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 is the value of $|2B(adj(3A))|$?

What is the sum of all four-digit numbers formed by using all digits 0, 1, 4, 5 without repetition of digits?

If $x, y$ and $z$ are the cube roots of unity, then what is the value of $xy + yz + zx$?

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 invite 3 women and 3 men so that 3 of them are man's relatives and 3 of them are his wife's relatives?

A triangle $PQR$ is such that 3 points lie on the side $PQ$, 4 points on $QR$ and 5 points on $RP$ respectively. Triangles are constructed using these points as vertices. What is the number of triangles so formed?

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 $a_0, a_1, a_2, a_3$ are integers, then which one of the following is correct?

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 $\operatorname{Re} \left(\frac{z_1}{z_2}\right) + 1$ equal to?

If $26! = n8^k$, where $k$ and $n$ are positive integers, then what is the maximum value of $k$?

Consider the following statements in respect of two non-singular matrices $A$ and $B$ of the same order $n$:

1. 1.$adj(AB) = (adjA)(adjB)$


2. 2. $adj(AB) = adj(BA)$


3. 3. $(AB) adj(AB) - |AB| I_n$ is a null matrix of order $n$

Consider the following statements in respect of a non-singular matrix $A$ of order $n$:

1. $A(\text{adj}A^T) = A(\text{adj}A)^T$

2. If $A^2 = A$, then $A$ is identity matrix of order $n$

3. If $A^3 = A$, then $A$ is identity matrix of order $n$

Which of the statements given above are correct?

How many four-digit natural numbers are there such that all of the digits are even?

If $\omega \neq 1$ is a cube root of unity, then what are the solutions of $(z-100)^3 + 1000 = 0$?

What is $(1 + i)^4 + (1 - i)^4$ equal to, where $i=\sqrt{-1}$?

Consider the following statements in respect of a skew-symmetric matrix $A$ of order $3$:

1. All diagonal elements are zero.

2. The sum of all the diagonal elements of the matrix is zero.

3. $A$ is orthogonal matrix.

Which of the statements given above are correct?

Four digit numbers are formed by using the digits 1, 2, 3, 5 without repetition of digits. How many of them are divisible by 4?

What is the remainder when $2^{120}$ is divided by 7?

If ABC is a triangle, then what is the value of the determinant

$$ \left|\begin{array}{ccc} \cos C & \sin B & 0 \\ \tan A & 0 & \sin B \\ 0 & \tan (B+C) & \cos C \end{array}\right| ? $$

What is the number of different matrices, each having 4 entries that can be formed using 1, 2, 3, 4 (repetition is allowed)?

Let $A = \{x \in \mathbb{R} : -1 <x <1\}$. Which of the following is/are bijective functions from A to itself?

1. $f(x) = x|x|$

2. $g(x) = \cos(\pi x)$

Select the correct answer using the code given below:

Let $R$ be a relation on the open interval $(-1, 1)$ and is given by $R = \{(x, y) : |x + y| < 2\}$. Then which one of the following is correct?

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?

If $a, b, c$ are the sides of a triangle $ABC$, then what is $$ \begin{vmatrix} a^2 & b \sin A & c \sin A \\ b \sin A & 1 & \cos A \\ c \sin A & \cos A & 1 \end{vmatrix}$$ equal to?

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. $a, c,$ and $e$ are in GP

2. $\frac{1}{a}, \frac{1}{c}, \frac{1}{e}$ are in GP

Select the correct answer using the code given below:

What is the number of solutions of $\log_4(x - 1) = \log_2(x - 3)$?

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 $k$ can never be equal to :

What is the coefficient of $x^{10}$ in the expansion of $(1-x^2)^{20}\left(2-x^2-\frac{1}{x^2}\right)^{-5}$?

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 $mn$?

If $a$, $b$, and $c$ $(a > 0, c > 0)$ are in GP, then consider the following in respect of the equation $ax^2 + bx + c = 0$:

1. The equation has imaginary roots.

2. The ratio of the roots of the equation is $1 : \omega$ where $\omega$ is a cube root of unity.

3. The product of roots of the equation is $\left(\frac{b^2}{a^2}\right)$.

Which of the statements given above are correct?

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 be an integer

2. $n$ must be an integer

Select the correct answer using the code given below:

Let $A = \{1, 2, 3, 4, 5\}$ and $B = \{6, 7\}$. What is the number of onto functions from $A$ to $B$?

What is $\sin 9^\circ - \cos 9^\circ$ equal to?

