Let the complex number $z = x + iy$ be such that $\frac{2z - 3i}{2z + i}$ is purely imaginary. If $x + y^2 = 0$, then $y^4 + y^2 - y$ is equal to (1) $\frac{2}{3}$ (2) $\frac{3}{2}$ (3) $\frac{3}{4}$ (4) $\frac{1}{3}$
Let the first term $a$ and the common ratio $r$ of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to (1) 241 (2) 231 (3) 210 (4) 220
If the coefficient of $x^7$ in $\left(ax - \frac{1}{bx^2}\right)^{13}$ and the coefficient of $x^{-5}$ in $\left(ax + \frac{1}{bx^2}\right)^{13}$ are equal, then $a^4 b^4$ is equal to: (1) 11 (2) 44 (3) 22 (4) 33
A line segment $AB$ of length $\lambda$ moves such that the points $A$ and $B$ remain on the periphery of a circle of radius $\lambda$. Then the locus of the point, that divides the line segment $AB$ in the ratio $2:3$, is a circle of radius (1) $\frac{3}{5}\lambda$ (2) $\frac{2}{3}\lambda$ (3) $\frac{\sqrt{19}}{5}\lambda$ (4) $\frac{\sqrt{19}}{7}\lambda$
Let the ellipse $E$: $x^2 + 9y^2 = 9$ intersect the positive $x$- and $y$-axes at the points $A$ and $B$ respectively. Let the major axis of $E$ be a diameter of the circle $C$. Let the line passing through $A$ and $B$ meet the circle $C$ at the point $P$. If the area of the triangle with vertices $A$, $P$ and the origin $O$ is $\frac{m}{n}$, where $m$ and $n$ are coprime, then $m - n$ is equal to (1) 16 (2) 15 (3) 17 (4) 18
For the system of linear equations $$2x - y + 3z = 5$$ $$3x + 2y - z = 7$$ $$4x + 5y + \alpha z = \beta$$ which of the following is NOT correct? (1) The system has infinitely many solutions for $\alpha = -5$ and $\beta = 9$ (2) The system has infinitely many solutions for $\alpha = -6$ and $\beta = 9$ (3) The system is inconsistent for $\alpha = -5$ and $\beta = 8$ (4) The system has a unique solution for $\alpha \neq -5$ and $\beta = 8$
If $f(x) = \frac{\tan^{-1}x + \log_e 123}{x \log_e 1234 - \tan^{-1}x}$, $x > 0$, then the least value of $f(f(x)) + f\!\left(f\!\left(\frac{4}{x}\right)\right)$ is (1) 0 (2) 8 (3) 2 (4) 4
A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in $\mathrm{cm}^2$) is equal to (1) 800 (2) 675 (3) 1025 (4) 900
Let $f$ be a differentiable function such that $x^2 f(x) - x = 4\int_0^x t\, f(t)\, dt$, $f(1) = \frac{2}{3}$. Then $18\, f(3)$ is equal to (1) 210 (2) 160 (3) 150 (4) 180
The slope of tangent at any point $(x, y)$ on a curve $y = y(x)$ is $\frac{x^2 + y^2}{2xy}$, $x > 0$. If $y(2) = 0$, then a value of $y(8)$ is (1) $-4\sqrt{2}$ (2) $2\sqrt{3}$ (3) $-2\sqrt{3}$ (4) $4\sqrt{3}$
An arc $PQ$ of a circle subtends a right angle at its centre $O$. The mid point of the arc $PQ$ is $R$. If $\overrightarrow{OP} = \vec{u}$, $\overrightarrow{OR} = \vec{v}$ and $\overrightarrow{OQ} = \alpha\vec{u} + \beta\vec{v}$, then $\alpha$, $\beta^2$ are the roots of the equation (1) $x^2 + x - 2 = 0$ (2) $x^2 - x - 2 = 0$ (3) $3x^2 - 2x - 1 = 0$ (4) $3x^2 + 2x - 1 = 0$
Let $O$ be the origin and the position vector of the point $P$ be $-\hat{i} - 2\hat{j} + 3\hat{k}$. If the position vectors of the points $A$, $B$ and $C$ are $-2\hat{i} + \hat{j} - 3\hat{k}$, $2\hat{i} + 4\hat{j} - 2\hat{k}$ and $-4\hat{i} + 2\hat{j} - \hat{k}$ respectively, then the projection of the vector $\overrightarrow{OP}$ on a vector perpendicular to the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ is (1) 3 (2) $\frac{8}{3}$ (3) $\frac{7}{3}$ (4) $\frac{10}{3}$
Let two vertices of a triangle $ABC$ be $(2, 4, 6)$ and $(0, -2, -5)$, and its centroid be $(2, 1, -1)$. If the image of the third vertex in the plane $x + 2y + 4z = 11$ is $(\alpha, \beta, \gamma)$, then $\alpha\beta + \beta\gamma + \gamma\alpha$ is equal to (1) 70 (2) 76 (3) 74 (4) 72
Let $P$ be the point of intersection of the line $\frac{x+3}{3} = \frac{y+2}{1} = \frac{1-z}{2}$ and the plane $x + y + z = 2$. If the distance of the point $P$ from the plane $3x - 4y + 12z = 32$ is $q$, then $q$ and $2q$ are the roots of the equation (1) $x^2 - 18x - 72 = 0$ (2) $x^2 - 18x + 72 = 0$ (3) $x^2 + 18x + 72 = 0$ (4) $x^2 + 18x - 72 = 0$
Let $N$ denote the sum of the numbers obtained when two dice are rolled. If the probability that $2^N < N!$ is $\frac{m}{n}$ where $m$ and $n$ are coprime, then $4m - 3n$ is equal to (1) 6 (2) 12 (3) 10 (4) 8
Let $a, b, c$ be three distinct positive real numbers such that $2a^{\log_e a} = bc^{\log_e b}$ and $b^{\log_e 2} = a^{\log_e c}$. Then $6a + 5bc$ is equal to $\_\_\_\_$.
The number of permutations, of the digits $1, 2, 3, \ldots, 7$ without repetition, which neither contain the string 153 nor the string 2467, is $\_\_\_\_$.
Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840, then the total number of persons who participated in the tournament is $\_\_\_\_$.
Let $f: (-2, 2) \rightarrow \mathbb{R}$ be defined by $f(x) = \begin{cases} x\lfloor x\rfloor, & 0 \leq x < 2 \\ (x-1)\lfloor x\rfloor, & -2 < x < 0 \end{cases}$ where $\lfloor x \rfloor$ denotes the greatest integer function. If $m$ and $n$ respectively are the number of points in $(-2, 2)$ at which $y = f(x)$ is not continuous and not differentiable, then $m + n$ is equal to $\_\_\_\_$.
Let $y = p(x)$ be the parabola passing through the points $(-1, 0)$, $(0, 1)$ and $(1, 0)$. If the area of the region $\{(x, y) : (x+1)^2 + (y-1)^2 \leq 1,\; y \leq p(x)\}$ is $A$, then $12\pi - 4A$ is equal to $\_\_\_\_$.