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NDA 2023 (II), 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 zz‾=∣z+z‾∣z\overline{z} = |z + \overline{z}|, where z=x+iy, i=−1z = x + iy,\ i = \sqrt{-1}, then the locus of zz is a pair of:

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  2. 2
    If 1!+3!+5!+7!+⋯+199!1! + 3! + 5! + 7! + \cdots + 199! is divided by 24, what is the remainder?

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  3. 3
    What is the value of ∣12+5i∣+∣12−5i∣|12 + 5i| + |12 - 5i|, where i=−1i = \sqrt{-1}?

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  4. 4
    If A=(123)A = \begin{pmatrix}1\\2\\3\end{pmatrix}, then what is the value of det⁡(I+AA′)\det(I + AA'), where II is the 3×33 \times 3 identity matrix?

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  5. 5
    If AA, BB and CC are square matrices of order 3 and det⁡(BC)=2det⁡(A)\det(BC) = 2\det(A), then what is the value of det⁡(2A−1BC)\det(2A^{-1}BC)?

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  6. 6
    If the nnth term of a sequence is 2n+57\frac{2n+5}{7}, then what is the sum of its first 140 terms?

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  7. 7
    Let AA be a skew-symmetric matrix of order 3. What is the value of det⁡(4A4)−det⁡(3A3)+det⁡(2A2)−det⁡(A)+det⁡(−I)\det(4A^4) - \det(3A^3) + \det(2A^2) - \det(A) + \det(-I)?

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  8. 8
    If A=(034−305−4−50)A = \begin{pmatrix}0&3&4\\-3&0&5\\-4&-5&0\end{pmatrix}, then which one of the following statements is correct?

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  9. 9
    If A=(200030004)A = \begin{pmatrix}2&0&0\\0&3&0\\0&0&4\end{pmatrix}, then which of the following statements are correct? (1) AnA^n will always be singular for any positive integer nn. (2) AnA^n will always be a diagonal matrix for any positive integer nn. (3) AnA^n will always be a symmetric matrix for any positive integer nn.

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  10. 10
    If (a+b), 2b, (b+c)(a+b),\ 2b,\ (b+c) are in HP, then which one of the following is correct?

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  11. 11
    Let t1,t2,t3,…t_1, t_2, t_3, \ldots be in GP. What is t1t3⋯t21t_1 t_3 \cdots t_{21} equal to?

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  12. 12
    Which one of the following is a square root of −−1\sqrt{-\sqrt{-1}}?

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  13. 13
    What is the maximum number of points of intersection of 10 circles?

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  14. 14
    A set SS contains (2n+1)(2n+1) elements. There are 4096 subsets of SS which contain at most nn elements. What is nn equal to?

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  15. 15
    If ∣x2+3xx−1x+3x+1−2xx−4x−3x+43x∣=ax4+bx3+cx2+dx+e\begin{vmatrix}x^2+3x & x-1 & x+3\\x+1 & -2x & x-4\\x-3 & x+4 & 3x\end{vmatrix} = ax^4+bx^3+cx^2+dx+e, then what is the value of ee?

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  16. 16
    If all elements of a third order determinant are equal to 1 or -1, then the value of the determinant is:

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  17. 17
    If A=(2−10−130101)A = \begin{pmatrix}2&-1&0\\-1&3&0\\1&0&1\end{pmatrix}, then what is the value of det⁡[adj(adj A)]\det[\text{adj}(\text{adj}\,A)]?

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  18. 18
    If A=I3A = I_3 (identity), then what is 23A3−19A2−4A23A^3 - 19A^2 - 4A equal to?

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  19. 19
    The value of the determinant of a matrix AA of order 3 is 3. If CC is the matrix of cofactors of AA, then what is the value of det⁡(C2)\det(C^2)?

