If the center and radius of the circle $\left|\frac{z - 2}{z - 3}\right| = 2$ are respectively $(\alpha, \beta)$ and $\gamma$, then $3\alpha + \beta + \gamma$ is equal to (1) 11 (2) 9 (3) 10 (4) 12
The combined equation of the two lines $ax + by + c = 0$ and $a'x + b'y + c' = 0$ can be written as $(ax + by + c)(a'x + b'y + c') = 0$. The equation of the angle bisectors of the lines represented by the equation $2x^2 + xy - 3y^2 = 0$ is (1) $3x^2 + 5xy + 2y^2 = 0$ (2) $x^2 - y^2 + 10xy = 0$ (3) $3x^2 + xy - 2y^2 = 0$ (4) $x^2 - y^2 - 10xy = 0$
The mean and variance of 5 observations are 5 and 8 respectively. If 3 observations are $1, 3, 5$, then the sum of cubes of the remaining two observations is (1) 1072 (2) 1792 (3) 1216 (4) 1456
For a triangle $ABC$, the value of $\cos 2A + \cos 2B + \cos 2C$ is least. If its inradius is 3 and incentre is $M$, then which of the following is NOT correct? (1) Perimeter of $\triangle ABC$ is $18\sqrt{3}$ (2) $\sin 2A + \sin 2B + \sin 2C = \sin A + \sin B + \sin C$ (3) $\overrightarrow{MA} \cdot \overrightarrow{MB} = -18$ (4) area of $\triangle ABC$ is $\frac{27\sqrt{3}}{2}$
Let $S$ denote the set of all real values of $\lambda$ such that the system of equations $$\lambda x + y + z = 1$$ $$x + \lambda y + z = 1$$ $$x + y + \lambda z = 1$$ is inconsistent, then $\sum_{\lambda \in S} (\lambda^2 + \lambda)$ is equal to (1) 2 (2) 12 (3) 4 (4) 6
Let $S$ be the set of all solutions of the equation $\cos^{-1}(2x) - 2\cos^{-1}\left(\sqrt{1 - x^2}\right) = \pi$, $x \in \left[-\frac{1}{2}, \frac{1}{2}\right]$. Then $\sum_{x \in S} 2\sin^{-1}(x^2 - 1)$ is equal to (1) 0 (2) $\frac{-2\pi}{3}$ (3) $\pi - \sin^{-1}\frac{\sqrt{3}}{4}$ (4) $\pi - 2\sin^{-1}\frac{\sqrt{3}}{4}$
Let $f(x) = \begin{vmatrix} 1 + \sin^2 x & \cos^2 x & \sin 2x \\ \sin^2 x & 1 + \cos^2 x & \sin 2x \\ \sin^2 x & \cos^2 x & 1 + \sin 2x \end{vmatrix}$, $x \in \left[\frac{\pi}{6}, \frac{\pi}{3}\right]$. If $\alpha$ and $\beta$ respectively are the maximum and the minimum values of $f$, then (1) $\beta^2 - 2\sqrt{\alpha} = \frac{19}{4}$ (2) $\beta^2 + 2\sqrt{\alpha} = \frac{19}{4}$ (3) $\alpha^2 - \beta^2 = 4\sqrt{3}$ (4) $\alpha^2 + \beta^2 = \frac{9}{2}$
The area enclosed by the closed curve $C$ given by the differential equation $\frac{dy}{dx} + \frac{x + a}{y - 2} = 0$, $y(1) = 0$ is $4\pi$. Let $P$ and $Q$ be the points of intersection of the curve $C$ and the $y$-axis. If normals at $P$ and $Q$ on the curve $C$ intersect $x$-axis at points $R$ and $S$ respectively, then the length of the line segment $RS$ is (1) $2\sqrt{3}$ (2) $\frac{2\sqrt{3}}{3}$ (3) 2 (4) $\frac{4\sqrt{3}}{3}$
Q77
First order differential equations (integrating factor)View
If $y = y(x)$ is the solution curve of the differential equation $\frac{dy}{dx} + y\tan x = x\sec x$, $0 \leq x \leq \frac{\pi}{3}$, $y(0) = 1$, then $y\left(\frac{\pi}{6}\right)$ is equal to (1) $\frac{\pi}{12} - \frac{\sqrt{3}}{2}\log_e\frac{2}{e\sqrt{3}}$ (2) $\frac{\pi}{12} + \frac{\sqrt{3}}{2}\log_e\frac{2\sqrt{3}}{e}$ (3) $\frac{\pi}{12} - \frac{\sqrt{3}}{2}\log_e\frac{2\sqrt{3}}{e}$ (4) $\frac{\pi}{12} + \frac{\sqrt{3}}{2}\log_e\frac{2}{e\sqrt{3}}$
Let the image of the point $P(2, -1, 3)$ in the plane $x + 2y - z = 0$ be $Q$. Then the distance of the plane $3x + 2y + z + 29 = 0$ from the point $Q$ is (1) $\frac{22\sqrt{2}}{7}$ (2) $\frac{24\sqrt{2}}{7}$ (3) $2\sqrt{14}$ (4) $3\sqrt{14}$
In a binomial distribution $B(n, p)$, the sum and product of the mean and variance are 5 and 6 respectively, then $6(n + p - q)$ is equal to: (1) 51 (2) 52 (3) 53 (4) 50
The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is $\_\_\_\_$.
Let $a_1 = 8, a_2, a_3, \ldots, a_n$ be an A.P. If the sum of its first four terms is 50 and the sum of its last four terms is 170, then the product of its middle two terms is $\_\_\_\_$.
If $\int_0^1 (x^{21} + x^{14} + x^7)(2x^{14} + 3x^7 + 6)^{1/7}\, dx = \frac{1}{l} \cdot 11^{m/n}$ where $l, m, n \in \mathbb{N}$, $m$ and $n$ are co-prime, then $l + m + n$ is equal to $\_\_\_\_$.
Let $A$ be the area bounded by the curve $y = x(x - 3)$, the $x$-axis and the ordinates $x = -1$ and $x = 2$. Then $12A$ is equal to $\_\_\_\_$.
Q88
First order differential equations (integrating factor)View
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f'(x) + f(x) = \int_0^2 f(t)\, dt$. If $f(0) = e^{-2}$, then $2f(0) - f(2)$ is equal to $\_\_\_\_$.
Let $\vec{v} = \alpha\hat{i} + 2\hat{j} - 3\hat{k}$, $\vec{w} = 2\alpha\hat{i} + \hat{j} - \hat{k}$, and $\vec{u}$ be a vector such that $|\vec{u}| = \alpha > 0$. If the minimum value of the scalar triple product $[\vec{u}\, \vec{v}\, \vec{w}]$ is $-\alpha\sqrt{3401}$, and $|\vec{u} \cdot \hat{i}|^2 = \frac{m}{n}$ where $m$ and $n$ are coprime natural numbers, then $m + n$ is equal to $\_\_\_\_$.
$A(2, 6, 2)$, $B(-4, 0, \lambda)$, $C(2, 3, -1)$ and $D(4, 5, 0)$, $\lambda \leq 5$ are the vertices of a quadrilateral $ABCD$. If its area is 18 square units, then $5 - 6\lambda$ is equal to $\_\_\_\_$.