Mathematics
1. Let $A$ and $B$ be matrices of order $3 \times 3$. If $|A| = \frac{1}{2 \sqrt{2}}$ and $|B| = \frac{1}{729}$, then what 2. If $z$ is any complex number and $i z^3+z^2-z+i=0$, where $i=\sqrt{-1}$, then what is the value of $(|z|+1)^2$ ? 3. What is the sum of all four-digit numbers formed by using all digits 0, 1, 4, 5 without repetition of digits? 4. If $x, y$ and $z$ are the cube roots of unity, then what is the value of $xy + yz + zx$? 5. A man has 7 relatives (4 women and 3 men). His wife also has 7 relatives (3 women and 4 men). In how many ways can they 6. A triangle $PQR$ is such that 3 points lie on the side $PQ$, 4 points on $QR$ and 5 points on $RP$ respectively. Triangl 7. If $\log _b a=p, \log _d c=2 p$ and $\log _f e=3 p$, then what is $(a c e)^{\frac{1}{p}}$ equal to ? 8. If $\sqrt{2}$ and $\sqrt{3}$ are roots of the equation $a_0 + a_1 x + a_2 x^2 + a_3 x^3 + x^4 = 0$ where $a_0, a_1, a_2, 9. Let $z_1$ and $z_2$ be two complex numbers such that $\left|\frac{z_1 + z_2}{z_1 - z_2}\right| = 1$, then what is $\oper 10. If $26! = n8^k$, where $k$ and $n$ are positive integers, then what is the maximum value of $k$? 11. Consider the following statements in respect of two non-singular matrices $A$ and $B$ of the same order $n$:
1.$adj(AB) 12. Consider the following statements in respect of a non-singular matrix $A$ of order $n$:1. $A(\text{adj}A^T) = A(\text{ad 13. How many four-digit natural numbers are there such that all of the digits are even? 14. If $\omega \neq 1$ is a cube root of unity, then what are the solutions of $(z-100)^3 + 1000 = 0$? 15. What is $(1 + i)^4 + (1 - i)^4$ equal to, where $i=\sqrt{-1}$? 16. Consider the following statements in respect of a skew-symmetric matrix $A$ of order $3$:1. All diagonal elements are ze 17. Four digit numbers are formed by using the digits 1, 2, 3, 5 without repetition of digits. How many of them are divisibl 18. What is the remainder when $2^{120}$ is divided by 7? 19. For what value of $n$ is the determinant
$$
\left|\begin{array}{ccc}
C(9,4) & C(9,3) & C(10, n-2) \\
C(11,6) & C(11,5) & 20. If ABC is a triangle, then what is the value of the determinant
$$
\left|\begin{array}{ccc}
\cos C & \sin B & 0 \\
\tan 21. What is the number of different matrices, each having 4 entries that can be formed using 1, 2, 3, 4 (repetition is allow 22. Let $A = \{x \in \mathbb{R} : -1 <x <1\}$. Which of the following is/are bijective functions from A to itself?1. $ 23. Let $R$ be a relation on the open interval $(-1, 1)$ and is given by $R = \{(x, y) : |x + y| 24. For any three non-empty sets $A, B, C$, what is $$(A \cup B - \{(A - B) \cup (B - A) \cup (A \cap B)\})$$ equal to? 25. If $a, b, c$ are the sides of a triangle $ABC$, then what is $$ \begin{vmatrix} a^2 & b \sin A & c \sin A \\ b \sin A & 26. If $a, b, c$ are in AP; $b, c, d$ are in GP; $c, d, e$ are in HP, then which of the following is/are correct?1. $a, c,$ 27. What is the number of solutions of $\log_4(x - 1) = \log_2(x - 3)$? 28. For $x \geq y > 1$, let $\log_x\left( \frac{x}{y} \right) + \log_y\left(\frac{y}{x}\right) = k$, then the value of $k$ c 29. If $A=\left[\begin{array}{ccc}\sin 2 \theta & 2 \sin ^2 \theta-1 & 0 \\ \cos 2 \theta & 2 \sin \theta \cos \theta & 0 \\ 30. What is the coefficient of $x^{10}$ in the expansion of $(1-x^2)^{20}\left(2-x^2-\frac{1}{x^2}\right)^{-5}$? 31. If the 4th term in the expansion of $\left(mx + \frac{1}{x}\right)^n$ is $\frac{5}{2}$, then what is the value of $mn$? 32. If $a$, $b$, and $c$ $(a > 0, c > 0)$ are in GP, then consider the following in respect of the equation $ax^2 + bx + c 33. If $x^2 + mx + n$ is an integer for all integral values of $x$, then which of the following is/are correct?1. $m$ must b 34. In a binomial expansion of $(x+y)^{2 n+1}(x-y)^{2 n+1}$, the sum of middle terms is zero. What is the value of $\left(\f 35. Let $A = \{1, 2, 3, 4, 5\}$ and $B = \{6, 7\}$. What is the number of onto functions from $A$ to $B$? 36. $$
