The number of integers greater than 6,000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is: (1) 216 (2) 192 (3) 120 (4) 72
If $m$ is the A.M. of two distinct real numbers $l$ and $n$ $(l, n > 1)$ and $G_1, G_2$ and $G_3$ are three geometric means between $l$ and $n$, then $G_1^4 + 2G_2^4 + G_3^4$ equals: (1) $4l^2 m n$ (2) $4lm^2 n$ (3) $4lmn^2$ (4) $4l^2 m^2 n^2$
The set of all values of $\lambda$ for which the system of linear equations: $2x_1 - 2x_2 + x_3 = \lambda x_1$ $2x_1 - 3x_2 + 2x_3 = \lambda x_2$ $-x_1 + 2x_2 = \lambda x_3$ has a non-trivial solution: (1) is an empty set (2) is a singleton (3) contains two elements (4) contains more than two elements
The number of distinct real roots of $\begin{vmatrix} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{vmatrix} = 0$ in the interval $-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$ is: (1) 0 (2) 2 (3) 1 (4) 3
If $A = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{pmatrix}$ is a matrix satisfying the equation $AA^T = 9I$, where $I$ is $3 \times 3$ identity matrix, then the ordered pair $(a, b)$ is equal to: (1) $(2, -1)$ (2) $(-2, 1)$ (3) $(2, 1)$ (4) $(-2, -1)$
The area (in sq. units) of the region described by $\{(x, y) : y^2 \leq 2x \text{ and } y \geq 4x - 1\}$ is: (1) $\frac{7}{32}$ (2) $\frac{5}{64}$ (3) $\frac{15}{64}$ (4) $\frac{9}{32}$
Q68
First order differential equations (integrating factor)View
Let $y(x)$ be the solution of the differential equation $(x \log x) \frac{dy}{dx} + y = 2x \log x$, $(x \geq 1)$. Then $y(e)$ is equal to: (1) $e$ (2) $0$ (3) $2$ (4) $2e$
Let $f(x) = x^2$, $g(x) = \sin x$ for all $x \in \mathbb{R}$ and $h(x) = (gof)(x) = g(f(x))$. Statement I: $h$ is not differentiable at $x = 0$. Statement II: $(hog)(x) = \sin^2(\sin x)$. Which of the following is correct? (1) Statement I is false, Statement II is true (2) Statement I is true, Statement II is false (3) Both Statement I and Statement II are true (4) Both Statement I and Statement II are false
Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = \begin{cases} k - 2x, & \text{if } x \leq -1 \\ 2x + 3, & \text{if } x > -1 \end{cases}$. If $f$ has a local minimum at $x = -1$, then a possible value of $k$ is: (1) $0$ (2) $-\frac{1}{2}$ (3) $-1$ (4) $1$
If the function $g(x) = \begin{cases} k\sqrt{x+1}, & 0 \leq x \leq 3 \\ mx + 2, & 3 < x \leq 5 \end{cases}$ is differentiable, then the value of $k + m$ is: (1) $2$ (2) $\frac{16}{5}$ (3) $\frac{10}{3}$ (4) $4$
The normal to the curve $x^2 + 2xy - 3y^2 = 0$ at $(1, 1)$: (1) does not meet the curve again (2) meets the curve again in the second quadrant (3) meets the curve again in the third quadrant (4) meets the curve again in the fourth quadrant
If $12 \cot^2\theta - 31 \csc\theta + 32 = 0$, then the value of $\sin\theta$ is: (1) $\frac{3}{5}$ or $1$ (2) $\frac{2}{3}$ or $-\frac{2}{3}$ (3) $\frac{4}{5}$ or $\frac{3}{4}$ (4) $\pm\frac{1}{2}$
A complex number $z$ is said to be unimodular if $|z| = 1$. Suppose $z_1$ and $z_2$ are complex numbers such that $\frac{z_1 - 2z_2}{2 - z_1\bar{z}_2}$ is unimodular and $z_2$ is not unimodular. Then the point $z_1$ lies on a: (1) straight line parallel to $x$-axis (2) straight line parallel to $y$-axis (3) circle of radius 2 (4) circle of radius $\sqrt{2}$
Let $\alpha$ and $\beta$ be the roots of equation $px^2 + qx + r = 0$, $p \neq 0$. If $p$, $q$, $r$ are in A.P. and $\frac{1}{\alpha} + \frac{1}{\beta} = 4$, then the value of $|\alpha - \beta|$ is: (1) $\frac{\sqrt{61}}{9}$ (2) $\frac{2\sqrt{17}}{9}$ (3) $\frac{\sqrt{34}}{9}$ (4) $\frac{2\sqrt{13}}{9}$
The sum of coefficients of integral powers of $x$ in the binomial expansion of $(1 - 2\sqrt{x})^{50}$ is: (1) $\frac{1}{2}(3^{50} + 1)$ (2) $\frac{1}{2}(3^{50})$ (3) $\frac{1}{2}(3^{50} - 1)$ (4) $\frac{1}{2}(2^{50} + 1)$
The distance of the point $(1, 0, 2)$ from the point of intersection of the line $\frac{x-2}{3} = \frac{y+1}{4} = \frac{z-2}{12}$ and the plane $x - y + z = 16$, is: (1) $2\sqrt{14}$ (2) $8$ (3) $3\sqrt{21}$ (4) $13$
Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three non-zero vectors such that no two of them are collinear and $(\vec{a} \times \vec{b}) \times \vec{c} = \frac{1}{3}|\vec{b}||\vec{c}|\vec{a}$. If $\theta$ is the angle between vectors $\vec{b}$ and $\vec{c}$, then a value of $\sin\theta$ is: (1) $\frac{2\sqrt{2}}{3}$ (2) $-\frac{\sqrt{2}}{3}$ (3) $\frac{2}{3}$ (4) $-\frac{2\sqrt{3}}{3}$
The mean of the data set comprising of 16 observations is 16. If one of the observation valued 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data, is: (1) 16.8 (2) 15.8 (3) 14.0 (4) 16.0
If $A$ and $B$ are coefficients of $x^n$ in the expansions of $(1+x)^{2n}$ and $(1+x)^{2n-1}$ respectively, then $\frac{A}{B}$ equals: (1) $1$ (2) $2$ (3) $\frac{1}{2}$ (4) $\frac{1}{n}$
The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $(0, 0)$, $(0, 41)$ and $(41, 0)$, is: (1) 820 (2) 780 (3) 901 (4) 861
Locus of the image of the point $(2, 3)$ in the line $(2x - 3y + 4) + k(x - 2y + 3) = 0$, $k \in \mathbb{R}$, is a: (1) straight line parallel to $x$-axis (2) straight line parallel to $y$-axis (3) circle of radius $\sqrt{2}$ (4) circle of radius $\sqrt{3}$
The number of terms in an A.P. is even; the sum of the odd terms in it is 24 and that the even terms is 30. If the last term exceeds the first term by $10\frac{1}{2}$, then the number of terms in the A.P. is: (1) 4 (2) 8 (3) 12 (4) 16
If the angles of elevation of the top of a tower from three collinear points A, B and C, on a line leading to the foot of the tower, are $30^\circ$, $45^\circ$ and $60^\circ$ respectively, then the ratio $AB : BC$, is: (1) $\sqrt{3} : 1$ (2) $\sqrt{3} : \sqrt{2}$ (3) $1 : \sqrt{3}$ (4) $2 : 3$