NDA 2022 (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 $\Delta_1 = \begin{vmatrix}1 & p & q\\1 & q & r\\1 & r & p\end{vmatrix}$
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2 $(a-b)(b-c)(c-a) = 2$
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3 $\begin{vmatrix}a & b & c\\b & c & a\\c & a & b\end{vmatrix}$
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4 $A=[m\ \ n]$
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5 $A = \begin{pmatrix}2\sin\theta & \cos\theta & 0\\-2\cos\theta & \sin\theta & 0\\-1 & 1 & 1\end{pmatrix}$
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6 $k$
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7 $A$
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8 $A$
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9 $A$
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10 $A$
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11 $\left(1+\frac{2}{x}\right)^9\left(1-\frac{2}{x}\right)^9$
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12 $(x+y)^{10}$
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13 $C(3n, 2n) = C(3n, 2n-7)$
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14 $C(51,21)-C(51,22)+C(51,23)-C(51,24)+C(51,25)-C(51,26)+C(51,27)-C(51,28)+C(51,29)-C(51,30)$
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15 $How many odd numbers between 300 and 400 are there in which none of the digits is repeated?$
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16 $How many permutations are there of the letters of the word 'TIGER' in which the vowels should not occupy the even positions?$
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17 $\alpha$
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18 $\alpha$
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19 $x^2-ax-bx-cx+bc+ca=0$
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20 $\alpha$
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21 $n$
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22 $2(2\times1)+3(3\times2\times1)+4(4\times3\times2\times1)+5(5\times4\times3\times2\times1)+\ldots+9(9\times8\times7\times6\times5\times4\times3\times2\times1)+2$
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23 $A=\{\{1,2,3\}\}$
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24 $a, b, c$
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25 $\frac{a+b}{2}, b, \frac{b+c}{2}$
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26 $\cot^215^\circ+\tan^215^\circ$
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27 $ABC$
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28 $\alpha+\beta=\frac{\pi}{4}$
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29 $\tan(45°+\theta)=1+\sin2\theta$
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30 $\sin2\theta=\cos3\theta$
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31 $\tan\theta=-\frac{5}{12}$
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32 $\cos^4\frac{7\pi}{8}+\cos^4\frac{5\pi}{8}$
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33 $\sin^2\left(\frac{\pi}{4}+\theta\right)-\sin^2\left(\frac{\pi}{4}-\theta\right)$
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34 $h$
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35 $x$
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36 $\tan^{-1}\left(\frac{1}{2}\right)+\tan^{-1}\left(\frac{x}{3}\right)=\frac{\pi}{4}$
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37 $3\sin^{-1}x+\cos^{-1}x=\pi$
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38 $\tan\alpha+\tan\beta=1-\tan\alpha\cdot\tan\beta$
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39 $(1+\tan\theta)(1+\tan9\theta)=2$
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40 $\sin0°+\sin10°+\sin20°+\sin30°+\ldots+\sin360°$
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41 $A=\{1,2,3,4\}$
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42 $A$
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43 $A$
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44 $R$
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45 $R$
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46 $\frac{1}{1+i}$
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47 $\left(\frac{\sqrt{-3}}{2}-\frac{1}{2}\right)^{200}$
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48 $\frac{n!}{3!}$
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49 $In how many ways can a team of 5 players be selected out of 9 players so as to exclude two particular players?$
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50 $\left(x+\frac{1}{x}\right)^{2n}$
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51 $If the sum of the first 9 terms of an AP is equal to sum of the first 11 terms, then what is the sum of the first 20 terms?$
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52 $\frac{1}{10}$
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53 $(1110011)_2\div(10111)_2$
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54 $x^3+y^3=(100010111)_2$
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55 $5x-4y+12<0$
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56 $y=[x]$
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57 $1+\left(\frac{dy}{dx}\right)^2=\left(\frac{d^2y}{dx^2}\right)^{\frac{4}{3}}$
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58 $A radioactive substance decays at a rate proportional to the amount of substance present. If half of the substance decays in 100 years, then what is the decay constant (proportionality constant)?$
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59 $f(x)=\sqrt{1-(x-1)^2}$
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60 $y^2=4kx$
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61 $\int_0^{\frac{\pi}{4}}\frac{dx}{(\sin x+\cos x)^2}$
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62 $\int(\sin x)^{-1/2}(\cos x)^{-3/2}\,dx$
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63 $I_1=\int\frac{e^x\,dx}{e^x+e^{-x}}$
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64 $\int_{-2}^{-1}\frac{x}{|x|}\,dx$
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65 $\sin4x+2x$
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66 $f(x)=\frac{1}{\tan x+\cot x}$
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67 $4f(x)-f\left(\frac{1}{x}\right)=\left(2x+\frac{1}{x}\right)\left(2x-\frac{1}{x}\right)$
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68 $f(x)=4x+3$
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69 $x^y y^x=1$
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70 $y=(x^x)^x$
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71 $y=[x+1]$
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72 $\frac{dy}{dx}=(\ln5)y$
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73 $f(x)=10^x$
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74 $\lim_{x\to0}x^3(\text{cosec}\,x)^2$
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75 $\lim_{x\to1}\frac{x^3-1}{\sqrt{x}-1}$
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76 $f(x)=\frac{x^3}{3}-\frac{7x^2}{2}+6x+5$
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77 $f(x)=\frac{m}{x}+2nx+1$
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78 $y=x$
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79 $xy=4225$
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80 $x\frac{dy}{dx}-2y=0$
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81 $(-5,0)$
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82 $(1,-2)$
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83 $y-2=0$
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84 $(-3,0)$
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85 $x^2+2y^2=1$
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86 $a, b, c$
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87 $x+2y+2=0$
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88 $ax+by+c=0$
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89 $x+y=p$
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90 $(x,y)$
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Consider the following for the next three (03) items that follow: The plane $6x+ky+3z-12=0$ where $k eq0$ meets the coordinate axes at $A$ , $B$ and $C$ respectively. The equation of the sphere passing through the origin and $A$ , $B$ , $C$ is $x^2+y^2+z^2-2x-3y-4z=0$ .
