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NDA 2025 (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
    The sum of the first kk terms of a series SS is 3k2+5k3k^2+5k. Which one of the following is correct?

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  2. 2
    The sum of the first 8 terms of a GP is five times the sum of its first 4 terms. If r≠1r \neq 1 is the common ratio, then what is the number of possible real values of rr?

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  3. 3
    If one root of the equation x2−kx+k=0x^2 - kx + k = 0 exceeds the other by 232\sqrt{3}, then which one of the following is a value of kk?

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  4. 4
    If x+5y=4x + 5y = 4 and y+5x=−4y + 5x = -4, then what is (x+y)(x + y) equal to?

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  5. 5
    If 5th, 7th and 13th terms of an AP are in GP, then what is the ratio of its first term to its common difference?

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  6. 6
    If p,1,qp, 1, q are in AP and p,2,qp, 2, q are in GP, then which of the following statements is/are correct? I. p,4,qp, 4, q are in HP. II. 1p,14,1q\frac{1}{p}, \frac{1}{4}, \frac{1}{q} are in AP. Select the answer using the code given below.

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  7. 7
    If x=(1111)2, y=(1001)2x=(1111)_2,\ y=(1001)_2 and z=(110)2z=(110)_2, then what is x3−y3−z3−3xyzx^3-y^3-z^3-3xyz equal to?

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  8. 8
    If Δ=∣abcdefghi∣\Delta = \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} and A,B,C,D,GA, B, C, D, G are the cofactors of the elements a,b,c,d,ga, b, c, d, g respectively, then what is bB+cC−dD−gGbB + cC - dD - gG equal to?

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  9. 9
    Consider the following statements in respect of the determinant Δ=∣k(k+2)2k+112k+1k+21331∣\Delta = \begin{vmatrix} k(k+2) & 2k+1 & 1 \\ 2k+1 & k+2 & 1 \\ 3 & 3 & 1 \end{vmatrix}: (I). Δ\Delta is positive if k>0k>0. (II). Δ\Delta is negative if k<0k<0. (III). Δ\Delta is zero if k=0k=0. How many of the statements given above are correct?

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  10. 10
    If ∣23+i−13−i0i−1−1−1−i1∣=A+iB\begin{vmatrix} 2 & 3+i & -1 \\ 3-i & 0 & i-1 \\ -1 & -1-i & 1 \end{vmatrix} = A + iB where i=−1i=\sqrt{-1}, then what is A+BA+B equal to?

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  11. 11
    If A2+B2+C2=0A^2 + B^2 + C^2 = 0, then what is the value of ∣1cos⁡Ccos⁡Bcos⁡C1cos⁡Acos⁡Bcos⁡A1∣\begin{vmatrix} 1 & \cos C & \cos B \\ \cos C & 1 & \cos A \\ \cos B & \cos A & 1 \end{vmatrix}?

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  12. 12
    If ω\omega is a non-real cube root of unity, then what is a root of the following equation? ∣x+1ωω2ωx+ω21ω21x+ω∣=0\begin{vmatrix} x+1 & \omega & \omega^2 \\ \omega & x+\omega^2 & 1 \\ \omega^2 & 1 & x+\omega \end{vmatrix} = 0

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  13. 13
    What is (3+i3−i)3\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^3 equal to?

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  14. 14
    If x2−x+1=0x^2 - x + 1 = 0, then what is (x−1x)2+(x−1x4)+(x−1x8)\left(x - \frac{1}{x}\right)^2 + \left(x - \frac{1}{x^4}\right) + \left(x - \frac{1}{x^8}\right) equal to?

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  15. 15
    How many 7-letter words (with or without meaning) can be constructed using all the letters of the word CAPITAL so that all consonants come together in each word?

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  16. 16
    If z≠0z \neq 0 is a complex number, then what is amp(z)+amp(zˉ)\text{amp}(z) + \text{amp}(\bar{z}) equal to?

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  17. 17
    How many sides are there in a polygon which has 20 diagonals?