\text { What is } \frac{\sqrt{3} \cos 10^{\circ}-\sin 10^{\circ}}{\sin 25^{\circ} \cos 25^{\circ}} \text { equal to ? 37. What is $\sin 9^\circ - \cos 9^\circ$ equal to? 38. If in a triangle $ABC$, $\sin^3A + \sin^3B + \sin^3C = 3\sin A \sin B \sin C$, then what is the value of the determinant 39. If $\cos^{-1} x = \sin^{-1} x$, then which one of the following is correct? 40. What is the number of solutions of $(\sin \theta - \cos \theta)^2 = 2$ where $-\pi 41. $ABC$ is a triangle such that angle $C = 60^{\circ}$, then what is $\frac{\cos A + \cos B}{\cos \left(\frac{A - B}{2}\ri 42. What is $\sqrt{15 + \cot^2 \left( \frac \pi 4 - 2 \cot^{-1} 3 \right)}$ equal to? 43. What is the value of $\sin10^{\circ} \cdot \sin50^{\circ} + \sin50^{\circ} \cdot \sin250^{\circ} + \sin250^{\circ} \cdot 44. What is $\tan^{-1} \left( \frac{a}{b} \right) - \tan^{-1} \left( \frac{a - b}{a + b} \right)$ equal to? 45. Under which one of the following conditions does the equation $\left(\cos \beta-1\right)x^2+(\cos \beta)x+\sin \beta=0$ 46. In a triangle $ABC$, $AB=16 \text{ cm}, BC=63 \text{ cm}$ and $AC=65 \text{ cm}$. What is the value of $\cos 2A+\cos 2B+ 47. If $f(\theta)=\frac{1}{1+\tan \theta}$ and $\alpha+\beta=\frac{5\pi}{4}$, then what is the value of $f(\alpha) f(\beta)$ 48. If $\tan \alpha$ and $\tan \beta$ are the roots of the equation $x^2-6x+8=0$, then what is the value of $\cos(2 \alpha+2 49. What is the value of $\tan 65^\circ +2 \tan 45^\circ-2 \tan 40^\circ-\tan 25^\circ$? 50. Consider the following statements:1. In a triangle $ABC$, if $\cot A \cdot \cot B \cdot \cot C>0$, then the triangle is 51. If (a, b) is the centre and c is the radius of the circle $x^2 + y^2 + 2x + 6y + 1 = 0$, then what is the value of $a^2 52. If (1, −1, 2) and (2, 1, −1) are the end points of a diameter of a sphere $x^2 + y^2 + z^2 + 2ux + 2vy + 2wz − 1 = 0$, t 53. The number of points represented by the equation $x = 5$ on the $xy$-plane is 54. If $\langle l, m, n \rangle$ are the direction cosines of a normal to the plane $2x − 3y + 6z + 4 = 0$, then what is the 55. A line through $(1, −1, 2)$ with direction ratios $\langle 3, 2, 2 \rangle$ meets the plane $x + 2y + 3z = 18$. What is 56. If $p$ is the perpendicular distance from origin to the plane passing through $(1, 0, 0)$, $(0, 1, 0)$ and $(0, 0, 1)$, 57. If the direction cosines <l, m, n> of a line are connected by relation $l + 2m + n = 0, 2l - 2m + 3n = 0$, then wh 58. If a variable line passes through the point of intersection of the lines $x + 2y - 1 = 0$ and $2x - y - 1 = 0$ and meets 59. What is the equation to the straight line passing through the point $(-sin\theta, cos\theta)$ and perpendicular to the l 60. Two points $P$ and $Q$ lie on line $y = 2x + 3$. These two points $P$ and $Q$ are at a distance 2 units from another poi 61. If two sides of a square lie on the lines $2x + y - 3 = 0$ and $4x + 2y + 5 = 0$, then what is the area of the square in 62. ABC is a triangle with A(3, 5). The mid-points of sides AB, AC are at (-1, 2), (6, 4) respectively. What are the coordin 63. ABC is an acute angled isosceles triangle. Two equal sides AB and AC lie on the lines 7x - y - 3 = 0 and x + y - 5 = 0. 