Let $\omega$ be a complex number such that $2\omega + 1 = z$ where $z = \sqrt{-3}$. If $$\begin{vmatrix} 1 & 1 & 1 \\ 1 & -\omega^2 - 1 & \omega^2 \\ 1 & \omega^2 & \omega^7 \end{vmatrix} = 3k$$ Then $k$ can be equal to: (1) $-z$ (2) $\frac{1}{z}$ (3) $-1$ (4) $1$
A man $X$ has 7 friends, 4 of them are ladies and 3 are men. His wife $Y$ also has 7 friends, 3 of them are ladies and 4 are men. Assume $X$ and $Y$ have no common friends. Then the total number of ways in which $X$ and $Y$ together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of $X$ and $Y$ are in this party is: (1) 485 (2) 468 (3) 469 (4) 484
For any three positive real numbers $a, b$ and $c$. If $9(25a^2 + b^2) + 25(c^2 - 3ac) = 15b(3a + c)$. Then (1) $b,\ c$ and $a$ are in G.P. (2) $b,\ c$ and $a$ are in A.P. (3) $a,\ b$ and $c$ are in A.P. (4) $a,\ b$ and $c$ are in G.P.
Let $k$ be an integer such that the triangle with vertices $(k, -3)$, $(5, k)$ and $(-k, 2)$ has area 28 sq. units. Then the orthocenter of this triangle is at the point: (1) $\left(2, -\dfrac{1}{2}\right)$ (2) $\left(1, \dfrac{3}{4}\right)$ (3) $\left(1, -\dfrac{3}{4}\right)$ (4) $\left(2, \dfrac{1}{2}\right)$
The radius of a circle, having minimum area, which touches the curve $y = 4 - x^2$ and the lines $y = |x|$ is: (1) $2(\sqrt{2} + 1)$ (2) $2(\sqrt{2} - 1)$ (3) $4(\sqrt{2} - 1)$ (4) $4(\sqrt{2} + 1)$
The eccentricity of an ellipse whose centre is at the origin is $\dfrac{1}{2}$. If one of its directrices is $x = -4$, then the equation of the normal to it at $\left(1, \dfrac{3}{2}\right)$ is: (1) $2y - x = 2$ (2) $4x - 2y = 1$ (3) $4x + 2y = 7$ (4) $x + 2y = 4$
A hyperbola passes through the point $P(\sqrt{2}, \sqrt{3})$ and has foci at $(\pm 2, 0)$. Then the tangent to this hyperbola at $P$ also passes through the point (1) $(3\sqrt{2}, 2\sqrt{3})$ (2) $(2\sqrt{2}, 3\sqrt{3})$ (3) $(\sqrt{3}, \sqrt{2})$ (4) $(-\sqrt{2}, -\sqrt{3})$
The statement $(p \rightarrow q) \rightarrow (\sim p \rightarrow q) \rightarrow q$ is (1) A tautology (2) Equivalent to $\sim p \rightarrow q$ (3) Equivalent to $p \rightarrow \sim q$ (4) A fallacy
A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is: (1) $\dfrac{12}{5}$ (2) $6$ (3) $4$ (4) $\dfrac{6}{25}$
Let a vertical tower $AB$ have its end $A$ on the level ground. Let $C$ be the mid-point of $AB$ and $P$ be a point on the ground such that $AP = 2AB$. If $\angle BPC = \beta$, then $\tan\beta$ is equal to: (1) $\dfrac{6}{7}$ (2) $\dfrac{1}{4}$ (3) $\dfrac{2}{9}$ (4) $\dfrac{4}{9}$
If $S$ is the set of distinct values of $b$ for which the following system of linear equations $$\begin{aligned} & x + y + z = 1 \\ & x + ay + z = 1 \\ & ax + by + z = 0 \end{aligned}$$ has no solution, then $S$ is: (1) An empty set (2) An infinite set (3) A finite set containing two or more elements (4) A singleton
