The velocity $(v)$ and time $(t)$ graph of a body in a straight line motion is shown in the figure. The point $S$ is at 4.333 seconds. The total distance covered by the body in 6 s is: (1) $\frac{37}{3}$ m (2) 12 m (3) 11 m (4) $\frac{49}{4}$ m
A spaceship in space sweeps stationary interplanetary dust. As a result, its mass increases at a rate $\frac{dM(t)}{dt} = bv^2(t)$, where $v(t)$ is its instantaneous velocity. The instantaneous acceleration of the satellite is: (1) $-bv^3(t)$ (2) $\frac{-bv^3}{M(t)}$ (3) $-\frac{2bv^3}{M(t)}$ (4) $-\frac{bv^3}{2M(t)}$
The acceleration due to gravity on the earth's surface at the poles is $g$ and angular velocity of the earth about the axis passing through the pole is $\omega$. An object is weighed at the equator and at a height $h$ above the poles by using a spring balance. If the weights are found to be same, then $h$ is: ($h \ll R$, where $R$ is the radius of the earth) (1) $\frac{R^2\omega^2}{2g}$ (2) $\frac{R^2\omega^2}{g}$ (3) $\frac{R^2\omega^2}{4g}$ (4) $\frac{R^2\omega^2}{8g}$
A body of mass 2 kg is driven by an engine delivering a constant power of $1\,\mathrm{J\,s^{-1}}$. The body starts from rest and moves in a straight line. After 9 s, the body has moved a distance (in m) ...
A thin rod of mass 0.9 kg and length 1 m is suspended, at rest, from one end so that it can freely oscillate in the vertical plane. A particle of mass 0.1 kg moving in a straight line with velocity $80\,\mathrm{m\,s^{-1}}$ hits the rod at its bottom most point and sticks to it (see figure). The angular speed (in $\mathrm{rad\,s^{-1}}$) of the rod immediately after the collision will be $\_\_\_\_$
If $\alpha$ and $\beta$ are the roots of the equation, $7x^2 - 3x - 2 = 0$, then the value of $\frac{\alpha}{1-\alpha^2} + \frac{\beta}{1-\beta^2}$ is equal to: (1) $\frac{27}{32}$ (2) $\frac{1}{24}$ (3) $\frac{3}{8}$ (4) $\frac{27}{16}$
There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is: (1) 3000 (2) 1500 (3) 2255 (4) 2250
If the sum of the second, third and fourth terms of a positive term G.P. is 3 and the sum of its sixth, seventh and eighth terms is 243, then the sum of the first 50 terms of this G.P. is: (1) $\frac{1}{26}\left(3^{49}-1\right)$ (2) $\frac{1}{26}\left(3^{50}-1\right)$ (3) $\frac{2}{13}\left(3^{50}-1\right)$ (4) $\frac{1}{13}\left(3^{50}-1\right)$
If the sum of the first 20 terms of the series $\log_{(7^{1/2})}x + \log_{(7^{1/3})}x + \log_{(7^{1/4})}x + \ldots$ is 460, then $x$ is equal to: (1) $7^2$ (2) $7^{1/2}$ (3) $e^2$ (4) $7^{46/21}$
If the length of the chord of the circle, $x^2 + y^2 = r^2$ $(r > 0)$ along the line, $y - 2x = 3$ is $r$, then $r^2$ is equal to: (1) $\frac{9}{5}$ (2) 12 (3) $\frac{24}{5}$ (4) $\frac{12}{5}$
If the line $y = mx + c$ is a common tangent to the hyperbola $\frac{x^2}{100} - \frac{y^2}{64} = 1$ and the circle $x^2 + y^2 = 36$, then which one of the following is true? (1) $c^2 = 369$ (2) $5m = 4$ (3) $4c^2 = 369$ (4) $8m + 5 = 0$
$\lim_{x\rightarrow 0} \frac{x\left(e^{\left(\sqrt{1+x^2+x^4}-1\right)/x}-1\right)}{\sqrt{1+x^2+x^4}-1}$ (1) is equal to $\sqrt{e}$ (2) is equal to 1 (3) is equal to 0 (4) does not exist
If the mean and the standard deviation of the data $3, 5, 7, a, b$ are 5 and 2 respectively, then $a$ and $b$ are the roots of the equation: (1) $x^2 - 10x + 18 = 0$ (2) $2x^2 - 20x + 19 = 0$ (3) $x^2 - 10x + 19 = 0$ (4) $x^2 - 20x + 18 = 0$
If the system of linear equations $$x + y + 3z = 0$$ $$x + 3y + k^2z = 0$$ $$3x + y + 3z = 0$$ has a non-zero solution $(x, y, z)$ for some $k \in \mathrm{R}$, then $x + \left(\frac{y}{z}\right)$ is equal to: (1) $-3$ (2) $9$ (3) $3$ (4) $-9$
