NDA Mathematics 21 April 2024
Paper was held on Sun, Apr 21, 2024 4:30 AM
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Mathematics

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
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If $z$ is any complex number and $iz^2 + z - z + i = 0$, where $i = \sqrt{-1}$, then what is the value of $(|z| + 1)^2$?
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What is the sum of all four-digit numbers formed by using all digits 0, 1, 4, 5 without repetition of digits?
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If $x, y$ and $z$ are the cube roots of unity, then what is the value of $xy + yz + zx$?
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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
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A triangle $PQR$ is such that 3 points lie on the side $PQ$, 4 points on $QR$ and 5 points on $RP$ respectively. Triangl
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If $\log{a} = p$, $\log{c} = 2p$ and $\log{e} = 3p$, then what is $(ace)^\frac{1}{p}$ equal to?
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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,
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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
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If $26! = n8^k$, where $k$ and $n$ are positive integers, then what is the maximum value of $k$?
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Consider the following statements in respect of two non-singular matrices $A$ and $B$ of the same order $n$: 1.$adj(AB)
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Consider the following statements in respect of a non-singular matrix $A$ of order $n$:1. $A(\text{adj}A^T) = A(\text{ad
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How many four-digit natural numbers are there such that all of the digits are even?
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If $\omega \neq 1$ is a cube root of unity, then what are the solutions of $(z-100)^3 + 1000 = 0$?
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What is $(1 + i)^4 + (1 - i)^4$ equal to, where $i=\sqrt{-1}$?
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Consider the following statements in respect of a skew-symmetric matrix $A$ of order $3$:1. All diagonal elements are ze
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Four digit numbers are formed by using the digits 1, 2, 3, 5 without repetition of digits. How many of them are divisibl
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What is the remainder when $2^{120}$ is divided by 7?
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For what value of $n$ is the determinant $$ \left|\begin{array}{ccc} C(9,4) & C(9,3) & C(10, n-2) \\ C(11,6) & C(11,5) &
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If ABC is a triangle, then what is the value of the determinant $$ \left|\begin{array}{ccc} \cos C & \sin B & 0 \\ \tan
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What is the number of different matrices, each having 4 entries that can be formed using 1, 2, 3, 4 (repetition is allow
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Let $A = \{x \in \mathbb{R} : -1 <x <1\}$. Which of the following is/are bijective functions from A to itself?1. $
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Let $R$ be a relation on the open interval $(-1, 1)$ and is given by $R = \{(x, y) : |x + y|
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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?
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If $a, b, c$ are the sides of a triangle $ABC$, then what is $$ \begin{vmatrix} a^2 & b \sin A & c \sin A \\ b \sin A &
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If $a, b, c$ are in AP; $b, c, d$ are in GP; $c, d, e$ are in HP, then which of the following is/are correct?1. $a, c,$
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What is the number of solutions of $\log_4(x - 1) = \log_2(x - 3)$?
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For $x \geq y > 1$, let $\log_x\left( \frac{x}{y} \right) + \log_y\left(\frac{y}{x}\right) = k$, then the value of $k$ c
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If $A=\left[\begin{array}{ccc}\sin 2 \theta & 2 \sin ^2 \theta-1 & 0 \\ \cos 2 \theta & 2 \sin \theta \cos \theta & 0 \\
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What is the coefficient of $x^{10}$ in the expansion of $(1-x^2)^{20}\left(2-x^2-\frac{1}{x^2}\right)^{-5}$?
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If the 4th term in the expansion of $\left(mx + \frac{1}{x}\right)^n$ is $\frac{5}{2}$, then what is the value of $mn$?
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If $a$, $b$, and $c$ $(a > 0, c > 0)$ are in GP, then consider the following in respect of the equation $ax^2 + bx + c
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If $x^2 + mx + n$ is an integer for all integral values of $x$, then which of the following is/are correct?1. $m$ must b
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In a binomial expansion of $(x+y)^{2 n+1}(x-y)^{2 n+1}$, the sum of middle terms is zero. What is the value of $\left(\f
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Let $A = \{1, 2, 3, 4, 5\}$ and $B = \{6, 7\}$. What is the number of onto functions from $A$ to $B$?
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$$ \text { What is } \frac{\sqrt{3} \cos 10^{\circ}-\sin 10^{\circ}}{\sin 25^{\circ} \cos 25^{\circ}} \text { equal to ?
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What is $\sin 9^\circ - \cos 9^\circ$ equal to?
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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
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If $\cos^{-1} x = \sin^{-1} x$, then which one of the following is correct?
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What is the number of solutions of $(\sin \theta - \cos \theta)^2 = 2$ where $-\pi
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$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
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What is $\sqrt{15 + \cot^2 \left( \frac \pi 4 - 2 \cot^{-1} 3 \right)}$ equal to?
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What is the value of $\sin10^{\circ} \cdot \sin50^{\circ} + \sin50^{\circ} \cdot \sin250^{\circ} + \sin250^{\circ} \cdot
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What is $\tan^{-1} \left( \frac{a}{b} \right) - \tan^{-1} \left( \frac{a - b}{a + b} \right)$ equal to?
