequalities 5x+3<8x−95x + 3 < 8x - 9 and 2x+20>5x+22x + 20 > 5x + 2 ?

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  60. 60
    A shopkeeper gives three consecutive discounts 10%10\%, 20%20\% and 25%25\% after which he sells the article at a profit of 8%8\% on the cost price. Had he sold the article after the first discount, how much profit would he have got?

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  61. 61
    Let ADAD be the altitude of a triangle ABCABC. If (AB+AC)=p(AB + AC) = p, (AB−AC)=q(AB - AC) = q and (BD−CD)=r(BD - CD) = r, then what is BCBC equal to?

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  62. 62
    The sum of the height and the radius of a right circular cylinder is 21 cm, and the radius is less than the height. If the curved surface area of the cylinder is 616 cm2616\ \text{cm}^2, then what is the volume of the cylinder? (Take π=22/7\pi = 22/7)

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  63. 63
    ABCABC is a triangle right angled at BB. PP is the midpoint of ABAB and QQ is the midpoint of BCBC. Consider the following : I. AQ=73AQ = \sqrt{73} units II. CP=52CP = \sqrt{52} units Which of the above is/are required to determine the area of the triangle?

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  64. 64
    The length, breadth and height of a cuboid are consecutive integers. If the volume of the cuboid is 336 cubic units, then what is the total surface area of the cuboid?

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  65. 65
    In a circle of radius 14 cm, APBAPB is a shorter arc and PP is the midpoint of the arc. Let CC be the midpoint of the chord ABAB and PC=7PC = 7 cm. What is the length of the chord APAP ?

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  66. 66
    Two poles of heights 10 m and 15 m are 25 m apart. What is the height of the point of intersection of the lines joining the tip of each pole to the foot of the other pole?

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  67. 67
    ABCABC is a triangle right angled at BB. Further, (AB+BC)(AB + BC) exceeds ACAC by 10 units. If the perimeter of the triangle is 60 units, then what is the area of the triangle?

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  68. 68
    Two poles are situated 24 m apart and their heights differ by 10 m. What is the distance between their tips?

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  69. 69
    Let XX, YY and ZZ be the midpoints of the sides BCBC, CACA and ABAB of a triangle ABCABC respectively. Consider the following statements : I. The quadrilateral AZXYAZXY is a parallelogram. II. The area of the quadrilateral AZXYAZXY is half of the area of the triangle ABCABC. Which of the statements given above is/are correct?

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  70. 70
    Consider the following angles : I. 4∘4^\circ II. 5∘5^\circ III. 6∘6^\circ IV. 8∘8^\circ How many of the above can be the exterior angle of a regular polygon?

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  71. 71
    In a triangle ABCABC, ∠A=30∘\angle A = 30^\circ, AB=7AB = 7 cm and AC=12AC = 12 cm. What is the area of the triangle ABCABC?

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  72. 72
    ABCABC is a triangle right angled at BB. DD is a point on ACAC such that BDBD is perpendicular to ACAC. If AB=pAB = p and BC=3pBC = \sqrt{3}p, then what is BDBD equal to?

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  73. 73
    The difference between an interior angle and an exterior angle of a regular polygon is 120∘120^\circ. What is the number of sides of the polygon?

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  74. 74
    An angle θ\theta is exactly one-fourth of its complementary angle. What is the value of angle θ\theta ?

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  75. 75
    The sides of a triangle are 11 cm, 60 cm and 61 cm. What is the area of the triangle formed by joining the mid-points of the sides of the triangle?

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  76. 76
    A pendulum swings through an angle of 30∘30^\circ and its end describes an arc of length 55 cm. What is the length of the pendulum? (Take π=22/7\pi = 22/7)

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  77. 77
    A conical tent has an angle of 60∘60^\circ at the vertex. If the curved surface area is 100100 m2^2, then what is the volume of the tent?

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  78. 78
    A right circular cone and a hemisphere have equal base and equal volume. What is the ratio of the height of the cone to the radius of the hemisphere?

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  79. 79
    A wire is in the form of an equilateral triangle with an area of 36336\sqrt{3} cm2^2. If it is changed into a semicircle, then what is its radius?

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  80. 80
    Let the area of the largest possible square inscribed in a circle of unit radius be xx. Let the area of the largest possible circle inscribed in a square of unit side length be yy. What is the relation between xx and yy?

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  81. 81
    A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question : What is the remainder when x2n−y2n+1x^{2n} - y^{2n} + 1 is divided by xn+ynx^n + y^n, where nn is a natural number? Statement-I : nn is odd. Statement-II : nn is even. Which one of the following is correct in respect of the above Question and the Statements?