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  20. 20
    If Ak=(k−1kk−2k+1)A_k = \begin{pmatrix}k-1&k\\k-2&k+1\end{pmatrix}, then what is det⁡(A1)+det⁡(A2)+⋯+det⁡(A100)\det(A_1)+\det(A_2)+\cdots+\det(A_{100}) equal to?

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  21. 21
    The Cartesian product A×AA\times A has 16 elements among which are (0,2)(0,2) and (1,3)(1,3). Which of the following statements is/are correct? (1) It is possible to determine set AA. (2) A×AA\times A contains the element (3,2)(3,2).

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  22. 22
    Let A={1,2,…,20}A=\{1,2,\ldots,20\}. Define R={(x,y):4x−3y=1}, x,y∈AR=\{(x,y):4x-3y=1\},\ x,y\in A. Which statements are correct? (1) Domain of RR is {1,4,7,10,13,16}\{1,4,7,10,13,16\}. (2) Range of RR is {1,5,9,13,17}\{1,5,9,13,17\}. (3) Range of RR equals codomain.

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  23. 23
    Consider the statements: (1) f(x)=x3, 0≤x≤2f(x)=x^3,\ 0\le x\le2 and 4x, 2≤x≤84x,\ 2\le x\le8 is a function. (2) g(x)=x2, 0≤x≤4g(x)=x^2,\ 0\le x\le4 and 3x, 4≤x≤83x,\ 4\le x\le8 is a function. Which is/are correct?

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  24. 24
    Consider: (1) A=(A∪B)∪(A−B)A=(A\cup B)\cup(A-B). (2) A∪(B−A)=(A∪B)A\cup(B-A)=(A\cup B). (3) B=(A∪B)−(A−B)B=(A\cup B)-(A-B). Which are correct?

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  25. 25
    A function satisfies f(x−y)=f(x)f(y)f(x-y)=\frac{f(x)}{f(y)}, where f(y)≠0f(y)\neq0. If f(1)=0.5f(1)=0.5, then what is f(2)+f(3)+f(4)+f(5)+f(6)f(2)+f(3)+f(4)+f(5)+f(6) equal to?

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  26. 26
    What is 2cot⁡ ⁣(12cos⁡−1 ⁣53)2\cot\!\left(\tfrac{1}{2}\cos^{-1}\!\tfrac{\sqrt5}{3}\right) equal to?

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  27. 27
    If sec⁡−1p−csc⁡−1q=0\sec^{-1}p - \csc^{-1}q = 0, where p>0,q>0p>0,q>0, then what is the value of p−2+q−2p^{-2}+q^{-2}?

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  28. 28
    What is 1+sin⁡ ⁣(2cos⁡−1 ⁣317)\sqrt{1+\sin\!\left(2\cos^{-1}\!\frac{3}{\sqrt{17}}\right)} equal to?

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  29. 29
    If tan⁡(πcos⁡θ)=cot⁡(πsin⁡θ), 0<θ<π/2\tan(\pi\cos\theta)=\cot(\pi\sin\theta),\ 0<\theta<\pi/2; then what is the value of 8sin⁡2 ⁣(θ+π4)8\sin^2\!\left(\theta+\frac{\pi}{4}\right)?

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  30. 30
    If tan⁡α=17, sin⁡β=110; 0<α,β<π2\tan\alpha=\frac{1}{7},\ \sin\beta=\frac{1}{\sqrt{10}};\ 0<\alpha,\beta<\frac{\pi}{2}, then what is the value of cos⁡(α+2β)\cos(\alpha+2\beta)?

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  31. Consider the equation (1−x)4+(5−x)4=82(1-x)^4+(5-x)^4=82.
    31
    What is the number of real roots of the equation?

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  32. 32
    What is the sum of all the roots of the equation?

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  33. Consider equation-I: z3+2z2+2z+1=0z^3+2z^2+2z+1=0 and equation-II: z1985+z100+1=0z^{1985}+z^{100}+1=0.
    33
    What are the roots of equation-I?

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  34. 34
    Which one of the following is a root of equation-II?