The function $f : \mathbb{R} \rightarrow \left(-\dfrac{1}{2}, \dfrac{1}{2}\right)$ defined as $f(x) = \dfrac{x}{1 + x^2}$, is: (1) Invertible (2) Injective but not surjective (3) Surjective but not injective (4) Neither injective nor surjective
Let $a, b, c \in \mathbb{R}$. If $f(x) = ax^2 + bx + c$ is such that $a + b + c = 3$ and $f(x + y) = f(x) + f(y) + xy,\ \forall x, y \in \mathbb{R}$, then $\displaystyle\sum_{n=1}^{10} f(n)$ is equal to: (1) 330 (2) 165 (3) 190 (4) 255
Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is: (1) 12.5 (2) 10 (3) 25 (4) 30
The normal to the curve $y(x - 2)(x - 3) = x + 6$ at the point where the curve intersects the $y$-axis passes through the point: (1) $\left(-\dfrac{1}{2}, -\dfrac{1}{2}\right)$ (2) $\left(\dfrac{1}{2}, \dfrac{1}{2}\right)$ (3) $\left(\dfrac{1}{2}, -\dfrac{1}{3}\right)$ (4) $\left(\dfrac{1}{2}, \dfrac{1}{3}\right)$
The area (in sq. units) of the region $\{(x, y) : x \geq 0,\ x + y \leq 3,\ x^2 \leq 4y$ and $y \leq 1 + \sqrt{x}\}$ is (1) $\dfrac{59}{12}$ sq. units (2) $\dfrac{3}{2}$ sq. units (3) $\dfrac{7}{3}$ sq. units (4) $\dfrac{5}{2}$ sq. units
If $(2 + \sin x)\dfrac{dy}{dx} + (y + 1)\cos x = 0$ and $y(0) = 1$, then $y\left(\dfrac{\pi}{2}\right)$ is equal to (1) $\dfrac{1}{3}$ (2) $-\dfrac{2}{3}$ (3) $-\dfrac{1}{3}$ (4) $\dfrac{4}{3}$
Given, $\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k}$ and $\vec{b} = \hat{i} + \hat{j}$. Let $\vec{c}$ be a vector such that $|\vec{c} - \vec{a}| = 3$, $|(\vec{a} \times \vec{b}) \times \vec{c}| = 3$ and the angle between $\vec{c}$ and $\vec{a} \times \vec{b}$ be $30^\circ$. Then $\vec{a} \cdot \vec{c}$ is equal to: (1) $\dfrac{25}{8}$ (2) $2$ (3) $5$ (4) $\dfrac{1}{8}$
If the image of the point $P(1, -2, 3)$ in the plane $2x + 3y - 4z + 22 = 0$ measured parallel to the line $\dfrac{x}{1} = \dfrac{y}{4} = \dfrac{z}{5}$ is $Q$, then $PQ$ is equal to: (1) $3\sqrt{5}$ (2) $2\sqrt{42}$ (3) $\sqrt{42}$ (4) $6\sqrt{5}$
The distance of the point $(1, 3, -7)$ from the plane passing through the point $(1, -1, -1)$, having normal perpendicular to both the lines $\dfrac{x-1}{1} = \dfrac{y+2}{-2} = \dfrac{z-4}{3}$ and $\dfrac{x-2}{2} = \dfrac{y+1}{-1} = \dfrac{z+7}{-1}$, is: (1) $\dfrac{20}{\sqrt{74}}$ (2) $\dfrac{10}{\sqrt{83}}$ (3) $\dfrac{5}{\sqrt{83}}$ (4) $\dfrac{10}{\sqrt{74}}$
For three events $A$, $B$ and $C$, $P(\text{Exactly one of } A \text{ or } B \text{ occurs}) = P(\text{Exactly one of } B \text{ or } C \text{ occurs}) = P(\text{Exactly one of } C \text{ or } A \text{ occurs}) = \dfrac{1}{4}$ and $P(\text{All the three events occur simultaneously}) = \dfrac{1}{16}$. Then the probability that at least one of the events occurs, is: (1) $\dfrac{7}{32}$ (2) $\dfrac{7}{16}$ (3) $\dfrac{1}{64}$ (4) $\dfrac{3}{16}$
If two different numbers are taken from the set $\{0, 1, 2, 3, \ldots, 10\}$; then the probability that their sum as well as absolute difference are both multiples of 4, is: (1) $\dfrac{6}{55}$ (2) $\dfrac{12}{55}$ (3) $\dfrac{14}{45}$ (4) $\dfrac{7}{55}$