PYQ Vault

NDA 2018 (I), Paper I, Mathematics question paper

120 questions · 300 marks · 150 minutes · +2.5 per correct answer, -0.83 per wrong one

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  1. 1
    If n∈Nn \in N, then 121n−25n+1900n−(−4)n121^{n} - 25^{n} + 1900^{n} - (-4)^{n} is divisible by which one of the following?

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  2. 2
    If n=(2017)!n = (2017)!, then what is 1log⁡2n+1log⁡3n+1log⁡4n+⋯+1log⁡2017n\frac{1}{\log_{2}n} + \frac{1}{\log_{3}n} + \frac{1}{\log_{4}n} + \cdots + \frac{1}{\log_{2017}n} equal to?

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  3. 3
    In the expansion of (1+x)43(1+x)^{43}, if the coefficients of (2r+1)th(2r+1)^{\text{th}} and (r+2)th(r+2)^{\text{th}} terms are equal, then what is the value of rr (r≠1)(r \ne 1)?

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  4. 4
    What is the principal argument of (−1−i)(-1-i), where i=−1i = \sqrt{-1}?

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  5. 5
    Let α\alpha and β\beta be real numbers and zz be a complex number. If z2+αz+β=0z^2 + \alpha z + \beta = 0 has two distinct non-real roots with Re(z)=1\text{Re}(z) = 1, then it is necessary that

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  6. 6
    Let A and B be subsets of X and C=(A∩B′)∪(A′∩B)C = (A \cap B') \cup (A' \cap B), where A' and B' are complements of A and B in X. What is C equal to?

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  7. 7
    How many numbers between 100 and 1000 can be formed with digits 5, 6, 7, 8, 9 if repetition is not allowed?

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  8. 8
    The number of non-zero integral solutions of the equation ∣1−2i∣x=5x|1-2i|^x = 5^x is

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  9. 9
    If the ratio of AM to GM of two positive numbers a and b is 5:35:3, then a:ba:b is equal to

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  10. 10
    If the coefficients of ama^m and ana^n in the expansion of (1+a)m+n(1+a)^{m+n} are α\alpha and β\beta, then which one is correct?

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  11. 11
    If x+log⁡15(1+3x)=xlog⁡155+log⁡1512x + \log_{15}(1+3^x) = x\log_{15}5 + \log_{15}12, where x is an integer, then what is x equal to?

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  12. 12
    How many four-digit numbers divisible by 10 can be formed using 1, 5, 0, 6, 7 without repetition?

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  13. Consider the information given below and answer the two items (02) that follow: In a class, 54 students are good in Hindi only, 63 students are good in Mathematics only and 41 students are good in English only. There are 18 students who are good in both Hindi and Mathematics. 10 students are good in all three subjects.
    13
    What is the number of students who are good in either Hindi or Mathematics but not in English?

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  14. 14
    What is the number of students who are good in Hindi and Mathematics but not in English?

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  15. 15
    If α\alpha and β\beta are different complex numbers with ∣α∣=1|\alpha| = 1, then what is ∣α−β1−αβˉ∣\left|\frac{\alpha - \beta}{1 - \alpha\bar{\beta}}\right| equal to?

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  16. 16
    The equation ∣1−x∣+x2=5|1-x| + x^2 = 5 has

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  17. 17
    The binary number expression of the decimal number 31 is

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  18. 18
    What is i1000+i1001+i1002+i1003i^{1000} + i^{1001} + i^{1002} + i^{1003} equal to (where i=−1i = \sqrt{-1})?

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  19. 19
    What is 1log⁡2N+1log⁡3N+⋯+1log⁡100N\frac{1}{\log_{2}N} + \frac{1}{\log_{3}N} + \cdots + \frac{1}{\log_{100}N} equal to (N≠1)(N \ne 1)?

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  20. 20
    The modulus-amplitude form of 3+i\sqrt{3}+i, where i=−1i = \sqrt{-1} is

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  21. 21
    What is the number of non-zero terms in (1+23 x)11+(1−23 x)11(1+2\sqrt{3}\,x)^{11} + (1-2\sqrt{3}\,x)^{11} after simplification?