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  35. 35
    What is the number of common roots of equation-I and equation-II?

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  36. A quadratic equation is given by (a+b)x2−(a+b+c)x+k=0(a+b)x^2-(a+b+c)x+k=0, where a,b,ca,b,c are real.
    36
    If k=c/2 (c≠0)k=c/2\ (c\neq0), then the roots of the equation are:

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  37. 37
    If k=ck=c, then the roots of the equation are:

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  38. Let (1+x)n=1+T1x+T2x2+⋯+Tnxn(1+x)^n = 1 + T_1 x + T_2 x^2 + \cdots + T_n x^n.
    38
    What is T1+2T2+3T3+⋯+nTnT_1+2T_2+3T_3+\cdots+nT_n equal to?

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  39. 39
    What is 1−T1+2T2−3T3+⋯+(−1)nnTn1-T_1+2T_2-3T_3+\cdots+(-1)^n nT_n equal to?

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  40. 40
    What is T1+T2+T3+⋯+TnT_1+T_2+T_3+\cdots+T_n equal to?

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  41. Let f(x)=x2−1f(x)=x^2-1 and g∘f(x)=x−x+1g\circ f(x)=x-\sqrt{x}+1.
    41
    Which one of the following is a possible expression for g(x)g(x)?

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  42. 42
    What is g(15)g(15) equal to?

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  43. Let ff be defined on R∖{0}\mathbb{R}\setminus\{0\} and 2f(x)+f ⁣(1x)=x+32f(x)+f\!\left(\frac{1}{x}\right)=x+3.
    43
    What is f(0.5)f(0.5) equal to?

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  44. 44
    If ff is differentiable, then what is f′(0.5)f'(0.5) equal to?

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  45. A function is defined by f(x)=(x+1)2/3(2x+4)6/3(6x+9)f(x)=(x+1)^{2/3}(2x+4)^{6/3}(6x+9).
    45
    The function is decreasing on:

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  46. 46
    The function attains local minimum value at:

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  47. Given that 4x2+y2=94x^2+y^2=9.
    47
    What is the maximum value of yy?

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  48. 48
    What is the maximum value of xyxy?

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  49. A function is defined by f(x)=π+sin⁡2xf(x)=\pi+\sin^2 x.
    49
    What is the range of the function?

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  50. 50
    What is the period of the function?

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  51. A parabola passes through (1,2)(1,2) and satisfies dydx=2yx, x>0,y>0\frac{dy}{dx}=\frac{2y}{x},\ x>0,y>0.
    51
    What is the directrix of the parabola?

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  52. 52
    What is the length of latus rectum of the parabola?

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  53. Let f(x)=ax−1+bx−12f(x)=\dfrac{a^{x-1}+b^{x-1}}{2} and g(x)=x−1g(x)=x-1.
    53
    What is lim⁡x→1f(x)−1g(x)\lim_{x\to1}\frac{f(x)-1}{g(x)} equal to?

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  54. 54
    What is lim⁡x→1f(x)1/g(x)\lim_{x\to1}f(x)^{1/g(x)} equal to?

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  55. Let f(x)=2−x+2+xf(x)=\sqrt{2-x}+\sqrt{2+x}.
    55
    What is the domain of the function?

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  56. 56
    What is the greatest value of the function?

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  57. Let f(x)=∣x∣f(x)=|x| and g(x)=[x]−1g(x)=[x]-1. Let h(x)=f(g(x))g(f(x))h(x)=\frac{f(g(x))}{g(f(x))}.
    57
    What is lim⁡x→0+h(x)\lim_{x\to0^+}h(x) equal to?

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  58. 58
    What is lim⁡x→0−h(x)\lim_{x\to0^-}h(x) equal to?

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  59. Let f(x)=x−3∣x−3∣+a; x<3f(x)=\frac{x-3}{|x-3|}+a;\ x<3, a−b; x=3a-b;\ x=3, x−3∣x−3∣+b; x>3\frac{x-3}{|x-3|}+b;\ x>3. f(x)f(x) is continuous at x=3x=3.
    59
    What is the value of aa?