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  18. 18
    In how many ways can the letters of the word DELHI be arranged keeping the positions of vowels and consonants unchanged?

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  19. 19
    What is the number of positive integer solutions of x+y+z=5x + y + z = 5?

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  20. 20
    What is the number of rational terms in the expansion of (312+514)12\left(3^{\frac{1}{2}} + 5^{\frac{1}{4}}\right)^{12}?

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  21. 21
    If the sum of binomial coefficients in the expansion of (x+y)n(x+y)^n is 256, then the greatest binomial coefficient occurs in which one of the following terms?

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  22. 22
    If k<(2+1)3<k+2k < (\sqrt{2}+1)^3 < k+2, where kk is a natural number, then what is the value of kk?

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  23. 23
    If (x11)(123456789)(11x)=(45)\begin{pmatrix} x & 1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \\ x \end{pmatrix} = \begin{pmatrix} 45 \end{pmatrix} then which one of the following is a value of xx?

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  24. 24
    If A=(yzxzxyxyz)A = \begin{pmatrix} y & z & x \\ z & x & y \\ x & y & z \end{pmatrix} where x,y,zx, y, z are integers, is an orthogonal matrix, then what is the value of x2+y2+z2x^2 + y^2 + z^2?

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  25. 25
    Consider the following in respect of a non-singular matrix MM: (I). M2=∣M∣2M^2 = |M|^2 (II). ∣M∣=M−1|M| = M^{-1} (III). ∣M∣=MT|M| = M^T How many of the above are correct?

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  26. 26
    If f(θ)=(cos⁡θsin⁡θ−sin⁡θcos⁡θ)f(\theta) = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} then what is {f(π)}2\{f(\pi)\}^2 equal to?

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  27. 27
    If A=(122212221)A = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{pmatrix} then what is A2−4AA^2 - 4A equal to? (where I3I_3 is the identity matrix of order 3)

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  28. 28
    If the number of selections of rr as well as (n+r)(n+r) things from 5n5n different things are equal, then what is the value of rr?

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  29. 29
    What is the number of selections of at most 3 things from 6 different things?

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  30. 30
    If A=(xyzyzxzxy)A = \begin{pmatrix} x & y & z \\ y & z & x \\ z & x & y \end{pmatrix} where x,y,zx, y, z are integers, is an orthogonal matrix, then what is A2A^2 equal to?

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  31. Consider the following for the three (03) items that follow: Let p=sin⁡35°, q=sin⁡25°p = \sin 35°,\ q = \sin 25° and r=sin⁡(−95°)r = \sin(-95°).
    31
    What is (p+q+r)(p + q + r) equal to?

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  32. 32
    What is (pq+qr+rp)(pq + qr + rp) equal to?

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  33. 33
    What is p2+q2+r2p^2 + q^2 + r^2 equal to?

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  34. Let p=sin⁡α−sin⁡(α−90°)p = \sin\alpha - \sin(\alpha - 90°).
    34
    What is the minimum value of pp?

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  35. 35
    What is the maximum value of pp?

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  36. The sides of a triangle ABC are AB = 3 cm, BC = 5 cm and CA = 7 cm.
    36
    Consider the following statements: (I). The triangle is obtuse-angled triangle. (II). The sum of acute angles of the triangle is also acute. Which of the statements given above is/are correct?

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  37. 37
    What is ∠B\angle B equal to?

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  38. 38
    What is the area of the triangle?

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  39. The top (M) of a tower is observed from three points P, Q and R lying in a horizontal straight line which passes directly along the foot (N) of the tower. The angles of elevations of M from P, Q and R are 30°, 45° and 60° respectively. Let PQ = a and QR = b.
    39
    What is PN equal to?

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  40. 40
    What is MN equal to?

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  41. Let p=tan⁡2α−tan⁡αp = \tan 2\alpha - \tan\alpha and q=cot⁡α−cot⁡2αq = \cot\alpha - \cot 2\alpha.
    41
    What is pq\frac{p}{q} equal to?

