NDA 2020 (Combined I & 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 $A = \begin{pmatrix} 1-i & i \\ -i & 1-i \end{pmatrix}$
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2 $x$
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3 $(1+2x-x^2)^6 = a_0 + a_1x + a_2x^2 + \ldots + a_{12}x^{12}$
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4 $C(20, n+2) = C(20, n-2)$
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5 $k$
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6 $(1101101 + 1011011)_2$
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7 $\frac{1}{10}\log_5 1024 - \log_5 10 + \frac{1}{5}\log_5 3125$
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8 $x = \log_c(ab)$
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9 $A = \begin{pmatrix}x+y & y \\ 2x & x-y\end{pmatrix}$
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10 $1.5 \leq x \leq 4.5$
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11 $S=\{1,2,3,\ldots\}$
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12 $\begin{vmatrix}i & i^2 & i^3 \\ i^4 & i^6 & i^8 \\ i^9 & i^{12} & i^{15}\end{vmatrix}$
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13 $A = \begin{pmatrix}a & h & g \\ h & b & f \\ g & f & c\end{pmatrix}$
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14 $What is the number of ways in which the letters of the word 'ABLE' can be arranged so that the vowels occupy even places?$
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15 $What is the maximum number of points of intersection of 5 non-overlapping circles?$
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Directions for the following three (03) items: Consider the following Venn diagram, where X, Y and Z are three sets. Let the number of elements in Z be denoted by n(Z) which is equal to 90.
16 $If the number of elements in Y and Z are in the ratio 4 : 5, then what is the value of b?$
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17 $n(X)+n(Y)+n(Z)-n(X\cap Y)-n(Y\cap Z)-n(X\cap Z)+n(X\cap Y\cap Z)$
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18 $If the number of elements belonging to neither X, nor Y, nor Z is equal to p, then what is the number of elements in the complement of X?$
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Directions for the following two (02) items: Read the following information and answer the two items that follow: Let $\frac{\tan 3A}{\tan A} = K$ , where $\tan A \neq 0$ and $K \neq \frac{1}{3}$ .
19 $\tan^2 A$
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20 $\tan A$
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Directions for the following two (02) items: Read the following information and answer the two items that follow: ABCD is a trapezium such that AB and CD are parallel and BC is perpendicular to them. Let ∠ADB = θ, ∠ABD = α, BC = p and CD = q.
21 $AD\sin\theta = AB\sin\alpha$
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22 $What is AB equal to?$
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23 $\tan\theta=\frac{\cos 17°-\sin 17°}{\cos 17°+\sin 17°}$
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24 $\cos 2B = 3\sin^2 A$
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25 $\sin 3x + \cos 3x + 4\sin^3 x - 3\sin x + 3\cos x - 4\cos^3 x$
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26 $y = 2 + \cos x$
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27 $8\cos 10°\cdot\cos 20°\cdot\cos 40°$
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28 $\cos 48°-\cos 12°$
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29 $\sin B=\frac{1}{3}$
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30 $A=2B$
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Directions for the following three (03) items: Read the following information and answer the three items that follow: Let $a\sin^2 x + b\cos^2 x = c$ ; $b\sin^2 y + a\cos^2 y = d$ and $p\tan x = q\tan y$ .
31 $\tan^2 x$
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32 $\frac{d-a}{b-d}$
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33 $\frac{p^2}{q^2}$
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Directions for the following three (03) items: Read the following information and answer the three items that follow: Let $t_n = \sin^n\theta + \cos^n\theta$ .
34 $\frac{t_3-t_5}{t_5-t_7}$
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35 $t_1^2 - t_2$
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36 $t_{10}$
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Directions for the following three (03) items: Read the following information and answer the three items that follow: Let $\alpha = \beta = 15°$ .
