A particle is moving with speed $v = b \sqrt { x }$ along positive $x$ - axis. Calculate the speed of the particle at time $t = \tau$ (assume that the particle is at origin at $t = 0$ ) (1) $b ^ { 2 } \tau$ (2) $\frac { b ^ { 2 } \tau } { \sqrt { 2 } }$ (3) $\frac { b ^ { 2 } \tau } { 2 }$ (4) $\frac { b ^ { 2 } \tau } { 4 }$
Two particles are projected from the same point with the same speed $u$ such that they have the same range $R$, but different maximum heights, $\mathrm { h } _ { 1 }$ and $\mathrm { h } _ { 2 }$. Which of the following is correct? (1) $R ^ { 2 } = h _ { 1 } h _ { 2 }$ (2) $R ^ { 2 } = 4 h _ { 1 } h _ { 2 }$ (3) $R ^ { 2 } = 2 h _ { 1 } h _ { 2 }$ (4) $R ^ { 2 } = 16 h _ { 1 } h _ { 2 }$
A block of mass 5 kg is (i) pushed in case (A) and (ii) pulled in case (B), by a force $\mathrm { F } = 20 \mathrm {~N}$, making an angle of $30 ^ { \circ }$ with the horizontal, as shown in the figures. The coefficient of friction between the block and floor is $\mu = 0.2$. The difference between the accelerations of the block, in case ( B ) and case ( A ) will be: ( $\mathrm { g } = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ ) (1) $3.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ (2) $0 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ (3) $0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ (4) $0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
A smooth wire of length $2 \pi r$ is bent into a circle and kept in a vertical plane. A bead can slide smoothly on the wire. When the circle is rotating with angular speed $\omega$ about the vertical diameter AB, as shown in figure, the bead is at rest with respect to the circular ring at position P as shown. Then the value of $\omega ^ { 2 }$ is equal to: (1) $2 \mathrm {~g} / \mathrm { r }$ (2) $\frac { \sqrt { 3 } \mathrm {~g} } { 2 \mathrm { r } }$ (3) $2 g / ( r \sqrt { 3 } )$ (4) $( \mathrm { g } \sqrt { 3 } ) / \mathrm { r }$
If $\alpha , \beta$ and $\gamma$ are three consecutive terms of a non-constant G.P. Such that the equations $\alpha x ^ { 2 } + 2 \beta x + \gamma = 0$ and $x ^ { 2 } + x - 1 = 0$ have a common root, then $\alpha ( \beta + \gamma )$ is equal to: (1) $\beta \gamma$ (2) $\alpha \beta$ (3) $\alpha \gamma$ (4) 0
Let $z \in C$ with $\operatorname { Im } ( z ) = 10$ and it satisfies $\frac { 2 z - n } { 2 z + n } = 2 i - 1$ for some natural number $n$. Then (1) $n = 20$ and $\operatorname { Re } ( z ) = 10$ (2) $n = 40$ and $\operatorname { Re } ( z ) = 10$ (3) $n = 20$ and $\operatorname { Re } ( z ) = - 10$ (4) $n = 40$ and $\operatorname { Re } ( z ) = - 10$
A group of students comprises of 5 boys and n girls. If the number of ways, in which a team of 3 students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is 1750, then n is equal to (1) 24 (2) 27 (3) 25 (4) 28
If $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ are in A.P. such that $a _ { 1 } + a _ { 7 } + a _ { 16 } = 40$, then the sum of the first 15 terms of this A.P. is