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  42. 42
    What is (p+q)(p + q) equal to?

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  43. 43
    What is tan⁡2α\tan^2\alpha equal to?

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  44. Let 2sin⁡α+cos⁡α=22\sin\alpha + \cos\alpha = 2, where 0<α<90°0 < \alpha < 90°.
    44
    What is tan⁡α\tan\alpha equal to?

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  45. 45
    What is 2sin⁡2α+cos⁡2α2\sin 2\alpha + \cos 2\alpha equal to?

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  46. In a triangle ABC, two sides BC and CA are in the ratio 2:1 and their opposite corresponding angles are in the ratio 3:1.
    46
    One of the angles of the triangle is

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  47. 47
    Consider the following statements: (I). The triangle is right-angled. (II). One of the sides of the triangle is 3\sqrt{3} times the other. (III). The angles A, C and B of the triangle are in AP. Which of the statements given above is/are correct?

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  48. 48
    A man at M, standing 100 m away from the base (P) of a chimney of height 50 m, observes the angle of elevation of the highest point (Q) of the smoke to be 45°. The highest point of the chimney is at R. Further P, R and Q are in a straight line and the straight line is perpendicular to PM. What is the angle RMQ equal to?

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  49. 49
    If kk is a root of x2−4x+1=0x^2 - 4x + 1 = 0, then what is tan⁡−1k+tan⁡−11k\tan^{-1}k + \tan^{-1}\frac{1}{k} equal to?

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  50. 50
    If tan⁡−1k+tan⁡−112=π4\tan^{-1}k + \tan^{-1}\frac{1}{2} = \frac{\pi}{4}, then what is the value of kk?

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  51. 51
    Under what condition will the lines m2x+ny−1=0m^2 x + ny - 1 = 0 and n2x−my+2=0n^2 x - my + 2 = 0 be perpendicular?

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  52. 52
    If pp and qq are real numbers between 0 and 1 such that the points (p,1),(1,q)(p, 1), (1, q) and (0,0)(0, 0) form an equilateral triangle, then what is (p+q)(p + q) equal to?

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  53. 53
    The vertices of a triangle are A(1,1),B(0,0)A(1,1), B(0,0) and C(2,0)C(2,0). The angular bisectors of the triangle meet at PP. What are the coordinates of PP?

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  54. 54
    Let A(3,−1)A(3,-1) and B(1,1)B(1,1) be the end points of line segment AB. Let P be the middle point of the line segment AB. Let Q be the point situated at a distance 2 units from P on the perpendicular bisector line of AB. What are the possible coordinates of Q?

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  55. 55
    ABC is an equilateral triangle and AD is the altitude on BC. If the coordinates of A are (1,2)(1, 2) and that of D are (−2,6)(-2, 6), then what is the equation of BC?

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  56. 56
    What is the equation of the circle whose diameter is 10 cm and the equations of two of its diameters are x+y=0x + y = 0 and x−y=0x - y = 0?

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  57. 57
    A square is inscribed in a circle x2+y2+2x+2y+1=0x^2 + y^2 + 2x + 2y + 1 = 0 and its sides are parallel to coordinate axes. Which one of the following is a vertex of the square?

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  58. 58
    A tangent to the parabola y2=4xy^2 = 4x is inclined at an angle 45°45° with the positive direction of xx-axis. What is the point of contact of the tangent and the parabola?

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  59. 59
    What is the distance between the two foci of the hyperbola 25x2−75y2=22525x^2 - 75y^2 = 225?

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  60. 60
    If any point on an ellipse is (3sin⁡α,5cos⁡α)(3\sin\alpha, 5\cos\alpha), then what is the eccentricity of the ellipse?

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  61. 61
    If a line in 3 dimensions makes angles α,β\alpha, \beta and γ\gamma with the positive directions of the coordinate axes, then what is cos⁡(α+β)cos⁡(α−β)\cos(\alpha+\beta)\cos(\alpha-\beta) equal to?