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 $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}$, where $a$, $b$, $c$ are sides of the triangle?

If $\cos^{-1} x = \sin^{-1} x$, then which one of the following is correct?

What is the number of solutions of $(\sin \theta - \cos \theta)^2 = 2$ where $-\pi < \theta < \pi$?

$ABC$ is a triangle such that angle $C = 60^{\circ}$, then what is $\frac{\cos A + \cos B}{\cos \left(\frac{A - B}{2}\right)}$ equal to?

What is $\sqrt{15 + \cot^2 \left( \frac \pi 4 - 2 \cot^{-1} 3 \right)}$ equal to?

What is the value of $\sin10^{\circ} \cdot \sin50^{\circ} + \sin50^{\circ} \cdot \sin250^{\circ} + \sin250^{\circ} \cdot \sin10^{\circ}$ equal to?

What is $\tan^{-1} \left( \frac{a}{b} \right) - \tan^{-1} \left( \frac{a - b}{a + b} \right)$ equal to?

Under which one of the following conditions does the equation $\left(\cos \beta-1\right)x^2+(\cos \beta)x+\sin \beta=0$ in $x$ have a real root for $\beta \in [0, \pi]$?

In a triangle $ABC$, $AB=16 \text{ cm}, BC=63 \text{ cm}$ and $AC=65 \text{ cm}$. What is the value of $\cos 2A+\cos 2B+\cos 2C$?

If $f(\theta)=\frac{1}{1+\tan \theta}$ and $\alpha+\beta=\frac{5\pi}{4}$, then what is the value of $f(\alpha) f(\beta)$?

If $\tan \alpha$ and $\tan \beta$ are the roots of the equation $x^2-6x+8=0$, then what is the value of $\cos(2 \alpha+2 \beta)$?

What is the value of $\tan 65^\circ +2 \tan 45^\circ-2 \tan 40^\circ-\tan 25^\circ$?

Consider the following statements:

1. In a triangle $ABC$, if $\cot A \cdot \cot B \cdot \cot C>0$, then the triangle is an acute-angled triangle.

2. In a triangle $ABC$, if $\tan A \cdot \tan B \cdot \tan C > 0$, then the triangle is an obtuse-angled triangle.

Which of the statements given above is/are correct?

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 $a^2 + b^2 + c^2$?

If (1, −1, 2) and (2, 1, −1) are the end points of a diameter of a sphere $x^2 + y^2 + z^2 + 2ux + 2vy + 2wz − 1 = 0$, then what is $u + v + w$ equal to?

The number of points represented by the equation $x = 5$ on the $xy$-plane is

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 value of $49(7l^2 + m^2 − n^2)$?

A line through $(1, −1, 2)$ with direction ratios $\langle 3, 2, 2 \rangle$ meets the plane $x + 2y + 3z = 18$. What is the point of intersection of line and plane?

If $p$ is the perpendicular distance from origin to the plane passing through $(1, 0, 0)$, $(0, 1, 0)$ and $(0, 0, 1)$, then what is $3p^2$ equal to?

If the direction cosines <l, m, n> of a line are connected by relation $l + 2m + n = 0, 2l - 2m + 3n = 0$, then what is the value of $l^{2} + m^{2} - n^{2}$?

If a variable line passes through the point of intersection of the lines $x + 2y - 1 = 0$ and $2x - y - 1 = 0$ and meets the coordinate axes in $A$ and $B$, then what is the locus of the mid-point of $AB$?

What is the equation to the straight line passing through the point $(-sin\theta, cos\theta)$ and perpendicular to the line $xcos\theta + ysin\theta = 9$?

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 point $R(1, 5)$. What are the coordinates of the points $P$ and $Q$?

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 square units?

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 coordinates of centroid of the triangle ABC?

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. If θ is one of the equal angles, then what is cotθ equal to?

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/are correct?

1. The coordinates of $P$ can be $\left(6, 4\sqrt{3}\right)$.

2. The perpendicular distance of $P$ from the directrix of parabola is 8 units.

Select the correct answer using the code given below:

What is the eccentricity of the ellipse if the angle between the straight lines joining the foci to an extremity of the minor axis is 90°?

Let $\vec{a} = \hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$. If $\vec{a} \times (\vec{b} \times \vec{a}) = \alpha \hat{i} - \beta \hat{j} + \gamma \hat{k}$, then what is the value of $\alpha + \beta + \gamma$?

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 acute angle $\theta$ with $4\hat{k}$, then what are the components of the vector?

