A boy can throw a stone up to a maximum height of 10 m. The maximum horizontal distance that the boy can throw the same stone up to will be (1) $20\sqrt{2}$ m (2) 10 m (3) $10\sqrt{2}$ m (4) 20 m
If the lines $\frac{x-1}{2} = \frac{y+1}{3} = \frac{z-1}{4}$ and $\frac{x-3}{1} = \frac{y-k}{2} = \frac{z}{1}$ intersect, then $k$ is equal to (1) $\frac{2}{9}$ (2) $\frac{9}{2}$ (3) 0 (4) $-1$
If $\vec{a} = \frac{1}{\sqrt{10}}(3\hat{i}+\hat{k})$ and $\vec{b} = \frac{1}{7}(2\hat{i}+3\hat{j}-6\hat{k})$, then the value of $(2\vec{a}-\vec{b})\cdot[(\vec{a}\times\vec{b})\times(\vec{a}+2\vec{b})]$ is (1) $-5$ (2) $-3$ (3) 5 (4) 3
The population $p(t)$ at time $t$ of a certain mouse species satisfies the differential equation $\frac{dp(t)}{dt} = 0.5\,p(t) - 450$. If $p(0) = 850$, then the time at which the population becomes zero is (1) $\ln 18$ (2) $\ln 9$ (3) $\frac{1}{2}\ln 18$ (4) $2\ln 18$
Let $\hat{a}$ and $\hat{b}$ be two unit vectors. If the vectors $\vec{c} = \hat{a} + 2\hat{b}$ and $\vec{d} = 5\hat{a} - 4\hat{b}$ are perpendicular to each other, then the angle between $\hat{a}$ and $\hat{b}$ is (1) $\frac{\pi}{6}$ (2) $\frac{\pi}{2}$ (3) $\frac{\pi}{3}$ (4) $\frac{\pi}{4}$
If $z \neq 1$ and $\frac{z^{2}}{z-1}$ is real, then the point represented by the complex number $z$ lies (1) either on the real axis or on a circle passing through the origin (2) on a circle with centre at the origin (3) either on the real axis or on a circle not passing through the origin (4) on the imaginary axis
If $n$ is a positive integer, then $(\sqrt{3}+1)^{2n} - (\sqrt{3}-1)^{2n}$ is (1) an irrational number (2) an odd positive integer (3) an even positive integer (4) a rational number other than positive integers
In a $\triangle PQR$, if $3\sin P + 4\cos Q = 6$ and $4\sin Q + 3\cos P = 1$, then the angle $R$ is equal to (1) $\frac{5\pi}{6}$ (2) $\frac{\pi}{6}$ (3) $\frac{\pi}{4}$ (4) $\frac{3\pi}{4}$
An ellipse is drawn by taking a diameter of the circle $(x-1)^{2}+y^{2}=1$ as its semi-minor axis and a diameter of the circle $x^{2}+(y-2)^{2}=4$ as its semi-major axis. If the centre of the ellipse is the origin and its axes are the coordinate axes, then the equation of the ellipse is (1) $4x^{2}+y^{2}=4$ (2) $x^{2}+4y^{2}=8$ (3) $4x^{2}+y^{2}=8$ (4) $x^{2}+4y^{2}=16$
A line is drawn through the point $(1,2)$ to meet the coordinate axes at $P$ and $Q$ such that it forms a triangle of area $\frac{9}{2}$ sq. units with the coordinate axes. The equation of the line $PQ$ is (1) $x+2y=5$ (2) $3x+y=5$ (3) $x+2y=4$ (4) $2x+y=4$
The equation of the circle passing through the point $(1,0)$ and $(0,1)$ and having the smallest radius is (1) $x^{2}+y^{2}-2x-2y+1=0$ (2) $x^{2}+y^{2}+2x+2y-7=0$ (3) $x^{2}+y^{2}-x-y=0$ (4) $x^{2}+y^{2}+x+y-2=0$
ABCD is a trapezium such that AB and CD are parallel and $BC \perp CD$. If $\angle ADB = \theta$, $BC = p$ and $CD = q$, then $AB$ is equal to (1) $\frac{(p^{2}+q^{2})\sin\theta}{p\cos\theta+q\sin\theta}$ (2) $\frac{p^{2}+q^{2}\cos\theta}{p\cos\theta+q\sin\theta}$ (3) $\frac{p^{2}+q^{2}}{p^{2}\cos\theta+q^{2}\sin\theta}$ (4) $\frac{(p^{2}+q^{2})\sin\theta}{(p\cos\theta+q\sin\theta)^{2}}$
If the integral $\displaystyle\int_{0}^{10} \frac{\lfloor x \rfloor e^{x}}{e^{\lfloor x \rfloor}} dx = \alpha(e-1)$, then $\alpha$ is equal to (where $\lfloor x \rfloor$ denotes the greatest integer function) (1) $\frac{1}{e-1}$ (2) $\frac{10}{e-1}$ (3) $\frac{e}{e-1}$ (4) $\frac{e^{10}-1}{e-1}$