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  62. 62
    A(1,2,-1), B(2,5,-2) and C(4,4,-3) are three vertices of a rectangle. What is the area of the rectangle?

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  63. 63
    ABC is a triangle right-angled at B. If A(k,1,−1),B(2k,0,2)A(k,1,-1), B(2k,0,2) and C(2+2k,k,1)C(2+2k,k,1) are the vertices of the triangle, then what is the value of kk?

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  64. 64
    If a line x+1p=y−1q=z−2r\frac{x+1}{p} = \frac{y-1}{q} = \frac{z-2}{r}, where p=2q=3rp = 2q = 3r, makes an angle θ\theta with the positive direction of yy-axis, then what is cos⁡2θ\cos 2\theta equal to?

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  65. 65
    What is the equation of the plane passing through the point (1,1,1)(1,1,1) and perpendicular to the line whose direction ratios are ⟨3,2,1⟩\langle 3,2,1 \rangle?

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  66. 66
    A line makes angles α,β\alpha, \beta and γ\gamma with the positive directions of the coordinate axes. If a⃗=sin⁡2α i^+sin⁡2β j^+sin⁡2γ k^\vec{a} = \sin^2\alpha\,\hat{i} + \sin^2\beta\,\hat{j} + \sin^2\gamma\,\hat{k} and b⃗=i^+j^+k^\vec{b} = \hat{i}+\hat{j}+\hat{k}, then what is a⃗⋅b⃗\vec{a}\cdot\vec{b} equal to?

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  67. 67
    Consider the following statements in respect of a vector d⃗=(a⃗×b⃗)×c⃗\vec{d} = (\vec{a}\times\vec{b})\times\vec{c}: (I). d⃗\vec{d} is coplanar with a⃗\vec{a} and b⃗\vec{b}. (II). d⃗\vec{d} is perpendicular to c⃗\vec{c}. Which of the statements given above is/are correct?

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  68. 68
    The position vectors of three points A, B and C are a⃗,b⃗\vec{a}, \vec{b} and c⃗\vec{c} respectively such that 3a⃗−4b⃗+c⃗=0⃗3\vec{a} - 4\vec{b} + \vec{c} = \vec{0}. What is AB : BC equal to?

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  69. 69
    The position vectors of three points A, B and C are a⃗,b⃗\vec{a}, \vec{b} and c⃗\vec{c} respectively, where c⃗=cos⁡2θ a⃗+sin⁡2θ b⃗\vec{c} = \cos^2\theta\,\vec{a} + \sin^2\theta\,\vec{b}. What is (a⃗×b⃗)+(b⃗×c⃗)+(c⃗×a⃗)(\vec{a}\times\vec{b})+(\vec{b}\times\vec{c})+(\vec{c}\times\vec{a}) equal to?

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  70. 70
    Let a⃗,b⃗,(a⃗×b⃗)\vec{a}, \vec{b}, (\vec{a}\times\vec{b}) be unit vectors. What is (a⃗⋅b⃗)(\vec{a}\cdot\vec{b}) equal to?

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  71. Let x=sec⁡θ−cos⁡θx = \sec\theta - \cos\theta and y=sec⁡4θ−cos⁡4θy = \sec^4\theta - \cos^4\theta.
    71
    What is (dydx)2\left(\frac{dy}{dx}\right)^2 equal to?

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  72. 72
    What is (x2+4y2+4)dydx[(x2+4)d2ydx2−16y]\left(\dfrac{x^2+4}{y^2+4}\right)\dfrac{dy}{dx}\left[(x^2+4)\dfrac{d^2y}{dx^2}-16y\right] equal to?

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  73. Let ABC be a triangle right-angled at B and AB + AC = 3 units.
    73
    What is ∠A\angle A equal to if the area of the triangle is maximum?

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  74. 74
    What is the maximum area of the triangle?

