A particle of mass $m$ projected with a velocity $u$ making an angle of $30 ^ { \circ }$ with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height $h$ is : (1) $\frac { \sqrt { 3 } } { 16 } \frac { m u ^ { 3 } } { g }$ (2) $\frac { \sqrt { 3 } } { 2 } \frac { m u ^ { 2 } } { g }$ (3) $\frac { m u ^ { 3 } } { \sqrt { 2 } g }$ (4) zero
All surfaces shown in figure are assumed to be frictionless and the pulleys and the string are light. The acceleration of the block of mass 2 kg is: (1) $g$ (2) $\frac { g } { 3 }$ (3) $\frac { g } { 2 }$ (4) $\frac { g } { 4 }$
A particle is placed at the point A of a frictionless track ABC as shown in figure. It is gently pushed towards right. The speed of the particle when it reaches the point $B$ is: (Take $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ ). (1) $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ (2) $\sqrt { 10 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ (3) $2 \sqrt { 10 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ (4) $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
A spherical body of mass 100 g is dropped from a height of 10 m from the ground. After hitting the ground, the body rebounds to a height of 5 m . The impulse of force imparted by the ground to the body is given by: (given $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ ) (1) $4.32 \mathrm {~kg} \mathrm {~m} \mathrm {~s} ^ { - 1 }$ (2) $43.2 \mathrm {~kg} \mathrm {~m} \mathrm {~s} ^ { - 1 }$ (3) $23.9 \mathrm {~kg} \mathrm {~m} \mathrm {~s} ^ { - 1 }$ (4) $2.39 \mathrm {~kg} \mathrm {~m} \mathrm {~s} ^ { - 1 }$
The displacement and the increase in the velocity of a moving particle in the time interval of $t$ to $( t + 1 )$ s are 125 m and $50 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, respectively. The distance travelled by the particle in $( t + 2 ) ^ { \text {th} } \mathrm { s }$ is $\_\_\_\_$ m.
Consider a disc of mass 5 kg , radius 2 m , rotating with angular velocity of $10 \mathrm { rad } \mathrm { s } ^ { - 1 }$ about an axis perpendicular to the plane of rotation. An identical disc is kept gently over the rotating disc along the same axis. The energy dissipated so that both the discs continue to rotate together without slipping is $\_\_\_\_$ J.
If $z = x + i y , x y \neq 0$, satisfies the equation $z ^ { 2 } + i \bar { z } = 0$, then $\left| z ^ { 2 } \right|$ is equal to: (1) 9 (2) 1 (3) 4 (4) $\frac { 1 } { 4 }$
Let $S _ { a }$ denote the sum of first $n$ terms an arithmetic progression. If $S _ { 20 } = 790$ and $S _ { 10 } = 145$, then $S _ { 15 } - S _ { 5 }$ is (1) 395 (2) 390 (3) 405 (4) 410
A line passing through the point $A ( 9,0 )$ makes an angle of $30 ^ { \circ }$ with the positive direction of $x$-axis. If this line is rotated about $A$ through an angle of $15 ^ { \circ }$ in the clockwise direction, then its equation in the new position is (1) $\frac { y } { \sqrt { 3 } - 2 } + x = 9$ (2) $\frac { x } { \sqrt { 3 } - 2 } + y = 9$ (3) $\frac { x } { \sqrt { 3 } + 2 } + y = 9$ (4) $\frac { y } { \sqrt { 3 } + 2 } + x = 9$
The maximum area of a triangle whose one vertex is at $( 0,0 )$ and the other two vertices lie on the curve $y = - 2 x ^ { 2 } + 54$ at points $( x , y )$ and $( - x , y )$ where $\mathrm { y } > 0$ is : (1) 88 (2) 122 (3) 92 (4) 108
If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is : (1) $\frac { \sqrt { 5 } } { 3 }$ (2) $\frac { \sqrt { 3 } } { 2 }$ (3) $\frac { 1 } { \sqrt { 3 } }$ (4) $\frac { 2 } { \sqrt { 5 } }$
Consider the system of linear equation $x + y + z = 4 \mu , x + 2 y + 2 \lambda z = 10 \mu , x + 3 y + 4 \lambda ^ { 2 } z = \mu ^ { 2 } + 15$, where $\lambda , \mu \in \mathrm { R }$. Which one of the following statements is NOT correct? (1) The system has unique solution if $\lambda \neq \frac { 1 } { 2 }$ and $\mu \neq 1$ (2) The system is inconsistent if $\lambda = \frac { 1 } { 2 }$ and $\mu \neq 1, 15$ (3) The system has infinite number of solutions if $\lambda = \frac { 1 } { 2 }$ and $\mu = 15$ (4) The system is consistent if $\lambda \neq \frac { 1 } { 2 }$
Let $g : R \rightarrow R$ be a non constant twice differentiable such that $g ^ { \prime } \left( \frac { 1 } { 2 } \right) = g ^ { \prime } \left( \frac { 3 } { 2 } \right)$. If a real valued function $f$ is defined as $f ( x ) = \frac { 1 } { 2 } [ g ( x ) + g ( 2 - x ) ]$, then (1) $f ^ { \prime \prime } ( x ) = 0$ for atleast two $x$ in $( 0,2 )$ (2) $f ^ { \prime \prime } ( x ) = 0$ for exactly one $x$ in $( 0,1 )$ (3) $f ^ { \prime \prime } ( x ) = 0$ for no $x$ in $( 0,1 )$ (4) $f ^ { \prime } \left( \frac { 3 } { 2 } \right) + f ^ { \prime } \left( \frac { 1 } { 2 } \right) = 1$