A goods train accelerating uniformly on a straight railway track, approaches an electric pole standing on the side of track. Its engine passes the pole with velocity $u$ and the guard's room passes with velocity $v$. The middle wagon of the train passes the pole with a velocity. (1) $\frac { u + v } { 2 }$ (2) $\frac { 1 } { 2 } \sqrt { u ^ { 2 } + v ^ { 2 } }$ (3) $\sqrt { u v }$ (4) $\sqrt { \left( \frac { u ^ { 2 } + v ^ { 2 } } { 2 } \right) }$
Sand is being dropped on a conveyer belt at the rate of 2 kg per second. The force necessary to keep the belt moving with a constant speed of $3 \mathrm {~ms} ^ { - 1 }$ will be (1) 12 N (2) 6 N (3) zero (4) 18 N
A block of weight $W$ rests on a horizontal floor with coefficient of static friction $\mu$. It is desired to make the block move by applying minimum amount of force. The angle $\theta$ from the horizontal at which the force should be applied and magnitude of the force $F$ are respectively. (1) $\theta = \tan ^ { - 1 } ( \mu ) , F = \frac { \mu W } { \sqrt { 1 + \mu ^ { 2 } } }$ (2) $\theta = \tan ^ { - 1 } \left( \frac { 1 } { \mu } \right) , F = \frac { \mu W } { \sqrt { 1 + \mu ^ { 2 } } }$ (3) $\theta = 0 , F = \mu W$ (4) $\theta = \tan ^ { - 1 } \left( \frac { \mu } { 1 + \mu } \right) , F = \frac { \mu W } { 1 + \mu }$
A moving particle of mass $m$, makes a head on elastic collision with another particle of mass $2m$, which is initially at rest. The percentage loss in energy of the colliding particle on collision, is close to (1) $33 \%$ (2) $67 \%$ (3) $90 \%$ (4) $10 \%$
Let $p , q , r \in R$ and $r > p > 0$. If the quadratic equation $p x ^ { 2 } + q x + r = 0$ has two complex roots $\alpha$ and $\beta$, then $| \alpha | + | \beta |$ is (1) equal to 1 (2) less than 2 but not equal to 1 (3) greater than 2 (4) equal to 2
Consider a quadratic equation $a x ^ { 2 } + b x + c = 0$, where $2 a + 3 b + 6 c = 0$ and let $g ( x ) = a \frac { x ^ { 3 } } { 3 } + b \frac { x ^ { 2 } } { 2 } + c x$. Statement 1: The quadratic equation has at least one root in the interval $( 0,1 )$. Statement 2: The Rolle's theorem is applicable to function $g ( x )$ on the interval $[ 0,1 ]$. (1) Statement 1 is false, Statement 2 is true. (2) Statement 1 is true, Statement 2 is false. (3) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1. (4) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
Let $Z$ and $W$ be complex numbers such that $| Z | = | W |$, and $\arg Z$ denotes the principal argument of $Z$. Statement 1: If $\arg Z + \arg W = \pi$, then $Z = - \bar { W }$. Statement 2: $| Z | = | W |$, implies $\arg Z - \arg \bar { W } = \pi$. (1) Statement 1 is true, Statement 2 is false. (2) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1. (3) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1. (4) Statement 1 is false, Statement 2 is true.
The number of arrangements that can be formed from the letters $a , b , c , d , e , f$ taken 3 at a time without repetition and each arrangement containing at least one vowel, is (1) 96 (2) 128 (3) 24 (4) 72
If $n = { } ^ { m } C _ { 2 }$, then the value of ${ } ^ { n } C _ { 2 }$ is given by (1) $3 \left( { } ^ { m + 1 } C _ { 4 } \right)$ (2) ${ } ^ { m - 1 } C _ { 4 }$ (3) ${ } ^ { m + 1 } C _ { 4 }$ (4) $2 \left( { } ^ { m + 2 } C _ { 4 } \right)$
Let $L$ be the line $y = 2 x$, in the two dimensional plane. Statement 1: The image of the point $( 0,1 )$ in $L$ is the point $\left( \frac { 4 } { 5 } , \frac { 3 } { 5 } \right)$. Statement 2: The points $( 0,1 )$ and $\left( \frac { 4 } { 5 } , \frac { 3 } { 5 } \right)$ lie on opposite sides of the line $L$ and are at equal distance from it. (1) Statement 1 is true, Statement 2 is false. (2) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1. (3) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1. (4) Statement 1 is false, Statement 2 is true.
If the line $y = m x + 1$ meets the circle $x ^ { 2 } + y ^ { 2 } + 3 x = 0$ in two points equidistant from and on opposite sides of $x$-axis, then (1) $3 m + 2 = 0$ (2) $3 m - 2 = 0$ (3) $2 m + 3 = 0$ (4) $2 m - 3 = 0$