NDA 2023 (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 $\omega$
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2 $What is the number of 6-digit numbers that can be formed only by using 0, 1, 2, 3, 4 and 5 (each once); and divisible by 6?$
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3 $What is the binary number equivalent to decimal number 1011?$
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4 $A$
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5 $\alpha$
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6 $A$
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7 $A$
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8 $x+2y+z=4$
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9 $AX=B$
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10 $\begin{vmatrix}0&x-a&x-b\\0&0&x-c\\x+b&x+c&1\end{vmatrix}=0$
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11 $2-i\sqrt{3}$
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12 $z=\dfrac{1+i\sqrt{3}}{1-i\sqrt{3}}$
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13 $a$
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14 $\log_x a$
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15 $2^{\frac{1}{c}}$
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16 $\frac{5}{2}$
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17 $k$
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18 $x$
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19 $(25)!+1$
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20 $z$
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21 $|x-4|+|x-7|=15$
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22 $f:A\to B$
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23 $\alpha$
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24 $\left(x^{-\frac{8}{3}}+x^{2}\log_{10}x\right)^{8}$
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25 $(3x-y)^{4}(x+3y)^{4}$
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26 $p$
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27 $n$
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28 $m$
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29 $x$
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30 $5-digit numbers are formed using the digits 0, 1, 2, 4, 5 without repetition. What is the percentage of numbers which are greater than 50,000?$
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Consider the following for the next two (02) items that follow: Let $\sin?eta$ be the GM of $\sin?lpha$ and $\cos?lpha$ ; $\tan\gamma$ be the AM of $\sin?lpha$ and $\cos?lpha$ .
31 $\cos2\beta$
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32 $\sec2\gamma$
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Consider the following for the next two (02) items that follow: A flagstaff 20 m long standing on a pillar 10 m high subtends an angle $\tan^{-1}(0.5)$ at a point $P$ on the ground. Let $\theta$ be the angle subtended by the pillar at this point $P$ .
33 $x$
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34 $\tan\theta$
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Consider the following for the next two (02) items that follow: The perimeter of a triangle $ABC$ is 6 times the AM of sine of angles of the triangle. Further $BC=\sqrt{3}$ and $CA=1$ .
35 $What is the perimeter of the triangle?$
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36 $ABC$
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Consider the following for the next two (02) items that follow: Let $x=\dfrac{\sin^{2}A+\sin A+1}{\sin A}$ where $0<A\leq\dfrac{\pi}{2}$ .
37 $x$
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38 $A$
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Consider the following for the next two (02) items that follow: In the triangle $ABC$ , $a^{2}+b^{2}+c^{2}=ac+\sqrt{3}\,bc$ .
39 $What is the nature of the triangle?$
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40 $c=8$
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Consider the following for the next two (02) items that follow: Consider the function $f(x)=|x-2|+|3-x|+|4-x|$ , where $x\in R$ .
41 $x$
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42 $What is the minimum value of the function?$
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Consider the following for the next two (02) items that follow: Consider the sum $S=0!+1!+2!+3!+4!+\ldots+100!$ .
43 $S$
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44 $S$
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Consider the following for the next two (02) items that follow: In a triangle $PQR$ , $P$ is the largest angle and $\cos P=\frac{1}{3}$ . Further the in-circle of the triangle touches the sides $PQ$ , $QR$ and $RP$ at $N$ , $L$ and $M$ respectively such that the lengths $PN$ , $QL$ and $RM$ are $n$ , $n+2$ , $n+4$ respectively where $n$ is an integer.
45 $n$
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46 $What is the length of the smallest side?$
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Consider the following for the next two (02) items that follow: Given that $\sin x+\cos x+\tan x+\cot x+\sec x+\cosec x=7$ .
47 $The given equation can be reduced to$
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48 $\sin2x=a-b\sqrt{c}$
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Consider the following for the next two (02) items that follow: A quadratic equation is given by $(3+2\sqrt{2})x^{2}-(4+2\sqrt{3})x+(8+4\sqrt{3})=0$ .
49 $What is the HM of the roots of the equation?$
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50 $What is the GM of the roots of the equation?$
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Consider the following for the next two (02) items that follow: Let $\Delta(a,b,c,\alpha)=\begin{vmatrix}a&b&a\alpha+b\\b&c&b\alpha+c\\a\alpha+b&b\alpha+c&0\end{vmatrix}$ .
51 $\Delta(a,b,c,\alpha)=0$
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52 $\Delta(7,4,2,\alpha)=0$
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Consider the following for the next two (02) items that follow: Given that $m(\theta)=\cot^{2}\theta+n^{2}\tan^{2}\theta+2n$ , where $n$ is a fixed positive real number.
53 $m(\theta)$
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54 $m$
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Consider the following for the next two (02) items that follow: A quadrilateral is formed by the lines $x=0$ , $y=0$ , $x+y=1$ and $6x+y=3$ .
55 $What is the equation of diagonal through origin?$
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56 $What is the equation of other diagonal?$
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Consider the following for the next two (02) items that follow: $P(x,y)$ is any point on the ellipse $x^{2}+4y^{2}=1$ . Let $E$ , $F$ be the foci of the ellipse.
57 $PE+PF$
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58 $\left(\frac{\sqrt{3}}{2},0\right)$
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Consider the following for the next two (02) items that follow: The line $y=x$ partitions the circle $(x-a)^{2}+y^{2}=a^{2}$ in two segments.
59 $What is the area of minor segment?$
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60 $What is the area of major segment?$
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Consider the following for the next two (02) items that follow: Let $A(1,-1,2)$ and $B(2,1,-1)$ be the end points of the diameter of the sphere $x^{2}+y^{2}+z^{2}+2ux+2vy+2wz-1=0$ .