64. In the parabola $y^2 = 8x$, the focal distance of a point P lying on it is 8 units. Which of the following statements is 65. What is the eccentricity of the ellipse if the angle between the straight lines joining the foci to an extremity of the 66. Let $\vec{a} = \hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$. If $\vec{a} \times (\vec{b} \t 67. If a vector of magnitude 2 units makes an angle $\frac{\pi}{3}$ with $2\hat{i}$, $\frac{\pi}{4}$ with $3\hat{j}$ and an 68. Consider the following in respect of moment of a force:1. The moment of force about a point is independent of point of a 69. For any vector $\vec{r}$, what is $\left(\vec{r}\cdot\hat{i}\right)\left(\vec{r}\times\hat{i}\right) + \left(\vec{r}\cdo 70. Let $\vec{a}$ and $\vec{b}$ be two vectors of magnitude 4 inclined at an angle $\frac{\pi}{3}$, then what is the angle b 71. Let $y_1(x)$ and $y_2(x)$ be two solutions of the differential equation $\frac{dy}{dx} = x$. If $y_1(0) = 0$ and $y_2(0) 72. The differential equation, representing the curve $y = e^{x}(a\cos{x} + b\sin{x})$ where $a$ and $b$ are arbitrary const 73. If $f(x)=ax-b$ and $g(x)=cx+d$ are such that $f(g(x))=g(f(x))$, then which one of the following holds? 74. What is $\int^{1}_{-1}(3\sin x-\sin 3x)\cos^2 xdx$ equal to? 75. What are the order and degree respectively of the differential equation
$$
\left\{2-\left(\frac{d y}{d x}\right)^2\right 76. If $\frac{dy}{dx} = 2e^xy^3$, $y(0)= \frac{1}{2}$ then what is $4y^2(2-e^x)$ equal to? 77. Let $p=\int_a^b f(x) d x$ and $q=\int_a^b|f(x)| d x$. If $f(x)=e^{-x}$, then which one of the following is correct ? 78. What is $\int^{\pi/2}_0 \frac{a+\sin x}{2a+\sin x+\cos x} dx$ equal to? 79. The non-negative values of $b$ for which the function $\frac{16x^3}{3} - 4bx^2 + x$ has neither maximum nor minimum in t 80. Which one of the following is correct in respect of $f(x) = \frac{1}{\sqrt{|x| - x}}$ and $g(x) = \frac{1}{\sqrt{x - |x| 81. What is the value of $\alpha$ ? 82. What is the value of $\beta$ ? 83. Consider the following statements :1. $f(x)$ is increasing in the interval $(e, \infty)$2. $f(x)$ is decreasing in the i 84. Consider the following statements :1. $f''(e) = \frac{1}{e}$2. $f(x)$ attains local minimum value at $x = e$3. A local m 85. What is $g[f(x) - 3x]$ equal to? 86. What is $f''(x)$ equal to? 87. Consider the following statements:1. $f(x)$ is differentiable for all $x 2. $g(x)$ is continuous at $x = 0.0001$3. The d 88. What is $\lim\limits_{x \to 0-} h(x) + \lim\limits_{x \to 0+} h(x)$ equal to? 89. What is $\varphi(a)$ equal to? 90. What is $\varphi'(a)$ equal to? 91. Which of the following is/are correct?1. $f'(0) = 0$2. $f''(0) Select the correct answer using the code given below: 92. The function $y$ has a relative maxima at $x = 0$ for 93. What is $\int\limits_{-1}^{0} h(x) dx$ equal to? 94. What is $\int_{0}^{2} h(x) dx$ equal to? 95. What is the value of $\alpha$? 96. What is the value of $\beta$? 97. What is the value of $A_1$? 98. What is the value of $\frac{2(A_1 + A_2)}{ A_1 - 3A_2}$? 99. What is $f(x)$ equal to? 100. What is $8\int_1^2 f(x)dx$ equal to? 101. A bag contains 5 black and 4 white balls. A man selects two balls at random. What is the probability that both of these 102. If a random variable $(x)$ follows binomial distribution with mean 5 and variance 4 , and $5^{23} P(X=3)=\lambda 4^\lamb 103. From data (-4, 1), (-1, 2), (2, 7) and (3, 1), the regression line of y on x is obtained as $y = a + bx$, then what is t 104. Let $x+2 y+1=0$ and $2 x+3 y+4=0$ are two lines of regression computed from some bivariate data. If $\theta$ is the acut 105. If $a, b, c$ are in HP, then what is $\frac{1}{b-a} + \frac{1}{b-c}$ equal to?