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  75. Let (x+y)p+q=xpyq(x+y)^{p+q} = x^p y^q, where p,qp, q are positive integers.
    75
    The derivative of yy with respect to xx

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  76. 76
    If p+q=10p + q = 10, then what is dydx\frac{dy}{dx} equal to?

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  77. The slope of the tangent to the curve y=f(x)y = f(x) at (x,f(x))(x, f(x)) is 4 for every real number xx and the curve passes through the origin.
    77
    What is the nature of the curve?

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  78. 78
    What is the area bounded by the curve, the xx-axis and the line x=4x = 4?

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  79. Let f(x)={x3,x2<1x2,x2≥1f(x) = \begin{cases} x^3, & x^2 < 1 \\ x^2, & x^2 \geq 1 \end{cases}
    79
    What is lim⁡x→0f′(x)\lim_{x \to 0} f'(x) equal to?

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  80. 80
    Consider the following statements: (I). The function is continuous at x=−1x = -1. (II). The function is differentiable at x=1x = 1. Which of the statements given above is/are correct?

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  81. Let the function y=(1−cos⁡x)−1y = (1 - \cos x)^{-1}, where x≠2nπx \neq 2n\pi and nn is an integer.
    81
    What is the range of the function?

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  82. 82
    What is ∫y dx\int y\,dx equal to?

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  83. Let the function f(x)=sin⁡[x]f(x) = \sin[x], where [⋅][\cdot] is the greatest integer function and g(x)=∣x∣g(x) = |x|.
    83
    What is lim⁡x→0{f(x)⋅g(x)}\lim_{x\to 0}\{f(x) \cdot g(x)\} equal to?

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  84. 84
    What is lim⁡x→0f(x)g(x)\lim_{x\to 0} \frac{f(x)}{g(x)} equal to?

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  85. Let the curve f(x)=∣x−3∣f(x) = |x - 3|.
    85
    What is the domain of the function f(x)f(x)?

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  86. 86
    What is the area bounded by the curve f(x)f(x) and y=3y = 3?

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  87. Let f={(1,1),(2,4),(3,7),(4,10)}f = \{(1,1),(2,4),(3,7),(4,10)\}.
    87
    If f(x)=px+qf(x) = px + q, then what is the value of (p+q)(p + q)?

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  88. 88
    Consider the following statements: (I). ff is one-one function. (II). ff is onto function if the codomain is the set of natural numbers. Which of the statements given above is/are correct?

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  89. Let the function f(x)=x2−1f(x) = x^2 - 1.
    89
    What is lim⁡x→1{f∘f(x)}\lim_{x\to1}\{f \circ f(x)\} equal to?

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  90. 90
    What is the area bounded by the function f(x)f(x) and the xx-axis?

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  91. Let y=sin⁡−1(x−4x327)y = \sin^{-1}\left(\frac{x - 4x^3}{27}\right).
    91
    What is yy equal to?

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  92. 92
    What is dydx\frac{dy}{dx} equal to?

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  93. Let the function f(x)=x2+9f(x) = x^2 + 9.
    93
    What is lim⁡x→0f(x)−3f(x)+7−4\lim_{x\to 0} \dfrac{\sqrt{f(x)}-3}{\sqrt{f(x)+7}-4} equal to?

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  94. 94
    Consider the following statements: (I). f(x)f(x) is an increasing function. (II). f(x)f(x) has local maximum at x=0x = 0. Which of the statements given above is/are correct?

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  95. The function f(x)f(x) satisfies f(xy)=f(x)f(y)f(xy) = f(x)f(y) for all positive real values of xx and yy, and f(2)=3f(2) = 3.
    95
    What is f(16)f(16) equal to?

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  96. 96
    What is f(1)f(4)\frac{f(1)}{f(4)} equal to?

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  97. A function ff is such that f(xy)=f(x+y)f(xy) = f(x+y) for all real values of xx and yy, and f(5)=10f(5) = 10.
    97
    What is f(0)f(0) equal to?

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  98. 98
    What is f(20)+f(−20)f(20) + f(-20) equal to?