61 $u+v+w$
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62 $P(x,y,z)$
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Consider the following for the next two (02) items that follow: Consider two lines whose direction ratios are $(2,-1,2)$ and $(k,3,5)$ . They are inclined at an angle $\dfrac{\pi}{4}$ .
63 $k$
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64 $What are the direction ratios of a line which is perpendicular to both the lines?$
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Consider the following for the next two (02) items that follow: Let $\vec{a}=3\hat{i}+3\hat{j}+3\hat{k}$ and $\vec{c}=\hat{j}-\hat{k}$ . Let $\vec{b}$ be such that $\vec{a}\cdot\vec{b}=27$ and $\vec{a}\times\vec{b}=9\vec{c}$ .
65 $\vec{b}$
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66 $(\vec{a}+\vec{b})$
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Consider the following for the next two (02) items that follow: Let a vector $\vec{a}=4\hat{i}-8\hat{j}+\hat{k}$ make angles $\alpha$ , $\beta$ , $\gamma$ with the positive directions of $x$ , $y$ , $z$ axes respectively.
67 $\cos\alpha$
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68 $\cos2\beta+\cos2\gamma$
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Consider the following for the next two (02) items that follow: The position vectors of two points $A$ and $B$ are $\hat{i}-\hat{j}$ and $\hat{j}+\hat{k}$ respectively.
69 $(-1,-3,1)$
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70 $\overrightarrow{AB}$
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Consider the following for the next three (03) items that follow: Let $f(x)=Pe^{x}+Qe^{2x}+Re^{3x}$ , where $P$ , $Q$ , $R$ are real numbers. Further $f(0)=6$ , $f'(\ln3)=282$ and $\int_{0}^{\ln2}f(x)\,dx=11$ .
71 $Q$
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72 $R$
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73 $f'(0)$
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Consider the following for the next two (02) items that follow: Suppose $E$ is the differential equation representing family of curves $y^{2}=2cx+2c\sqrt{c}$ where $c$ is a positive parameter.
74 $What is the order of the differential equation?$
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75 $What is the degree of the differential equation?$
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Consider the following for the next three (03) items that follow: Let $f(x)=\begin{vmatrix}\cos x & x & 1\\2\sin x & x^{2} & 2x\\\tan x & x & 1\end{vmatrix}$ .
76 $f(0)$
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77 $\lim_{x\to0}\dfrac{f(x)}{x}$
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78 $\lim_{x\to0}\dfrac{f(x)}{x^{2}}$
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Consider the following for the next two (02) items that follow: Let $f(x)=\sin[\pi^{2}]x+\cos[-\pi^{2}]x$ where $[\cdot]$ is a greatest integer function.
79 $f\!\left(\dfrac{\pi}{2}\right)$
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80 $f\!\left(\dfrac{\pi}{4}\right)$
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Consider the following for the next three (03) items that follow: Let $I_{1}=\int_{0}^{\pi}\dfrac{x}{1+\cos^{2}x}\,dx$ and $I_{2}=\int_{0}^{\pi}\dfrac{1}{1+\sin^{2}x}\,dx$ .
81 $\dfrac{I_{1}+I_{2}}{I_{1}-I_{2}}$
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82 $8I_{1}^{2}$
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83 $I_{2}$
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Consider the following for the next two (02) items that follow: Let $I=\int_{a}^{b}\dfrac{|x|}{x}\,dx$ , $a<b$ .
84 $I$
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85 $I$
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Consider the following for the next three (03) items that follow: Let $f(x)=|\ln x|,\, x\neq1$ .
86 $f(x)$
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87 $f(x)$
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88 $f\circ f(x)$
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Consider the following for the next two (02) items that follow: Let $f(x)=\begin{cases}x+6, & x\leq1\\px+q, & 1<x<2\\5x, & x\geq2\end{cases}$ and $f(x)$ is continuous.
89 $p$
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90 $q$
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91 $f(x)=\ln x$
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92 $\sin^{2}x$
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93 $m$
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94 $\cosec(x^{\circ})$
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95 $\left(\dfrac{dy}{dx}\right)^{2}-x\dfrac{dy}{dx}=0$
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96 $f(x)=x^{2}+2$
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97 $f(x)=x+|x|$
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98 $f(x)=x(4x^{2}-3)$
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99 $\lim_{x\to5}\dfrac{5-x}{|x-5|}$
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100 $\lim_{x\to1}\dfrac{x^{9}-1}{x^{3}-1}$
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101 $The mean and variance of five observations are 14 and 13.2 respectively. Three of the five observations are 11, 16 and 20. What are the other two observations?$
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102 $A$
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103 $\frac{1}{4}$
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104 $A coin is biased so that heads comes up thrice as likely as tails. In four independent tosses of the coin, what is probability of getting exactly three heads?$
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105 $X$
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106 $x+4y+1=0$
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107 $p$
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108 $m$
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109 $(-1,1)$
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110 $A die is thrown 10 times and obtained the following outputs: 1, 2, 1, 1, 2, 1, 4, 6, 5, 4. What will be the mode of data so obtained?$
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111 $x$
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112 $-1,1,4,3,8,12,17,19,9,11$
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113 $P$
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114 $G$
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115 $H$
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116 $P$
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117 $One bag contains 3 white and 2 black balls, another bag contains 2 white and 3 black balls. Two balls are drawn from the first bag and put into the second bag and then a ball is drawn from the second bag. What is the probability that it is white?$
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118 $Three dice are thrown. What is the probability that each face shows only multiples of 3?$
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119 $What is the probability that the month of December has 5 Sundays?$
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120 $n$
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$x$	1	2	3	5
$f$	4	6	9	7

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