37 $\sin\alpha + \cos\beta$
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38 $\sin 7\alpha - \cos 7\beta$
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39 $\sin(\alpha+1°)+\cos(\beta+1°)$
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40 $\sin x+\sin y=\cos y-\cos x$
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41 $A$
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42 $p^2$
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43 $\cot\alpha$
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44 $\alpha$
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45 $\frac{1-i\sqrt{3}}{1+i\sqrt{3}}$
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46 $\frac{\cos\theta+i\sin\theta}{\cos\theta-i\sin\theta}$
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47 $\{1,2,3,4\}$
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48 $p$
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49 $(1+x)^9$
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50 $A(\text{adj }A)=(\text{adj }A)A$
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51 $(x-2a)(x-2b)+(y-2c)(y-2d)=0$
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52 $(1,-1)$
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53 $P(x,y)$
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54 $x+2=0$
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55 $y^2=x$
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56 $(a,b)$
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57 $D(2,5)$
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58 $(0,k)$
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59 $2-\sqrt{3}$
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60 $3x-4y-5=0$
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61 $(1,-2,3)$
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62 $(2,3,4)$
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63 $\langle a+b, b+c, c+a\rangle$
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64 $Into how many compartments do the coordinate planes divide the space?$
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65 $What is the equation of the plane which cuts an intercept 5 units on the z-axis and is parallel to xy-plane?$
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66 $\hat{a}$
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67 $|\overrightarrow{OA}|=12$
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68 $2\hat{i}-4\hat{j}+5\hat{k}$
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69 $|\vec{a}\times\vec{b}|^2+|\vec{a}\cdot\vec{b}|^2=144$
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70 $\vec{a}=2\hat{i}-3\hat{j}+\hat{k}$
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71 $\lim_{x\to1}\frac{x+x^2+x^3-3}{x-1}$
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72 $0.7$
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73 $\lim_{x\to1}\frac{x^4-1}{x-1}=\lim_{x\to k}\frac{x^3-k^3}{x^2-k^2}$
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74 $k\frac{dy}{dx}=\int\left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{2}{3}}dx$
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75 $\lim_{x\to0}\frac{\sin x\log(1-x)}{x^2}$
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76 $f(x)=3x^2-5x+p$
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77 $e^{\theta\varphi}=c+4\theta\varphi$
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78 $p(x)=(4e)^{2x}$
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79 $\int_0^{\pi/4}(\tan^3 x+\tan x)\,dx$
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80 $y=3x^2+2$
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81 $f(x)=\frac{\sin x}{x}$
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82 $dy=(1+y^2)\,dx$
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83 $\int(e^{\log x}+\sin x)\cos x\,dx$
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84 $f(x)=\cos^{-1}(x-2)$
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85 $y^2=2x$
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86 $f(x)=2x-x^2$
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87 $x^m y^n=a^{m+n}$
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88 $\int\frac{dx}{x(x^n+1)}$
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89 $|x-1|$
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90 $k$
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91 $f(x)=e^{-|x|}$
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92 $\sin x\cdot\cos x$
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93 $\lim_{x\to0}\frac{3^x+3^{-x}-2}{x}$
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94 $\tan^{-1}x$
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95 $u(x,y)=c$
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96 $3\cos\left(A+\frac{\pi}{3}\right)$
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97 $f(x)=\ln x$
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98 $y=\frac{1}{x-1}$
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99 $\ln\left(\frac{dy}{dx}\right)=x$
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100 $l$
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101 $x$
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102 $If three dice are rolled under the condition that no two dice show the same face, then what is the probability that one of the faces is having the number 6?$
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103 $P(A\cup B)=\frac{5}{6}$
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104 $n$
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105 $n$
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Directions for the following three (03) items: Read the following information and answer the three items that follow:
Marks Physics Mathematics
10-20 8 10
20-30 11 21
30-40 30 38
40-50 26 15
50-60 15 10
60-70 10 6
106 $The difference between number of students under Physics and Mathematics is largest for the interval$
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107 $30-40$
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108 $What is the mean of marks in Physics?$
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109 $-\sqrt{6}, -\sqrt{5}, -\sqrt{4}, -1, 1, \sqrt{4}, \sqrt{5}, \sqrt{6}$
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110 $\sum x_i=20$
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111 $What is the probability that February of a leap year selected at random, will have five Sundays?$
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112 $The arithmetic mean of 100 observations is 40. Later, it was found that an observation '53' was wrongly read as '83'. What is the correct arithmetic mean?$
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113 $\frac{1}{7}$
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114 $A dealer has a stock of 15 gold coins out of which 6 are counterfeits. A person randomly picks 4 of the 15 gold coins. What is the probability that all the coins picked will be counterfeits?$
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115 $A committee of 3 is to be formed from a group of 2 boys and 2 girls. What is the probability that the committee consists of 2 boys and 1 girl?$
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116 $In a lottery of 10 tickets numbered 1 to 10, two tickets are drawn simultaneously. What is the probability that both the tickets drawn have prime numbers?$
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117 $\bar{X}=65$
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118 $n=10$
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119 $P(A)=0.6$
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120 $Three cooks X, Y and Z bake a special kind of cake, and with respective probabilities 0\cdot02, 0\cdot03 and 0\cdot05, it fails to rise. In the restaurant where they work, X bakes 50%, Y bakes 30% and Z bakes 20% of cakes. What is the proportion of failures caused by X?$
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Marks	Physics	Mathematics
10-20	8	10
20-30	11	21
30-40	30	38
40-50	26	15
50-60	15	10
60-70	10	6

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