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  99. Let f(x)=[x2]f(x)=[x^2], where [⋅][\cdot] is the greatest integer function.
    99
    What is ∫23f(x) dx\int_{\sqrt{2}}^{\sqrt{3}} f(x)\,dx equal to?

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  100. 100
    What is ∫22f(x) dx\int_{\sqrt{2}}^{2} f(x)\,dx equal to?

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  101. The frequency distribution of height of students of a class is given below: Height (cm): 160–162, 162–164, 164–166, 166–168 Number of Students: 12, 15, 24, 13
    101
    What is the total number of students whose height is less than or equal to 165 cm?

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  102. 102
    What is the median height of the class?

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  103. 103
    The height which occurs most frequently in the class is

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  104. 104
    The most appropriate graphical representation of the given frequency distribution is

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  105. 105
    The sum and the sum of squares of observations corresponding to length X (in cm) and weight Y (in gm) of 50 tropical tubers are given as ΣX=200,ΣY=250,ΣX2=900\Sigma X = 200, \Sigma Y = 250, \Sigma X^2 = 900 and ΣY2=1400\Sigma Y^2 = 1400. Which one of the following is correct?

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  106. 106
    The sum and the sum of squares of observations corresponding to length X (in cm) and weight Y (in gm) of 50 tropical tubers are given as ΣX=200,ΣY=250,ΣX2=900\Sigma X = 200, \Sigma Y = 250, \Sigma X^2 = 900 and ΣY2=1400\Sigma Y^2 = 1400. Which one of the following statements is correct?

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  107. Let X be a random variable following binomial distribution with parameters n=6n = 6 and p=kp = k. Further, 9P(X=4)=P(X=2)9P(X=4) = P(X=2).
    107
    What is the value of kk?

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  108. 108
    What is the value of P(X=3)P(X=3)?

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  109. A committee of 6 members is formed from a group of 7 gentlemen and 4 ladies.
    109
    What is the probability that the committee includes exactly 3 gentlemen?

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  110. 110
    What is the probability that the committee includes at least 2 ladies?

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  111. The probabilities that A, B and C become managers are 310,12\frac{3}{10}, \frac{1}{2} and 45\frac{4}{5} respectively. The probabilities that bonus scheme will be introduced if A, B and C become managers are 49,29\frac{4}{9}, \frac{2}{9} and 13\frac{1}{3} respectively.
    111
    What is the probability that the bonus scheme will be introduced?

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  112. 112
    If the bonus scheme has been introduced, then what is the probability that the manager appointed was B?

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  113. 113
    The arithmetic mean of 100 observations is 50. If 5 is subtracted from each observation and then divided by 20, then what is the new arithmetic mean?

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  114. 114
    The standard deviation of 100 observations is 10. If 5 is added to each observation and then divided by 20, then what will be the new standard deviation?

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  115. 115
    If P(A)=13,P(B)=12P(A)=\frac{1}{3}, P(B)=\frac{1}{2} and P(A∩B)=14P(A\cap B)=\frac{1}{4}, then what is the value of P(B∣Ac)P(B|A^c)?

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  116. 116
    If P(A)=13,P(B)=12P(A)=\frac{1}{3}, P(B)=\frac{1}{2} and P(A∩B)=14P(A\cap B)=\frac{1}{4}, then what is the value of P(Ac∩Bc)P(A^c\cap B^c)?

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  117. 117
    If two fair dice are tossed, then what is the probability that the sum of the numbers on the faces of the dice is strictly greater than 7?

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  118. 118
    The probability of a man hitting a target is 15\frac{1}{5}. If the man fires 7 times, then what is the probability that he hits the target at least twice?

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  119. 119
    Let X be a random variable following binomial distribution whose mean and variance are 200 and 160 respectively. What is the value of the number of trials (nn)?

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  120. 120
    What is the arithmetic mean of 82,92,102,…,1528^2, 9^2, 10^2, \ldots, 15^2?

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