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  60. 60
    What is the value of bb?

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  61. Let I=∫−2π2πsin⁡4x+cos⁡4x1+3x dxI=\int_{-2\pi}^{2\pi}\frac{\sin^4x+\cos^4x}{1+3^x}\,dx.
    61
    What is ∫0π(sin⁡4x+cos⁡4x) dx\int_0^\pi(\sin^4x+\cos^4x)\,dx equal to?

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  62. 62
    What is II equal to?

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  63. Consider the following for the next two (02) items that follow: Let f(x)={ax(x+1)+b,x<1x−1,1≤x≤2f(x)=\begin{cases}ax(x+1)+b, & x<1\\ x-1, & 1\leq x\leq 2\end{cases}
    63
    If f(x)f(x) is differentiable at x=1x=1, then what is the value of (a+b)(a+b)?

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  64. 64
    What is lim⁡x→0f(x)\lim_{x\to0}f(x) equal to?

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  65. 65
    If f(x)=∣ln⁡∣x∣∣f(x)=|\ln|x|| where 0<x<10<x<1, then what is f′(0.5)f'(0.5) equal to?

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  66. 66
    If f′(x)=cos⁡(ln⁡x)f'(x)=\cos(\ln x) and y=f ⁣(2x−3x)y=f\!\left(\frac{2x-3}{x}\right), then what is dydx\frac{dy}{dx} equal to?

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  67. 67
    What is ∫08π∣sin⁡x∣ dx\int_0^{8\pi}|\sin x|\,dx equal to?

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  68. 68
    What is the area between f(x)=x∣x∣f(x)=x|x| and the xx-axis for x∈[−1,1]x\in[-1,1]?

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  69. 69
    What are the order and degree respectively of x2 ⁣(d3ydx3)2+(dydx)4+sin⁡x=0x^2\!\left(\frac{d^3y}{dx^3}\right)^2+\left(\frac{dy}{dx}\right)^4+\sin x=0?

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  70. 70
    What is the differential equation of all parabolas of the type y2=4a(x−b)y^2=4a(x-b)?

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  71. Let a1,a2,a3,…a_1,a_2,a_3,\ldots be in AP such that a1+a5+a10+a15+a20+a25+a30+a34=300a_1+a_5+a_{10}+a_{15}+a_{20}+a_{25}+a_{30}+a_{34}=300.
    71
    What is a1+a5−a10−a15−a20−a25+a30+a34a_1+a_5-a_{10}-a_{15}-a_{20}-a_{25}+a_{30}+a_{34} equal to?

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  72. 72
    What is ∑n=134an\sum_{n=1}^{34}a_n equal to?

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  73. Let p=cos⁡π5cos⁡2π5p=\cos\frac{\pi}{5}\cos\frac{2\pi}{5} and q=cos⁡4π5cos⁡8π5q=\cos\frac{4\pi}{5}\cos\frac{8\pi}{5}.
    73
    What is the value of p+qp+q?

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  74. 74
    What is the value of pqpq?

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  75. Let p=13−tan⁡3xtan⁡xp=\dfrac{1}{3}-\dfrac{\tan 3x}{\tan x} and q=1−3tan⁡2x, 0<x<π, x≠π2q=1-3\tan^2 x,\ 0<x<\pi,\ x\neq\dfrac{\pi}{2}.
    75
    What is pqpq equal to?

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  76. 76
    For how many values of xx does 1p\frac{1}{p} become zero?

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  77. Let sin⁡x+sin⁡y=3(cos⁡y−cos⁡x)\sin x+\sin y=\sqrt{3}(\cos y-\cos x); x+y=π2x+y=\frac{\pi}{2}, 0<x,y<π20<x,y<\frac{\pi}{2}.
    77
    What is a value of sin⁡3x+sin⁡3y\sin3x+\sin3y?