91 $k$
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92 $p$
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93 $What is the equation of the line through the origin and the centre of the sphere?$
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Consider the following for the next two (02) items that follow: Let the plane $\frac{2x}{k}+\frac{2y}{3}+\frac{z}{3}=2$ pass through the point $(2,3,-6)$ .
94 $What are the direction ratios of a normal to the plane?$
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95 $p$
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96 $4\hat{i}+\hat{j}-3\hat{k}$
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97 $\vec{a}, \vec{b}, \vec{c}$
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98 $Consider the following statements: 1. Dot product over vector addition is distributive 2. Cross product over vector addition is distributive 3. Cross product of vectors is associative Which of the above statements is/are correct?$
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99 $\vec{a}, \vec{b}, \vec{c}$
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100 $\vec{a}$
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101 $Two digits out of 1, 2, 3, 4, 5 are chosen at random and multiplied together. What is the probability that the last digit in the product appears as 0?$
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102 $The frequency curve (assuming unimodal) corresponding to the data obtained in an experiment is skewed to the left. What conclusion can be drawn from the curve?$
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103 $The variance of five positive observations is 3.6. If four of the observations are 2, 2, 4, 5 then what is the remaining observation?$
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104 $What is the arithmetic mean of 50 terms of an AP with first term 4 and common difference 4?$
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105 $What is the coefficient of mean deviation of 21, 34, 23, 39, 26, 37, 40, 20, 33, 27 (taken from mean)?$
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Consider the following for the next three (03) items that follow: The algebraic sum of the deviations of a set of values $x_1, x_2, x_3, \ldots, x_n$ measured from 100 is $-20$ and the algebraic sum of the deviations of the same set of values measured from 92 is 140.
106 $What is the mean of the values?$
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107 $What is the algebraic sum of the deviations of the same set of values measured from 99?$
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108 $y$
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Consider the following data for the next three (03) items that follow: The marks obtained by 51 students in a class are in AP with its first term 4 and common difference 3.
109 $What is the mean of the marks?$
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110 $What is the median of the marks?$
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111 $What is the sum of the deviations measured from the median?$
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Consider the following data for the next three (03) items that follow: There are 90 applicants for a job. Some of them are graduates. Some of them have less than three years experience.
Number of graduates Number of non-graduates
At least 3 years experience 18 9
Less than 3 years experience 36 27
Let G be the event that the first applicant interviewed is a graduate and T be the event that first applicant interviewed has at least 3 years experience.
112 $P\left(G\cap\overline{T}\right)$
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113 $P\left(G|\overline{T}\right)$
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114 $P\left(\overline{T}|\overline{G}\right)$
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Consider the following data for the next three (03) items that follow: The incidence of suffering from a disease among workers in an industry has a chance of $33\frac{1}{3}\%$ .
115 $What is the probability that exactly 3 out of 6 workers suffer from a disease?$
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116 $What is the probability that no one out of 6 workers suffers from a disease?$
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117 $What is the probability that at least one out of 6 workers suffers from a disease?$
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Consider the following frequency distribution for the next three (03) items that follow: Class: 0-20, 20-40, 40-60, 60-80, 80-100; Frequency: 17, $p+q$ , 32, $p-3q$ , 19. The total frequency is 120. The mean is 50.
118 $p$
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119 $q$
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120 $If the frequency of each class is doubled, then what would be the mean?$
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	Number of graduates	Number of non-graduates
At least 3 years experience	18	9
Less than 3 years experience	36	27

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