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  22. 22
    What is the greatest integer among the following by which 55+755^5 + 7^5 is divisible?

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  23. 23
    If x=1−y+y2−y3+…x = 1 - y + y^2 - y^3 + \ldots where ∣y∣<1|y| < 1, then which is correct?

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  24. 24
    What is the inverse of the matrix A=(cos⁡θsin⁡θ0−sin⁡θcos⁡θ0001)A = \begin{pmatrix}\cos\theta & \sin\theta & 0 \\ -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1\end{pmatrix}?

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  25. 25
    If A is a 2×32 \times 3 matrix and AB is a 2×52 \times 5 matrix, then B must be a

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  26. 26
    If A=(1223)A = \begin{pmatrix}1 & 2 \\ 2 & 3\end{pmatrix} and A2−kA−I2=OA^2 - kA - I_2 = O, then what is k?

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  27. 27
    What is the number of triangles that can be formed by choosing vertices from 12 points, 7 of which lie on the same line?

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  28. 28
    What is C(n,r)+2C(n,r−1)+C(n,r−2)C(n,r) + 2C(n,r-1) + C(n,r-2) equal to?

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  29. 29
    Let [x] denote the greatest integer function. What is the number of solutions of x2−4x+[x]=0x^2 - 4x + [x] = 0 in [0,2][0, 2]?

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  30. 30
    A survey of 850 students yields 680 like music and 215 like dance. What is the least number who like both?

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  31. 31
    What is the sum of all two-digit numbers which when divided by 3 leave 2 as remainder?

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  32. 32
    If 0<a<10 < a < 1, the value of log⁡10a\log_{10}a is negative. This is justified by

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  33. 33
    The third term of a GP is 3. What is the product of the first five terms?

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  34. 34
    If x, 3/2, z are in AP; x, 3, z are in GP; then which is in HP?

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  35. 35
    What is ∑n=211(in+in+1)\sum_{n=2}^{11}(i^n + i^{n+1}), where i=−1i = \sqrt{-1}?

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  36. 36
    If sin⁡x=15\sin x = \frac{1}{\sqrt{5}}, sin⁡y=110\sin y = \frac{1}{\sqrt{10}}, where 0<x,y<π20 < x, y < \frac{\pi}{2}, then what is (x+y)(x+y) equal to?

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  37. 37
    What is sin⁡5x−sin⁡3xcos⁡5x+cos⁡3x\frac{\sin 5x - \sin 3x}{\cos 5x + \cos 3x} equal to?

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  38. 38
    What is sin⁡105°+cos⁡105°\sin 105° + \cos 105° equal to?

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  39. 39
    In triangle ABC if a=2a=2, b=3b=3 and sin⁡A=2/3\sin A = 2/3, then what is angle B?

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  40. 40
    What is the principal value of sin⁡−1 ⁣(sin⁡2π3)\sin^{-1}\!\left(\sin\frac{2\pi}{3}\right)?

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  41. 41
    If x, x−yx-y and x+yx+y are angles of a triangle (not equilateral) such that tan⁡(x−y)\tan(x-y), tan⁡x\tan x, tan⁡(x+y)\tan(x+y) are in GP, then x equals?

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  42. 42
    ABC is inscribed in a circle with centre O. Let α=∠BAC\alpha = \angle BAC, β=∠BOC\beta = \angle BOC. Which is correct?

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  43. 43
    A flag-staff of 6 m on top of a tower throws shadow of 232\sqrt{3} m along the ground. What is the angle the sun makes with the ground?

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  44. 44
    What is tan⁡−1 ⁣(14)+tan⁡−1 ⁣(35)\tan^{-1}\!\left(\frac{1}{4}\right) + \tan^{-1}\!\left(\frac{3}{5}\right) equal to?

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  45. 45
    A spherical balloon of radius r subtends angle α\alpha at observer's eye, elevation of centre is β\beta. Height of centre is?