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  78. 78
    What is a value of cos⁡3x+cos⁡3y\cos^3x+\cos^3y?

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  79. The angles AA, BB and CC of triangle ABCABC are in the ratio 3:5:43:5:4.
    79
    What is a+b+2 ca+b+\sqrt{2}\,c equal to?

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  80. 80
    What is the ratio a2:b2:c2a^2:b^2:c^2?

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  81. 81
    What is the equation of directrix of parabola y2=4bxy^2=4bx, where b<0b<0 and b2+b−2=0b^2+b-2=0?

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  82. 82
    The points (−a,−b)(-a,-b), (0,0)(0,0), (a,b)(a,b) and (a2,ab)(a^2,ab) are:

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  83. 83
    Given that 16p2+49q2−4r2−56pq=016p^2+49q^2-4r^2-56pq=0. Which one of the following is a point on the pair of straight lines (px+qy+r)(px+qy−r)=0(px+qy+r)(px+qy-r)=0?

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  84. 84
    If 3x+y−5=03x+y-5=0 is the equation of a chord of the circle x2+y2−25=0x^2+y^2-25=0, then what are the coordinates of the mid-point of the chord?

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  85. 85
    Consider x224−k+y2k−16=2\frac{x^2}{24-k}+\frac{y^2}{k-16}=2. (1) Represents ellipse if k=19k=19. (2) Represents hyperbola if k=12k=12. (3) Represents circle if k=20k=20. How many statements are correct?

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  86. 86
    Consider hyperbola x2cos⁡2θ−y2sin⁡2θ=1\frac{x^2}{\cos^2\theta}-\frac{y^2}{\sin^2\theta}=1. (1) Two foci are independent of θ\theta. (2) Eccentricity is sec⁡θ\sec\theta. (3) Distance between foci is 2. How many statements are correct?

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  87. 87
    Consider 4x2+4y2−4ax−4ay+a2=04x^2+4y^2-4ax-4ay+a^2=0. (1) Circle touches both axes. (2) Diameter is 2a2a. (3) Centre lies on x+y=ax+y=a. How many statements are correct?

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  88. 88
    For what values of kk is the line (k−3)x−(5−k2)y+(k2−7k+6)=0(k-3)x-(5-k^2)y+(k^2-7k+6)=0 parallel to the line x+y=1x+y=1?

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  89. 89
    The line x+y=4x+y=4 cuts the line joining P(−1,1)P(-1,1) and Q(5,7)Q(5,7) at RR. What is PR:RQPR:RQ equal to?

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  90. 90
    What is the sum of intercepts of the line whose perpendicular distance from origin is 4 units and the normal makes 15°15° with positive xx-axis?

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  91. 91
    What is the length of projection of the vector i^+2j^+3k^\hat{i}+2\hat{j}+3\hat{k} on the vector 2i^+3j^−2k^2\hat{i}+3\hat{j}-2\hat{k}?

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  92. 92
    If (a⃗×b⃗)2+(a⃗⋅b⃗)2=144(\vec{a}\times\vec{b})^2+(\vec{a}\cdot\vec{b})^2=144 and ∣b⃗∣=4|\vec{b}|=4, then what is the value of ∣a⃗∣|\vec{a}|?

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  93. 93
    If θ\theta is the angle between a⃗\vec{a} and b⃗\vec{b} such that a⃗⋅b⃗≥0\vec{a}\cdot\vec{b}\geq0, then which one is correct?

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  94. 94
    The vectors 60i^+3j^60\hat{i}+3\hat{j}, 40i^−8j^40\hat{i}-8\hat{j} and βi^−52j^\beta\hat{i}-52\hat{j} are collinear if:

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  95. 95
    Consider vectors a⃗=(0,1,1)\vec{a}=(0,1,1) and b⃗=(1,0,1)\vec{b}=(1,0,1). (1) Number of unit vectors perpendicular to both is only one. (2) Angle between them is π/3\pi/3. Which is/are correct?