If $a + x = b + y = c + z + 1$, where $a, b, c, x, y, z$ are non-zero distinct real numbers, then $\left|\begin{array}{lll} x & a+y & x+a \\ y & b+y & y+b \\ z & c+y & z+c \end{array}\right|$ is equal to: (1) $y(b-a)$ (2) $y(a-b)$ (3) $0$ (4) $y(a-c)$
The derivative of $\tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ with respect to $\tan^{-1}\left(\frac{2x\sqrt{1-x^2}}{1-2x^2}\right)$ at $x = \frac{1}{2}$ is: (1) $\frac{2\sqrt{3}}{5}$ (2) $\frac{\sqrt{3}}{12}$ (3) $\frac{2\sqrt{3}}{3}$ (4) $\frac{\sqrt{3}}{10}$
If $x = 1$ is a critical point of the function $f(x) = (3x^2 + ax - 2 - a)e^x$, then (1) $x = 1$ and $x = -\frac{2}{3}$ are local minima of $f$ (2) $x = 1$ and $x = -\frac{2}{3}$ is a local maxima of $f$ (3) $x = 1$ is a local maxima and $x = -\frac{2}{3}$ is a local minima of $f$ (4) $x = 1$ is a local minima and $x = -\frac{2}{3}$ are local maxima of $f$
Which of the following points lies on the tangent to the curve $x^4 e^y + 2\sqrt{y+1} = 3$ at the point $(1, 0)$? (1) $(2, 2)$ (2) $(2, 6)$ (3) $(-2, 6)$ (4) $(-2, 4)$
If $\int \frac{\cos\theta}{5 + 7\sin\theta - 2\cos^2\theta}\,d\theta = A\log_e|B(\theta)| + C$, where $C$ is a constant of integration, then $\frac{B(\theta)}{A}$ can be: (1) $\frac{2\sin\theta+1}{\sin\theta+3}$ (2) $\frac{2\sin\theta+1}{5(\sin\theta+3)}$ (3) $\frac{5(\sin\theta+3)}{2\sin\theta+1}$ (4) $\frac{5(2\sin\theta+1)}{\sin\theta+3}$
The area (in sq. units) of the region $A = \{(x,y) : (x-1)[x] \leq y \leq 2\sqrt{x},\, 0 \leq x \leq 2\}$, where $[t]$ denotes the greatest integer function, is: (1) $\frac{8}{3}\sqrt{2} - \frac{1}{2}$ (2) $\frac{4}{3}\sqrt{2} + 1$ (3) $\frac{8}{3}\sqrt{2} - 1$ (4) $\frac{4}{3}\sqrt{2} - \frac{1}{2}$
Q69
First order differential equations (integrating factor)View
Let $y = y(x)$ be the solution of the differential equation $\cos x\frac{dy}{dx} + 2y\sin x = \sin 2x$, $x \in \left(0, \frac{\pi}{2}\right)$. If $y(\pi/3) = 0$, then $y(\pi/4)$ is equal to: (1) $2 - \sqrt{2}$ (2) $2 + \sqrt{2}$ (3) $\sqrt{2} - 2$ (4) $\frac{1}{\sqrt{2}} - 1$
If for some $\alpha \in \mathrm{R}$, the lines $L_1: \frac{x+1}{2} = \frac{y-2}{-1} = \frac{z-1}{1}$ and $L_2: \frac{x+2}{\alpha} = \frac{y+1}{5-\alpha} = \frac{z+1}{1}$ are coplanar, then the line $L_2$ passes through the point: (1) $(10, 2, 2)$ (2) $(2, -10, -2)$ (3) $(10, -2, -2)$ (4) $(-2, 10, 2)$
Let $A = \{a, b, c\}$ and $B = \{1, 2, 3, 4\}$. Then the number of elements in the set $C = \{f: A \rightarrow B \mid 2 \in f(A)$ and $f$ is not one-one$\}$ is ...
If the lines $x + y = a$ and $x - y = b$ touch the curve $y = x^2 - 3x + 2$ at the points where the curve intersects the $x$-axis, then $\frac{a}{b}$ is equal to ...
Let the vectors $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ be such that $|\overrightarrow{\mathrm{a}}| = 2$, $|\overrightarrow{\mathrm{b}}| = 4$ and $|\overrightarrow{\mathrm{c}}| = 4$. If the projection of $\overrightarrow{\mathrm{b}}$ on $\overrightarrow{\mathrm{a}}$ is equal to the projection of $\overrightarrow{\mathrm{c}}$ on $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ is perpendicular to $\overrightarrow{\mathrm{c}}$, then the value of $|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{c}}|$ is ...
In a bombing attack, there is $50\%$ chance that a bomb will hit the target. At least two independent hits are required to destroy the target completely. Then the minimum number of bombs, that must be dropped to ensure that the probability of the target being destroyed is at least $0.99$, is ...