Consider the following in respect of moment of a force:

1. The moment of force about a point is independent of point of application of force.

2. The moment of a force about a line is a vector quantity.

Which of the statements given above is/are correct?

For any vector $\vec{r}$, what is $\left(\vec{r}\cdot\hat{i}\right)\left(\vec{r}\times\hat{i}\right) + \left(\vec{r}\cdot\hat{j}\right)\left(\vec{r}\times\hat{j}\right) + \left(\vec{r}\cdot\hat{k}\right)\left(\vec{r}\times\hat{k}\right)$ equal to?

Let $\vec{a}$ and $\vec{b}$ be two vectors of magnitude 4 inclined at an angle $\frac{\pi}{3}$, then what is the angle between $\vec{a}$ and $\vec{a} - \vec{b}$?

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 $y_2(0) = 4$, then what is the number of points of intersection of the curves $y_1(x)$ and $y_2(x)$?

The differential equation, representing the curve $y = e^{x}(a\cos{x} + b\sin{x})$ where $a$ and $b$ are arbitrary constants, is

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?

What is $\int^{1}_{-1}(3\sin x-\sin 3x)\cos^2 xdx$ equal to?

If $\frac{dy}{dx} = 2e^xy^3$, $y(0)= \frac{1}{2}$ then what is $4y^2(2-e^x)$ equal to?

What is $\int^{\pi/2}_0 \frac{a+\sin x}{2a+\sin x+\cos x} dx$ equal to?

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

Which one of the following is correct in respect of $f(x) = \frac{1}{\sqrt{|x| - x}}$ and $g(x) = \frac{1}{\sqrt{x - |x|}}$?

What is the value of $\alpha$ ?

What is the value of $\beta$ ?

Consider the following statements :
1. $f(x)$ is increasing in the interval $(e, \infty)$
2. $f(x)$ is decreasing in the interval $(1, e)$
3. $9 \ln 7 > 7 \ln 9$
Which of the statements given above are correct ?

Consider the following statements :
1. $f''(e) = \frac{1}{e}$
2. $f(x)$ attains local minimum value at $x = e$
3. A local minimum value of $f(x)$ is $e$
Which of the statements given above are correct ?

What is $g[f(x) - 3x]$ equal to?

What is $f''(x)$ equal to?

Consider the following statements:

1. $f(x)$ is differentiable for all $x < 0$

2. $g(x)$ is continuous at $x = 0.0001$

3. The derivative of $g(x)$ at $x = 2.5$ is 1

Which of the statements given above are correct?

What is $\lim\limits_{x \to 0-} h(x) + \lim\limits_{x \to 0+} h(x)$ equal to?

What is $\varphi(a)$ equal to?

What is $\varphi'(a)$ equal to?

Which of the following is/are correct?

1. $f'(0) = 0$

2. $f''(0) < 0$

Select the correct answer using the code given below:

The function $y$ has a relative maxima at $x = 0$ for

What is $\int\limits_{-1}^{0} h(x) dx$ equal to?

What is $\int_{0}^{2} h(x) dx$ equal to?

What is the value of $\alpha$?

What is the value of $\beta$?

What is the value of $A_1$?

What is the value of $\frac{2(A_1 + A_2)}{ A_1 - 3A_2}$?

What is $f(x)$ equal to?

What is $8\int_1^2 f(x)dx$ equal to?

A bag contains 5 black and 4 white balls. A man selects two balls at random. What is the probability that both of these are of the same colour?

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 the value of $2a + 15b$?

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. $\frac{1}{a} + \frac{1}{c}$

3. $\frac{1}{2} \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right)$

Select the correct answer using the code given below:

An edible oil is sold at the rates 150, 200, 250, 300 rupees per litre in four consecutive years. Assuming that an equal amount of money is spent on oil by a family in every year during these years, what is the average price of oil in rupees (approximately) per litre?

If the letters of the word 'TIRUPATI' are written down at random, then what is the probability that both Ts are always consecutive?

Let $m = 77^n$. The index $n$ is given a positive integral value at random. What is the probability that the value of $m$ will have $1$ in the units place?

Three different numbers are selected at random from the first 15 natural numbers. What is the probability that the product of two of the numbers is equal to third number?

What is the minimum value of $P(A) + P(B)$?

What is the maximum value of $P(A) + P(B)$?

What is the minimum value of $P(B \cap C)$?

What is the maximum value of $P(B \cap C)$?

What is the value of $n$?

What is the value of $p + q$?

What is the mean of the distribution?

What is the mean deviation of the largest five observations?

What is the variance of the largest five observations?