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  96. 96
    If LL is the line with direction ratios ⟨3,−2,6⟩\langle3,-2,6\rangle passing through (1,−1,1)(1,-1,1), what are points on LL at distance 2 from (1,−1,1)(1,-1,1)?

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  97. 97
    Which one of the planes is parallel to the line x−23=y−34=z−45\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}?

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  98. 98
    What is the angle between the lines 2x=3y=−z2x=3y=-z and 6x=−y=−4z6x=-y=-4z?

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  99. 99
    What is the equation of the sphere concentric with x2+y2+z2−2x−6y−8z−5=0x^2+y^2+z^2-2x-6y-8z-5=0 and passing through the origin?

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  100. 100
    A point PP lies on the line joining A(1,2,3)A(1,2,3) and B(2,10,1)B(2,10,1). If zz-coordinate of PP is 7, what is the sum of the other two coordinates?

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  101. 101
    The sum of deviations of nn numbers from 10 and 20 are pp and qq respectively. If (p−q)2=10000(p-q)^2=10000, then what is the value of nn?

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  102. 102
    If Xˉ=20\bar{X}=20 is the mean of 10 observations x1,x2,…,x10x_1, x_2, \ldots, x_{10}, then what is the value of ∑i=110(3xi−45)\displaystyle\sum_{i=1}^{10}\left(\frac{3x_i-4}{5}\right)?

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  103. 103
    If the mean and sum of squares of 10 observations are 40 and 16160 respectively, then what is the standard deviation?

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  104. 104
    Three dice are thrown. What is the probability of getting a sum which is a perfect square?

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  105. 105
    A, B, C, D are mutually exclusive and exhaustive. If 2P(A)=3P(B)=4P(C)=5P(D)2P(A)=3P(B)=4P(C)=5P(D), then what is 77P(A)77P(A) equal to?

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  106. 106
    Two distinct natural numbers from 1 to 9 are picked at random. What is the probability that their product has 1 in its unit place?

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  107. 107
    Two dice are thrown. What is the probability that difference of numbers is 2 or 3?

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  108. 108
    What is the mean of the numbers 1,2,3,…,101,2,3,\ldots,10 with frequencies (90),(91),…,(99)\binom{9}{0},\binom{9}{1},\ldots,\binom{9}{9} respectively?

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  109. 109
    The probability that a person recovers from a disease is 0.8. What is the probability that exactly 2 persons out of 5 will recover?

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  110. 110
    A building may collapse due to faulty design (prob 0.1) or not (prob 0.9). Collapse prob: 0.95 if faulty, 0.45 if not. If building collapsed, what is probability it was due to faulty design?

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  111. 111
    If rr is the correlation coefficient between xx and yy, then what is the correlation coefficient between (3x+4)(3x+4) and (−3y+3)(-3y+3)?

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  112. 112
    A fair coin is tossed 6 times. What is the probability of getting in the 6th toss a result different from those in the first five tosses?

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  113. 113
    If HH is the Harmonic Mean of (104)\binom{10}{4}, (105)\binom{10}{5}, and (106)\binom{10}{6}, then what is the value of 270H\frac{270}{H}?

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  114. 114
    In a class of nn students including P and Q, what is the probability that P and Q sit together if seats assigned randomly?

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  115. 115
    In B(n,p)B(n,p), n=6n=6 and 9P(X=4)=P(X=2)9P(X=4)=P(X=2). What is pp equal to?

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  116. Three boys P, Q, R and three girls S, T, U are to be arranged in a row for a group photograph.
    116
    What is the probability that all three boys sit together?

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  117. 117
    What is the probability that boys and girls sit alternatively?

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  118. 118
    What is the probability that no two girls sit together?

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  119. 119
    What is the probability that P and Q take the two end positions?

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  120. 120
    What is the probability that Q and U sit together?

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