1. $\frac{2}{b}$2. $\frac{1}{a} + \frac{1 106. If two random variables $X$ and $Y$ are connected by relation
$\frac{2 X-3 Y}{5 X+4 Y}=4$ and $X$ follows Binomial distr 107. An edible oil is sold at the rates 150, 200, 250, 300 rupees per litre in four consecutive years. Assuming that an equal 108. If the letters of the word 'TIRUPATI' are written down at random, then what is the probability that both Ts are always c 109. Let $m = 77^n$. The index $n$ is given a positive integral value at random. What is the probability that the value of $m 110. Three different numbers are selected at random from the first 15 natural numbers. What is the probability that the produ 111. What is the minimum value of $P(A) + P(B)$? 112. What is the maximum value of $P(A) + P(B)$? 113. What is the minimum value of $P(B \cap C)$? 114. What is the maximum value of $P(B \cap C)$? 115. What is the value of $n$? 116. What is the value of $p + q$? 117. $$
\text { What is } \sum_i^n x_i f_i \text { equal to? }
$$ 118. What is the mean of the distribution? 119. What is the mean deviation of the largest five observations? 120. What is the variance of the largest five observations?
1
NDA Mathematics 21 April 2024
MCQ (Single Correct Answer)
+2.5
-0.83
What is $\int^{\pi/2}_0 \frac{a+\sin x}{2a+\sin x+\cos x} dx$ equal to?
A
$\frac{\pi}{4}$
B
$\frac{\pi}{2}$
C
1
D
0
2
NDA Mathematics 21 April 2024
MCQ (Single Correct Answer)
+2.5
-0.83
The non-negative values of $b$ for which the function $\frac{16x^3}{3} - 4bx^2 + x$ has neither maximum nor minimum in the range $x>0$ is
A
0 < b < 1
B
1 < b < 2
C
b > 2
D
0 <= b < 1
3
NDA Mathematics 21 April 2024
MCQ (Single Correct Answer)
+2.5
-0.83
Which one of the following is correct in respect of $f(x) = \frac{1}{\sqrt{|x| - x}}$ and $g(x) = \frac{1}{\sqrt{x - |x|}}$?
A
$f(x)$ has some domain and $g(x)$ has no domain
B
$f(x)$ has no domain and $g(x)$ has some domain
C
$f(x)$ and $g(x)$ have the same domain
D
$f(x)$ and $g(x)$ do not have any domain
4
NDA Mathematics 21 April 2024
MCQ (Single Correct Answer)
+2.5
-0.83
Consider the following for the next two (02) items that follow:
Given that $\int \frac{3 \cos x + 4 \sin x}{2 \cos x + 5 \sin x} dx = \frac{\alpha x}{29} + \frac{\beta}{29} \ln |2 \cos x + 5 \sin x| + c$
What is the value of $\alpha$ ?
A
7
B
13
C
17
D
26
Paper analysis
Total Questions
Mathematics
120
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