Let $x_1, x_2, \ldots, x_{10}$ be ten observations such that $\sum_{i=1}^{10}(x_i - 2) = 30$, $\sum_{i=1}^{10}(x_i - \beta)^2 = 98$, $\beta > 2$, and their variance is $\frac{4}{5}$. If $\mu$ and $\sigma^2$ are respectively the mean and the variance of $2(x_1 - 1) + 4\beta$, $2(x_2 - 1) + 4\beta, \ldots, 2(x_{10} - 1) + 4\beta$, then $\frac{\beta\mu}{\sigma^2}$ is equal to: (1) 100 (2) 120 (3) 110 (4) 90
Consider an A.P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its $11^{\text{th}}$ term is: (1) 90 (2) 84 (3) 122 (4) 108
The number of solutions of the equation $\left(\frac{9}{x} - \frac{9}{\sqrt{x}} + 2\right)\left(\frac{2}{x} - \frac{7}{\sqrt{x}} + 3\right) = 0$ is: (1) 2 (2) 3 (3) 1 (4) 4
Define a relation R on the interval $\left[0, \frac{\pi}{2}\right)$ by $x\mathrm{R}y$ if and only if $\sec^2 x - \tan^2 y = 1$. Then R is: (1) both reflexive and transitive but not symmetric (2) an equivalence relation (3) reflexive but neither symmetric nor transitive (4) both reflexive and symmetric but not transitive
Two parabolas have the same focus $(4,3)$ and their directrices are the $x$-axis and the $y$-axis, respectively. If these parabolas intersect at the points $A$ and $B$, then $(AB)^2$ is equal to: (1) 392 (2) 384 (3) 192 (4) 96
Let P be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in P are formed by using the digits 1, 2 and 3 only, then the number of elements in the set $P$ is: (1) 173 (2) 164 (3) 158 (4) 161
Let $\overrightarrow{\mathrm{a}} = \hat{i} + 2\hat{j} + \hat{k}$ and $\overrightarrow{\mathrm{b}} = 2\hat{i} + 7\hat{j} + 3\hat{k}$. Let $\mathrm{L}_1: \overrightarrow{\mathrm{r}} = (-\hat{i} + 2\hat{j} + \hat{k}) + \lambda\overrightarrow{\mathrm{a}},\ \lambda \in \mathbf{R}$ and $\mathrm{L}_2: \overrightarrow{\mathrm{r}} = (\hat{j} + \hat{k}) + \mu\overrightarrow{\mathrm{b}},\ \mu \in \mathbf{R}$ be two lines. If the line $\mathrm{L}_3$ passes through the point of intersection of $\mathrm{L}_1$ and $\mathrm{L}_2$, and is parallel to $\vec{a} + \vec{b}$, then $\mathrm{L}_3$ passes through the point: (1) $(5, 17, 4)$ (2) $(2, 8, 5)$ (3) $(8, 26, 12)$ (4) $(-1, -1, 1)$
Let $\overrightarrow{\mathrm{a}} = 2\hat{i} - \hat{j} + 3\hat{k}$, $\overrightarrow{\mathrm{b}} = 3\hat{i} - 5\hat{j} + \hat{k}$ and $\overrightarrow{\mathrm{c}}$ be a vector such that $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}} = \overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{b}}$ and $(\vec{a} + \vec{c}) \cdot (\vec{b} + \vec{c}) = 168$. Then the maximum value of $|\vec{c}|^2$ is: (1) 462 (2) 77 (3) 154 (4) 308
Let the ellipse $\mathrm{E}_1: \frac{x^2}{\mathrm{a}^2} + \frac{y^2}{\mathrm{b}^2} = 1,\ \mathrm{a} > \mathrm{b}$ and $\mathrm{E}_2: \frac{x^2}{\mathrm{A}^2} + \frac{y^2}{\mathrm{B}^2} = 1,\ \mathrm{A} < \mathrm{B}$ have same eccentricity $\frac{1}{\sqrt{3}}$. Let the product of their lengths of latus rectums be $\frac{32}{\sqrt{3}}$, and the distance between the foci of $E_1$ be 4. If $E_1$ and $E_2$ meet at $A, B, C$ and $D$, then the area of the quadrilateral $ABCD$ equals: (1) $\frac{12\sqrt{6}}{5}$ (2) $6\sqrt{6}$ (3) $\frac{18\sqrt{6}}{5}$ (4) $\frac{24\sqrt{6}}{5}$
Let $\mathrm{L}_1: \frac{x-1}{1} = \frac{y-2}{-1} = \frac{z-1}{2}$ and $\mathrm{L}_2: \frac{x+1}{-1} = \frac{y-2}{2} = \frac{z}{1}$ be two lines. Let $L_3$ be a line passing through the point $(\alpha, \beta, \gamma)$ and be perpendicular to both $L_1$ and $L_2$. If $L_3$ intersects $\mathrm{L}_1$, then $|5\alpha - 11\beta - 8\gamma|$ equals: (1) 20 (2) 18 (3) 25 (4) 16
Let M and m respectively be the maximum and the minimum values of $$f(x) = \begin{vmatrix} 1+\sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2 x & 1+\cos^2 x & 4\sin 4x \\ \sin^2 x & \cos^2 x & 1+4\sin 4x \end{vmatrix}, \quad x \in \mathbf{R}.$$ Then $M^4 - m^4$ is equal to: (1) 1280 (2) 1295 (3) 1215 (4) 1040
