If $ \theta = \tan^{-1}\dfrac{1}{1+2} + \tan^{-1}\dfrac{1}{1+2\cdot3} + \tan^{-1}\dfrac{1}{1+3\cdot4} + \ldots + \tan^{-1}\dfrac{1}{1+n(n+1)} $, then $\tan\theta$ is equal to:

If $(1 + x - 2x^2)^6 = 1 + a_1 x + a_2 x^2 + \ldots + a_{12} x^{12}$, then the value of $a_2 + a_4 + a_6 + \ldots + a_{12}$ is:

A square with side $a$ is revolved about its centre through $45^\circ$. What is the area common to both the squares?

How many different paths in the $xy$-plane are there from $(1,3)$ to $(5,6)$, if a path proceeds one step at a time either right (R) or upward (U)?

If the distance of $(x,y)$ from the origin is defined as $d(x,y) = \max(|x|,|y|)$, then the locus of points where $d(x,y)=1$ is:

If $\sin^{-1}x + \cos^{-1}(1-x) = \sin^{-1}(1-x)$ then $x$ satisfies the equation:

A and B are independent witnesses. Probability A speaks the truth = $x$, Probability B speaks the truth = $y$. If both agree on a statement, the probability that the statement is true is:

If $A$ is a $3\times 3$ matrix with $\det(A)=3$, then $\det(\operatorname{adj}A)$ is:

A set contains $(2n+1)$ elements. If the number of subsets that contain at most $n$ elements is $4096$, then the value of $n$ is:

The total number of relations that exist from a set $A$ with $m$ elements into the set $A \times A$ is:

Water runs into a conical tank of radius $5$ ft and height $10$ ft at a constant rate of $2\text{ ft}^3/\text{min}$. How fast is the water level rising when the water is $6$ ft deep?

The probability that a man who is 85 yrs old will die before attaining the age of 90 is $1/3$. $A_1, A_2, A_3, A_4$ are four persons aged 85 yrs. The probability that $A_1$ will die before attaining 90 and will be the first to die is:

A box open at the top is made by cutting squares from the four corners of a $6 \times 6$ m sheet. The height of the box for maximum volume is:

If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors, then $|\vec{a}-\vec{b}|^2 + |\vec{b}-\vec{c}|^2 + |\vec{c}-\vec{a}|^2$ does not exceed:

Let $f(x) = \lfloor x^2 - 3 \rfloor$ where $\lfloor \cdot \rfloor$ is the greatest integer function. Number of points in $(1,2)$ where $f$ is discontinuous:

If $a + b + c \neq 0$, the system of equations: $(b+c)(y+z) - ax = b - c$ $(c+a)(z+x) - by = c - a$ $(a+b)(x+y) - cz = a - b$ has:

If $y = f(x)$ is odd and differentiable on $(-\infty,\infty)$ such that $f'(3) = -2$, then $f'(-3)$ equals:

The value of $\displaystyle \int_{0}^{\pi} \frac{x \sin x}{1+\cos^2 x},dx$ is:

If $\tan^{-1}(2x) + \tan^{-1}(3x) = \dfrac{\pi}{4}$, then $x$ is:

The equation $\sin^4 x + \cos^4 x + \sin 2x + \alpha = 0$ is solvable for:

If $x < -1$ and $2^{|x+1|} - 2^x = |2x - 1| + 1$ then the value of $x$ is:

The vector $\vec{B} = 3\vec{i} + 4\vec{k}$ is to be written as the sum of a vector $\vec{B_1}$ parallel to $\vec{A} = \vec{i} + \vec{j}$ and a vector $\vec{B_2}$ perpendicular to $\vec{A}$. Then $\vec{B_1}$ is:

Find $k$ in the equation $x^3 - 6x^2 + kx + 64 = 0$ if roots are in geometric progression.

If $P = {(4n - 3n - 1) : n \in N}$ and $Q = {(9n - 9) : n \in N}$, then $P \cup Q$ equals to:

If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, then $I + A + A^2 + A^3 + \cdots \infty$ equals:

$A_1, A_2, A_3, A_4$ are subsets of $U$ (75 elements). Each $A_i$ has 28 elements. Any two intersect in 12 elements. Any three intersect in 5 elements. All four intersect in 1 element. Find the number of elements belonging to none of the four subsets.

$ABC$ is isosceles with $AB = AC$. $BC$ is parallel to x-axis. $m_1, m_2$ are slopes of the medians from $B$ and $C$. Then:

The smaller area bounded by $y = 2 - x$ and $x^2 + y^2 = 4$ is:

There are 10 points, out of which 6 are collinear. Number of triangles formed:

Number of distinct integer values of $a$ satisfying $2^{2a} - 3(2^{a+2}) + 25 = 0$ is:

A man has 5 coins: 2 double-headed 1 double-tailed 2 normal He randomly picks a coin and tosses it. Probability that the lower face is a head is:

If $A = \cos^2\theta + \sin^4\theta$, then for all values of $\theta$:

From 50 students: 37 passed Math, 24 Physics, 43 Chemistry. At most 19 passed Math & Physics, at most 29 passed Math & Chemistry, at most 20 passed Physics & Chemistry. Intersection of all 3 is $x$. Find maximum possible value of $x$.

Number of solutions for $\tan^{-1}\sqrt{x(x+1)} + \sin^{-1}\sqrt{x^2 + x + 1} = \dfrac{\pi}{2}$ is:

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar unit vectors and $\vec{a} \times (\vec{b} \times \vec{c}) = \dfrac{\vec{b} + \vec{c}}{\sqrt{2}}$, then the angle between $\vec{a}$ and $\vec{b}$ is:

The straight lines $\dfrac{x}{a} + \dfrac{y}{b} = k$ and $\dfrac{x}{a} + \dfrac{y}{b} = \dfrac{1}{k}$ (with $k\neq0$) meet on:

Events $A$ and $B$ satisfy: $P(A \cup B) = \dfrac{1}{6}$, $P(A \cap B) = \dfrac{1}{4}$, $P(A) = \dfrac{1}{4}$ Then events $A$ and $B$ are:

An anti-aircraft gun fires a maximum of four shots. Probabilities of hitting in the 1st, 2nd, 3rd, and 4th shot are 0.4, 0.3, 0.2 and 0.1 respectively. Find the probability that the gun hits the plane.

If $2x^4 + x^3 - 11x^2 + x + 2 = 0$ then the values of $x + \dfrac{1}{x}$ are:

Find the value of $x$, if: $\left( 2^{\frac{1}{\log_x 4}} \right) \left( 2^{\frac{1}{\log_x 16}} \right) \left( 2^{\frac{1}{\log_x 256}} \right) \cdots = 2$