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  46. 46
    If sin⁡(x+y)sin⁡(x−y)=a+ba−b\frac{\sin(x+y)}{\sin(x-y)} = \frac{a+b}{a-b}, then what is tan⁡xtan⁡y\frac{\tan x}{\tan y} equal to?

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  47. 47
    If sin⁡α+sin⁡β=0=cos⁡α+cos⁡β\sin\alpha + \sin\beta = 0 = \cos\alpha + \cos\beta, where 0<β<α<2π0 < \beta < \alpha < 2\pi, then which is correct?

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  48. 48
    Suppose cos⁡A\cos A is given. If only one value of cos⁡(A/2)\cos(A/2) is possible, then A must be

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  49. 49
    If cos⁡α+cos⁡β+cos⁡γ=0\cos\alpha + \cos\beta + \cos\gamma = 0, where 0<α,β,γ≤π/20 < \alpha, \beta, \gamma \le \pi/2, then what is sin⁡α+sin⁡β+sin⁡γ\sin\alpha + \sin\beta + \sin\gamma?

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  50. 50
    The maximum value of sin⁡(x+π/5)+cos⁡(x+π/5)\sin(x+\pi/5) + \cos(x+\pi/5), where x∈(0,π/2)x \in (0, \pi/2), is attained at

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  51. 51
    What is the distance between points that divide (4,3)-(5,7) internally and externally in ratio 2:3?

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  52. 52
    What is the angle between (m2−mn)y=(mn+n2)x+n3(m^2-mn)y=(mn+n^2)x+n^3 and (mn+m2)y=(mn−n2)x+m3(mn+m^2)y=(mn-n^2)x+m^3, where m>nm>n?

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  53. 53
    What is the equation of the line cutting intercept 2 from negative y-axis direction, inclined at 30 degrees with positive x-axis?

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  54. 54
    What is the equation of the line through the intersection of x+2y−3=0x+2y-3=0 and 2x−y+5=02x-y+5=0, parallel to y−x+10=0y-x+10=0?

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  55. 55
    Consider the following statements about perpendicular length p from origin: 1. To line ax+by=c: p2=c2/(a2+b2)p^2=c^2/(a^2+b^2) 2. To line x/a+y/b=1: 1/p2=1/a2+1/b21/p^2=1/a^2+1/b^2 3. To line y=mx+c: 1/p2=(1+m2+c2)/c21/p^2=(1+m^2+c^2)/c^2 Which is/are correct?

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  56. 56
    What is the equation of the ellipse with vertices (plus/minus 5, 0) and foci at (plus/minus 4, 0)?

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  57. 57
    What is the equation of the line through (2,3) with y-intercept equal to twice its x-intercept?

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  58. 58
    Let A=(1,8,4), B=(0,-11,4), C=(2,-3,1). What are the coordinates of D, foot of perpendicular from A to BC?

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  59. 59
    What is the equation of the plane through (-2,6,-6), (-3,10,-9) and (-5,0,-6)?

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  60. 60
    A sphere of radius r through origin cuts axes at A,B,C. What is the locus of centroid of triangle ABC?

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  61. 61
    Vertices P=(1,-1,1), Q=(3,-2,2), R=(0,2,6). If angle RQP = theta, then angle PRQ equals?

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  62. 62
    Perpendiculars from any point on 2x+11y=5 to lines 24x+7y=20 and 4x-3y=2 are

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  63. 63
    Line whose intercept between axes is divided by (2,3) in ratio 3:2 has equation:

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  64. 64
    What is the distance between 3x+4y=93x+4y=9 and 6x+8y=156x+8y=15?

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  65. 65
    What is the equation of the sphere with centre (-2,3,4) and radius 6?

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  66. 66
    If ∣a⃗∣=2|\vec{a}|=2, ∣b⃗∣=7|\vec{b}|=7 and a⃗×b⃗=3i^+2j^+6k^\vec{a}\times\vec{b}=3\hat{i}+2\hat{j}+6\hat{k}, what is the acute angle between them?