If $100$ times the $100^{\text{th}}$ term of an AP with non-zero common difference equals the $50$ times its $50^{\text{th}}$ term, then the $150^{\text{th}}$ term of this AP is (1) $-150$ (2) 150 times its $50^{\text{th}}$ term (3) 150 (4) zero
If $g(x) = x^{2} + x - 2$ and $\frac{1}{2}\,g\circ f(x) = 2x^{2} - 5x + 2$, then $f(x)$ is equal to (1) $2x-3$ (2) $2x+3$ (3) $2x^{2}+3x+1$ (4) $2x^{2}-3x-1$
Let $x_{1}, x_{2}, \ldots, x_{n}$ be $n$ observations, and let $\bar{x}$ be their arithmetic mean and $\sigma^{2}$ be their variance. Statement 1: Variance of $2x_{1}, 2x_{2}, \ldots, 2x_{n}$ is $4\sigma^{2}$. Statement 2: Arithmetic mean of $2x_{1}, 2x_{2}, \ldots, 2x_{n}$ is $4\bar{x}$. (1) Statement 1 is false, Statement 2 is true (2) Statement 1 is true, Statement 2 is false (3) Statement 1 is true, Statement 2 is the correct explanation for Statement 1 (4) Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for Statement 1
Let $P$ and $Q$ be $3 \times 3$ matrices with $P \neq Q$. If $P^{3} = Q^{3}$ and $P^{2}Q = Q^{2}P$, then the determinant of $(P^{2}+Q^{2})$ is equal to (1) $-2$ (2) 1 (3) 0 (4) $-1$
The length of the diameter of the circle which touches the $x$-axis at the point $(1,0)$ and passes through the point $(2,3)$ is (1) $\frac{10}{3}$ (2) $\frac{3}{5}$ (3) $\frac{6}{5}$ (4) $\frac{5}{3}$
The area bounded between the parabolas $x^{2} = \frac{y}{4}$ and $x^{2} = 9y$, and the straight line $y = 2$ is (1) $20\sqrt{2}$ (2) $\frac{10\sqrt{2}}{3}$ (3) $\frac{20\sqrt{2}}{3}$ (4) $10\sqrt{2}$
Let $a, b \in \mathbb{R}$ be such that the function $f$ given by $f(x) = \ln|x| + bx^{2} + ax$, $x \neq 0$ has extreme values at $x = -1$ and $x = 2$. Statement 1: $f$ has local maximum at $x = -1$ and at $x = 2$. Statement 2: $a = \frac{1}{2}$ and $b = \frac{-1}{4}$. (1) Statement 1 is false, Statement 2 is true (2) Statement 1 is true, Statement 2 is false (3) Statement 1 is true, Statement 2 is the correct explanation for Statement 1 (4) Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for Statement 1
A spherical balloon is filled with 4500$\pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72\pi$ cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is (1) $\frac{9}{7}$ (2) $\frac{7}{9}$ (3) $\frac{2}{9}$ (4) $\frac{9}{2}$
If the lines $\frac{x-2}{1} = \frac{y-3}{1} = \frac{z-4}{-k}$ and $\frac{x-1}{k} = \frac{y-4}{2} = \frac{z-5}{1}$ are coplanar, then $k$ can be (1) $-1$ or $-3$ (2) $-1$ or $3$ (3) $1$ or $-1$ (4) $0$ or $-3$
The number of values of $k$ for which the linear equations $4x+ky+2z=0$, $kx+4y+z=0$, $2x+2y+z=0$ possess a non-zero solution is (1) 2 (2) 1 (3) zero (4) 3
The sum of first 20 terms of the sequence $0.7, 0.77, 0.777, \ldots$ is (1) $\frac{7}{81}(179-10^{-20})$ (2) $\frac{7}{9}(99-10^{-20})$ (3) $\frac{7}{81}(179+10^{-20})$ (4) $\frac{7}{9}(99+10^{-20})$
If $z$ is a complex number of unit modulus and argument $\theta$, then $\arg\left(\frac{1+z}{1+\bar{z}}\right)$ equals (1) $-\theta$ (2) $\frac{\pi}{2}-\theta$ (3) $\theta$ (4) $\pi-\theta$
The number of 3-digit numbers, with distinct digits, that can be formed using the digits $1, 2, 3, 4, 5, 6, 7$ and divisible by 3 is (1) 80 (2) 120 (3) 40 (4) 108
If $x, y, z$ are in AP and $\tan^{-1}x$, $\tan^{-1}y$ and $\tan^{-1}z$ are also in AP, then (1) $x = y = z$ (2) $2x = 3y = 6z$ (3) $6x = 3y = 2z$ (4) $6x = 4y = 3z$