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  82. 82
    A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question : The product of a natural number NN and the number MM written by the same digits of NN in the reverse order is 252. What is the number NN? Statement-I : N+M=33N + M = 33 Statement-II : N>MN > M Which one of the following is correct in respect of the above Question and the Statements?

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  83. 83
    A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question : The last digit in the expansion of the number (54D)100(54D)^{100} is 1. What is the value of the digit DD? Statement-I : D>5D > 5 Statement-II : DD is a multiple of 3. Which one of the following is correct in respect of the above Question and the Statements?

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  84. 84
    A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question : In a triangle ABCABC, ∠A=∠B−∠C\angle A = \angle B - \angle C. Is angle AA acute? Statement-I : ABCABC is not an obtuse-angled triangle. Statement-II : Angle CC is acute. Which one of the following is correct in respect of the above Question and the Statements?

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  85. 85
    A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question : In a triangle ABCABC right angled at BB, AC=20AC = 20 cm. What is the circumradius of the triangle? Statement-I : AB=12AB = 12 cm Statement-II : BC=16BC = 16 cm Which one of the following is correct in respect of the above Question and the Statements?

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  86. 86
    A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question : ABCDABCD is a parallelogram with ∠ABC=60∘\angle ABC = 60^\circ. If the area of the parallelogram is 737\sqrt{3} square units, then what is the perimeter of the parallelogram? Statement-I : The lengths of the sides ABAB and DADA are prime numbers. Statement-II : The lengths of the sides are natural numbers each greater than 1 unit. Which one of the following is correct in respect of the above Question and the Statements?

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  87. 87
    A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question : AB and CD are chords of a circle intersecting at P. If AP×PB=48AP \times PB = 48 square units, then what is CP×PDCP \times PD equal to? Statement-I : AP=8AP = 8 units Statement-II : CP=10CP = 10 units Which one of the following is correct in respect of the above Question and the Statements?

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  88. 88
    A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question : In a quadrilateral ABCD, AB=6AB = 6 units, BC=18BC = 18 units, CD=6CD = 6 units, DA=9DA = 9 units. What is the length of diagonal BD? Statement-I : The length of BD is an integer greater than 13. Statement-II : The length of BD is an even integer. Which one of the following is correct in respect of the above Question and the Statements?

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  89. 89
    A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question : ABC is an isosceles triangle with AB=AC=10AB = AC = 10 units. If the area of the triangle is 48 square units, then what is the length of the base BC? Statement-I : The length of BC is an even integer. Statement-II : The height of the triangle is greater than the length of half of the base. Which one of the following is correct in respect of the above Question and the Statements?

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  90. 90
    A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question : The diagonals of a rhombus ABCD are in the ratio 5:125 : 12. Is one of the diagonals equal to side of the rhombus? Statement-I : The sum of the diagonals =34= 34 cm. Statement-II : The length of a side =13= 13 cm. Which one of the following is correct in respect of the above Question and the Statements?

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  91. 91
    Which measure of central tendency is least affected by the presence of extreme observations in the data?

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  92. 92
    To find the average ratio like price/unit, work done/hour, kilometre/hour under certain conditions, the suitable measure of central tendency applicable is

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  93. 93
    The frequency distribution of marks of 100 candidates in a particular examination is as follows :
    MarksNumber of Candidates
    More than 10100
    More than 2075
    More than 3060
    More than 4040
    What are the average marks of the candidates?

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  94. 94
    The arithmetic mean of 200 observations is 60. If 5 is multiplied to each observation, then what will be the new arithmetic mean?

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  95. 95
    A distribution consists of 3 components with frequencies 45, 40 and 55 having their means 2, 2.5 and 2 respectively. What is the mean of the combined distribution?

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  96. 96
    Which one of the following is a positional average?

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  97. For the following two (02) items : The following data represent the distance covered (in metres) by two groups of athletic children. It is known that the median distance in the first group is 20.8 metres while the mean distance in the second group is 17.3 metres. Some frequencies in both the groups are missing :
    Distance ClassFirst GroupSecond Group
    0-5uu3u3u
    5-10vv2v2v
    10-151140
    15-205250
    20-257530
    25-302228
    97
    What is the value of uu?

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  98. 98
    What is the value of vv?

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  99. For the following two (02) items : Consider the following distribution having median value 24 :
    MarksNumber of Students
    Less than 105
    Less than 2030
    Less than 3030+k30 + k
    Less than 4048+k48 + k
    Less than 5055+k55 + k
    99
    What is the value of kk?

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  100. 100
    What is the mean of the distribution?

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