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Under which one of the following conditions does the equation $\left(\cos \beta-1\right)x^2+(\cos \beta)x+\sin \beta=0$
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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+
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If $f(\theta)=\frac{1}{1+\tan \theta}$ and $\alpha+\beta=\frac{5\pi}{4}$, then what is the value of $f(\alpha) f(\beta)$
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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
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What is the value of $\tan 65^\circ +2 \tan 45^\circ-2 \tan 40^\circ-\tan 25^\circ$?
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Consider the following statements:1. In a triangle $ABC$, if $\cot A \cdot \cot B \cdot \cot C>0$, then the triangle is
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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
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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
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The number of points represented by the equation $x = 5$ on the $xy$-plane is
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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
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A line through $(1, −1, 2)$ with direction ratios $\langle 3, 2, 2 \rangle$ meets the plane $x + 2y + 3z = 18$. What is
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If $p$ is the perpendicular distance from origin to the plane passing through $(1, 0, 0)$, $(0, 1, 0)$ and $(0, 0, 1)$,
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If the direction cosines <l, m, n> of a line are connected by relation $l + 2m + n = 0, 2l - 2m + 3n = 0$, then wh
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If a variable line passes through the point of intersection of the lines $x + 2y - 1 = 0$ and $2x - y - 1 = 0$ and meets
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What is the equation to the straight line passing through the point $(-sin\theta, cos\theta)$ and perpendicular to the l
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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
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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
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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
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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.
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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
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What is the eccentricity of the ellipse if the angle between the straight lines joining the foci to an extremity of the
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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
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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
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Consider the following in respect of moment of a force:1. The moment of force about a point is independent of point of a
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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
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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
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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)
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The differential equation, representing the curve $y = e^{x}(a\cos{x} + b\sin{x})$ where $a$ and $b$ are arbitrary const
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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?
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What is $\int^{1}_{-1}(3\sin x-\sin 3x)\cos^2 xdx$ equal to?
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What are the order and degree respectively of the differential equation $$ \left\{2-\left(\frac{d y}{d x}\right)^2\right
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If $\frac{dy}{dx} = 2e^xy^3$, $y(0)= \frac{1}{2}$ then what is $4y^2(2-e^x)$ equal to?
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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 ?
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What is $\int^{\pi/2}_0 \frac{a+\sin x}{2a+\sin x+\cos x} dx$ equal to?
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The non-negative values of $b$ for which the function $\frac{16x^3}{3} - 4bx^2 + x$ has neither maximum nor minimum in t
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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|
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What is the value of $\alpha$ ?
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What is the value of $\beta$ ?
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Consider the following statements :1. $f(x)$ is increasing in the interval $(e, \infty)$2. $f(x)$ is decreasing in the i
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Consider the following statements :1. $f''(e) = \frac{1}{e}$2. $f(x)$ attains local minimum value at $x = e$3. A local m
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What is $g[f(x) - 3x]$ equal to?
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What is $f''(x)$ equal to?
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Consider the following statements:1. $f(x)$ is differentiable for all $x 2. $g(x)$ is continuous at $x = 0.0001$3. The d
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What is $\lim\limits_{x \to 0-} h(x) + \lim\limits_{x \to 0+} h(x)$ equal to?
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What is $\varphi(a)$ equal to?
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What is $\varphi'(a)$ equal to?
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Which of the following is/are correct?1. $f'(0) = 0$2. $f''(0) Select the correct answer using the code given below:
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The function $y$ has a relative maxima at $x = 0$ for
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What is $\int_{-1}^{0} h(x) dx$ equal to?
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What is $\int_{0}^{2} h(x) dx$ equal to?
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What is the value of $\alpha$?
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What is the value of $\beta$?
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What is the value of $A_1$?
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What is the value of $\frac{2(A_1 + A_2)}{ A_1 - 3A_2}$?
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What is $f(x)$ equal to?
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What is $8\int_1^2 f(x)dx$ equal to?
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A bag contains 5 black and 4 white balls. A man selects two balls at random. What is the probability that both of these
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If a random variable $(x)$ follows binomial distribution with mean 5 and variance 4 , and $5^{23} P(X=3)=\lambda 4^\lamb
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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
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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
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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
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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
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An edible oil is sold at the rates 150, 200, 250, 300 rupees per litre in four consecutive years. Assuming that an equal
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If the letters of the word 'TIRUPATI' are written down at random, then what is the probability that both Ts are always c
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Let $m = 77^n$. The index $n$ is given a positive integral value at random. What is the probability that the value of $m
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Three different numbers are selected at random from the first 15 natural numbers. What is the probability that the produ
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What is the minimum value of $P(A) + P(B)$?
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What is the maximum value of $P(A) + P(B)$?
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What is the minimum value of $P(B \cap C)$?
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What is the maximum value of $P(B \cap C)$?
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What is the value of $n$?
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What is the value of $p + q$?
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$$ \text { What is } \sum_i^n x_i f_i \text { equal to? } $$
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What is the mean of the distribution?
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What is the mean deviation of the largest five observations?
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What is the variance of the largest five observations?
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