Let $ABC$ be a triangle formed by the lines $7x - 6y + 3 = 0$, $x + 2y - 31 = 0$ and $9x - 2y - 19 = 0$. Let the point $(h, k)$ be the image of the centroid of $\triangle ABC$ in the line $3x + 6y - 53 = 0$. Then $h^2 + k^2 + hk$ is equal to: (1) 47 (2) 37 (3) 36 (4) 40
The least value of $n$ for which the number of integral terms in the Binomial expansion of $(\sqrt[3]{7} + \sqrt[12]{11})^n$ is 183, is: (1) 2184 (2) 2196 (3) 2148 (4) 2172
Q18
First order differential equations (integrating factor)View
Let $y = y(x)$ be the solution of the differential equation $\cos x\left(\log_e(\cos x)\right)^2 \mathrm{d}y + \left(\sin x - 3y\sin x\log_e(\cos x)\right)\mathrm{d}x = 0$, $x \in \left(0, \frac{\pi}{2}\right)$. If $y\left(\frac{\pi}{4}\right) = \frac{-1}{\log_e 2}$, then $y\left(\frac{\pi}{6}\right)$ is equal to: (1) $\frac{1}{\log_e(3) - \log_e(4)}$ (2) $\frac{2}{\log_e(3) - \log_e(4)}$ (3) $\frac{1}{\log_e(4) - \log_e(3)}$ (4) $-\frac{1}{\log_e(4)}$
Let the line $x + y = 1$ meet the circle $x^2 + y^2 = 4$ at the points A and B. If the line perpendicular to $AB$ and passing through the mid point of the chord $AB$ intersects the circle at $C$ and $D$, then the area of the quadrilateral ADBC is equal to: (1) $\sqrt{14}$ (2) $3\sqrt{7}$ (3) $2\sqrt{14}$ (4) $5\sqrt{7}$
Let $f:(0,\infty) \rightarrow \mathbf{R}$ be a twice differentiable function. If for some $\mathrm{a} \neq 0$, $\int_0^1 f(\lambda x)\,\mathrm{d}\lambda = \mathrm{a}f(x)$, $f(1) = 1$ and $f(16) = \frac{1}{8}$, then $16 - f'\left(\frac{1}{16}\right)$ is equal to \_\_\_\_ .
The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is \_\_\_\_ .
Let $[t]$ be the greatest integer less than or equal to $t$. Then the least value of $p \in \mathbf{N}$ for which $$\lim_{x \rightarrow 0^+}\left(x\left(\left[\frac{1}{x}\right] + \left[\frac{2}{x}\right] + \ldots + \left[\frac{\mathrm{p}}{x}\right]\right) - x^2\left(\left[\frac{1}{x^2}\right] + \left[\frac{2^2}{x^2}\right] + \ldots + \left[\frac{9^2}{x^2}\right]\right)\right) \geq 1$$ is equal to \_\_\_\_ .
Two projectiles are fired with same initial speed from same point on ground at angles of $(45^\circ - \alpha)$ and $(45^\circ + \alpha)$, respectively, with the horizontal direction. The ratio of their maximum heights attained is: (1) $\frac{1 - \tan\alpha}{1 + \tan\alpha}$ (2) $\frac{1 - \sin 2\alpha}{1 + \sin 2\alpha}$ (3) $\frac{1 + \sin 2\alpha}{1 - \sin 2\alpha}$ (4) $\frac{1 + \sin\alpha}{1 - \sin\alpha}$
A body of mass $m$ connected to a massless and unstretchable string goes in a vertical circle of radius $R$ under gravity $g$. The other end of the string is fixed at the center of circle. If velocity at top of circular path is $n\sqrt{gR}$, where $n \geq 1$, then ratio of kinetic energy of the body at bottom to that at top of the circle is: (1) $\frac{n^2}{n^2 + 4}$ (2) $\frac{n^2 + 4}{n^2}$ (3) $\frac{n+4}{n}$ (4) $\frac{n}{n+4}$
The coordinates of a particle with respect to origin in a given reference frame is $(1,1,1)$ meters. If a force of $\overrightarrow{\mathrm{F}} = \hat{i} - \hat{j} + \hat{k}$ acts on the particle, then the magnitude of torque (with respect to origin) in the $z$-direction is \_\_\_\_ .
The maximum speed of a boat in still water is $27\,\mathrm{km/h}$. Now this boat is moving downstream in a river flowing at $9\,\mathrm{km/h}$. A man in the boat throws a ball vertically upwards with speed of $10\,\mathrm{m/s}$. Range of the ball as observed by an observer at rest on the river bank, is \_\_\_\_ cm. (Take $g = 10\,\mathrm{m/s^2}$)