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  67. 67
    Let p⃗\vec{p}, q⃗\vec{q} be position vectors of P,Q. R and S divide PQ internally/externally in ratio 2:3. If OR and OS are perpendicular, which is correct?

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  68. 68
    What is the moment about i^+2j^−k^\hat{i}+2\hat{j}-\hat{k} of force 3i^+k^3\hat{i}+\hat{k} through 2i^−j^+3k^2\hat{i}-\hat{j}+3\hat{k}?

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  69. 69
    If a⃗+2b⃗+3c⃗=0⃗\vec{a}+2\vec{b}+3\vec{c}=\vec{0} and a⃗×b⃗+b⃗×c⃗+c⃗×a⃗=λ(b⃗×c⃗)\vec{a}\times\vec{b}+\vec{b}\times\vec{c}+\vec{c}\times\vec{a}=\lambda(\vec{b}\times\vec{c}), then lambda is?

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  70. 70
    If vectors k and A are parallel, what is kk⃗×A⃗k\vec{k}\times\vec{A} equal to?

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  71. 71
    Which is correct for f:R→R+f:\mathbb{R}\to\mathbb{R}^+ defined as f(x)=∣x+1∣f(x)=|x+1|?

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  72. 72
    For f(x)=x21+x2f(x)=\frac{x^2}{1+x^2}, what is the range?

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  73. 73
    If f(x)=∣x∣+∣x−1∣f(x)=|x|+|x-1|, which is correct?

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  74. 74
    For f(x)=x2ln⁡∣x∣f(x)=x^2\ln|x| (x≠0x\ne0), f(0)=0f(0)=0, what is f′(0)f'(0)?

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  75. 75
    What is the area bounded by y2=6(x−1)y^2=6(x-1) and y2=3xy^2=3x?

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  76. Consider the following information for the next three (03) items that follow: Three sides of a trapezium are each equal to 6 cm. Let alpha in (0, pi/2) be the angle between a pair of adjacent sides.
    76
    If the area of the trapezium is the maximum possible, then what is alpha equal to?

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  77. 77
    If the area is maximum, what is the length of the fourth side?

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  78. 78
    What is the maximum area of the trapezium?

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  79. 79
    What is ∫0πexsin⁡x dx\int_{0}^{\pi}e^x\sin x\,dx equal to?

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  80. 80
    If f(x)=x2−9x2−2x−3f(x)=\frac{x^2-9}{x^2-2x-3}, x≠3x\ne3 is continuous at x=3, then?

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  81. 81
    What is ∫1exln⁡x dx\int_{1}^{e}x\ln x\,dx equal to?

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  82. 82
    What is ∫02[x2] dx\int_{0}^{\sqrt{2}}[x^2]\,dx equal to (where [.] is greatest integer function)?

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  83. 83
    What is the maximum value of 16sin⁡θ−12sin⁡2θ16\sin\theta-12\sin^2\theta?

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  84. 84
    If f:R→Sf:\mathbb{R}\to S, f(x)=4sin⁡x−3cos⁡x+1f(x)=4\sin x-3\cos x+1 is onto, what is S?

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  85. 85
    For f(x)=1∣x∣−xf(x)=\frac{1}{\sqrt{|x|-x}}, what is the domain of f?

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  86. 86
    What is the solution of the differential equation x dy - y dx = 0?

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  87. 87
    What is the derivative of f(x)=etan⁡x+ln⁡(sec⁡x)−eln⁡xf(x)=e^{\tan x}+\ln(\sec x)-e^{\ln x} at x=π/4x=\pi/4?

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  88. 88
    Which differential equation has a periodic solution?

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  89. 89
    What is the period of f(x) = sin x?

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  90. 90
    What is ∫dx2x−1\int\frac{dx}{2^x-1} equal to?

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  91. 91
    The order and degree of the DE y2=4a(x−a)y^2=4a(x-a), where a is arbitrary, are respectively

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  92. 92
    What is ∫−π/4π/4(sin⁡x−tan⁡x) dx\int_{-\pi/4}^{\pi/4}(\sin x-\tan x)\,dx?

