An object, moving with a speed of $6.25 \mathrm{~m}/\mathrm{s}$, is decelerated at a rate given by : $$\frac{\mathrm{dv}}{\mathrm{dt}} = -2.5\sqrt{\mathrm{v}}$$ where $v$ is the instantaneous speed. The time taken by the object, to come to rest, would be: (1) 2 s (2) 4 s (3) 8 s (4) 1 s
A water fountain on the ground sprinkles water all around it. If the speed of water coming out of the fountain is $v$, the total area around the fountain that gets wet is: (1) $\pi \frac{v^{4}}{g^{2}}$ (2) $\frac{\pi}{2}\frac{v^{4}}{g^{2}}$ (3) $\pi \frac{v^{2}}{g^{2}}$ (4) $\pi \frac{v^{4}}{g}$
A mass $m$ hangs with the help of a string wrapped around a pulley on a frictionless bearing. The pulley has mass $m$ and radius $R$. Assuming pulley to be a perfect uniform circular disc, the acceleration of the mass $m$, if the string does not slip on the pulley, is (1) g (2) $\frac{2}{3}\mathrm{~g}$ (3) $\frac{g}{3}$ (4) $\frac{3}{2}g$
A pulley of radius 2 m is rotated about its axis by a force $\mathrm{F} = \left(20\mathrm{t} - 5\mathrm{t}^{2}\right)$ Newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is $10\,\mathrm{kg\,m}^2$, the number of rotations made by the pulley before its direction of motion is reversed, is: (1) more than 3 but less than 6 (2) more than 6 but less than 9 (3) more than 9 (4) less than 3
A mass $M$, attached to a horizontal spring, executes S.H.M. with amplitude $A_{1}$. When the mass $M$ passes through its mean position then a smaller mass m is placed over it and both of them move together with amplitude $A_{2}$. The ratio of $\left(\frac{A_{1}}{A_{2}}\right)$ is: (1) $\frac{M+m}{M}$ (2) $\left(\frac{M}{M+m}\right)^{1/2}$ (3) $\left(\frac{M+m}{M}\right)^{1/2}$ (4) $\frac{M}{M+m}$
Let $\alpha, \beta$ be real and $z$ be a complex number. If $z^{2}+\alpha z+\beta=0$ has two distinct roots on the line $\operatorname{Re}z=1$, then it is necessary that (1) $\beta\in(-1,0)$ (2) $|\beta|=1$ (3) $\beta\in(1,\infty)$ (4) $\beta\in(0,1)$
This question has Statement-1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements. Statement-1: The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is ${}^{9}\mathrm{C}_{3}$. Statement-2: The number of ways of choosing any 3 places from 9 different places is ${}^{9}\mathrm{C}_{3}$. (1) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1. (2) Statement-1 is true, Statement-2 is false. (3) Statement-1 is false, Statement-2 is true. (4) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
A man saves Rs. 200 in each of the first three months of his service. In each of the subsequent months his saving increases by Rs. 40 more than the saving of immediately previous month. His total saving from the start of service will be Rs. 11040 after (1) 19 months (2) 20 months (3) 21 months (4) 18 months
If $A=\sin^{2}x+\cos^{4}x$, then for all real $x$ (1) $\frac{13}{16}\leq\mathrm{A}\leq 1$ (2) $1\leq A\leq 2$ (3) $\frac{3}{4}\leq\mathrm{A}\leq\frac{13}{16}$ (4) $\frac{3}{4}\leq\mathrm{A}\leq 1$
The lines $L_{1}: y-x=0$ and $L_{2}: 2x+y=0$ intersect the line $L_{3}: y+2=0$ at $P$ and $Q$ respectively. The bisector of the acute angle between $L_{1}$ and $L_{2}$ intersects $L_{3}$ at $R$. This question has Statement-1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements. Statement-1: The ratio $PR:RQ$ equals $2\sqrt{2}:\sqrt{5}$. Statement-2: In any triangle, bisector of an angle divides the triangle into two similar triangles. (1) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1. (2) Statement-1 is true, Statement-2 is false. (3) Statement-1 is false, Statement-2 is true. (4) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
Equation of the ellipse whose axes are the axes of coordinates and which passes through the point $(-3,1)$ and has eccentricity $\sqrt{\frac{2}{5}}$ is (1) $5x^{2}+3y^{2}-48=0$ (2) $3x^{2}+5y^{2}-15=0$ (3) $5x^{2}+3y^{2}-32=0$ (4) $3x^{2}+5y^{2}-32=0$
If the mean deviation about the median of the numbers $\mathrm{a},2\mathrm{a},\ldots,50\mathrm{a}$ is 50, then $|\mathrm{a}|$ equals (1) 3 (2) 4 (3) 5 (4) 2
Let $A$ and $B$ be two symmetric matrices of order 3. This question has Statement-1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements. Statement-1: $A(BA)$ and $(AB)A$ are symmetric matrices. Statement-2: $AB$ is symmetric matrix if matrix multiplication of $A$ and $B$ is commutative. (1) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1. (2) Statement-1 is true, Statement-2 is false. (3) Statement-1 is false, Statement-2 is true. (4) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
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) 0 (4) 3