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  93. 93
    If ∫abx3 dx=0\int_a^b x^3\,dx=0 and ∫abx2 dx=2/3\int_a^b x^2\,dx=2/3, then a and b are respectively?

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  94. 94
    What is ∫01x(1−x)9 dx\int_{0}^{1}x(1-x)^9\,dx equal to?

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  95. 95
    What is lim⁡x→0tan⁡xsin⁡2x\lim_{x\to0}\frac{\tan x}{\sin 2x} equal to?

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  96. 96
    What is lim⁡h→02x+3h−2x2h\lim_{h\to0}\frac{\sqrt{2x+3h}-\sqrt{2x}}{2h} equal to?

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  97. 97
    If f(x) is an even function, then which is correct?

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  98. 98
    If y=ex2sin⁡2xy=e^{x^2}\sin 2x, what is dy/dx at x=πx=\pi?

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  99. 99
    What is the solution of (1+2x) dy - (1-2y) dx = 0?

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  100. 100
    What are order and degree of (d3ydx3)2=y4+(dydx)5\left(\frac{d^3y}{dx^3}\right)^2=y^4+\left(\frac{dy}{dx}\right)^5?

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  101. 101
    In Binomial distribution, mean is three times variance. What is the probability of exactly 3 successes in 5 trials?

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  102. 102
    Which of the following probability statements are correct? 1. P(Aˉ∪B)=P(Aˉ)+P(B)−P(Aˉ∩B)P(\bar{A}\cup B) = P(\bar{A})+P(B)-P(\bar{A}\cap B) 2. P(A∩Bˉ)=P(B)−P(A∩B)P(A\cap\bar{B}) = P(B)-P(A\cap B) 3. P(A∩B)=P(B) P(A∣B)P(A\cap B) = P(B)\,P(A\mid B)

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  103. 103
    If r=0.6, cov=27, var(y)=25, then what is var(x)?

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  104. 104
    P(solves A)=0.4, P(solves B)=0.5. P(solves at least one)?

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  105. 105
    If xˉ\bar{x} is the mean of xix_i and xi=a+c yix_i = a + c\,y_i, then what is the mean of yiy_i?

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  106. 106
    Consider statements: 1. r=0 => regression lines parallel 2. r=+1 => regression lines perpendicular Which is/are correct?

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  107. 107
    If 4x-5y+33=0 and 20x-9y=107 are regression lines, then x-bar and y-bar are respectively?

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  108. 108
    Consider statements: 1. Mean is independent of change in scale and origin. 2. Variance is independent of change in scale but not in origin. Which is/are correct?

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  109. 109
    Consider statements: 1. Sum of deviations from mean is always zero. 2. Sum of absolute deviations is minimum around median. Which is/are correct?

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  110. 110
    Median of: 4.6, 0, 9.3, -4.8, 7.6, 2.3, 12.7, 3.5, 8.2, 6.1, 3.9, 5.2?

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  111. 111
    In a Maths test, 20% got first class. Central angle in Pie Chart?

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  112. 112
    Mean=5, SD=2. If 5 added to each value, what is CV of new set?

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  113. 113
    Train: 5 km at 30 km/hr, next 15 km at 45 km/hr. Average speed?

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  114. 114
    Two fair dice rolled. P(sum=7)?

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  115. 115
    A,B events with 2P(A)=3P(B), 0<P(A)<P(B)<1. Which ordering is correct?

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  116. 116
    Box has chits 0-9. First drawn, kept aside; second drawn. P(second is '9')?

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  117. 117
    One bag: 3W 2B; another: 5W 3B. Bag chosen randomly, ball drawn. P(white)?

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  118. 118
    Which probability statements are correct? 1. P(A not B)=P(A)-P(B) if B subset A 2. P(A alone or B alone)=P(A)+P(B)-P(A intersect B) 3. P(A union B)=P(A)+P(B) if mutually exclusive

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  119. 119
    Committee of 3 from 4 men and 5 women. P(exactly 2 men)?

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  120. 120
    SD sigma of first N natural numbers can be obtained using?

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