Two forces $P$ and $Q$, of magnitude $2F$ and $3F$, respectively, are at an angle $\theta$ with each other. If the force $Q$ is doubled, then their resultant also gets doubled. Then, the angle $\theta$ is: (1) $120 ^ { \circ }$ (2) $60 ^ { \circ }$ (3) $30 ^ { \circ }$ (4) $90 ^ { \circ }$
A particle starts from the origin at time $t = 0$ and moves along the positive $x$-axis. The graph of velocity with respect to time is shown in figure. What is the position of the particle at time $t = 5s$? (1) $10 m$ (2) $9 m$ (3) $6 m$ (4) $3 m$
A particle which is experiencing a force, given by $\vec { F } = 3 \hat { \mathrm { i } } - 12 \hat { \mathrm { j } }$, undergoes a displacement of $\vec { d } = 4 \hat { \mathrm { i } }$. If the particle had a kinetic energy of 3 J at the beginning of the displacement, what is its kinetic energy at the end of the displacement? (1) 9 J. (2) 15 J. (3) 12 J. (4) 10 J.
Two identical spherical balls of mass $M$ and radius $R$ each are stuck on two ends of a rod of length $2R$ and mass $M$ (see figure). The moment of inertia of the system about the axis passing perpendicularly through the centre of the rod is (1) $\frac { 209 } { 15 } M R ^ { 2 }$. (2) $\frac { 152 } { 15 } M R ^ { 2 }$. (3) $\frac { 137 } { 15 } M R ^ { 2 }$. (4) $\frac { 17 } { 5 } M R ^ { 2 }$.
A rigid massless rod of length $3l$ has two masses attached at each end as shown in the figure. The rod is pivoted at point $P$ on the horizontal axis. When released from the initial horizontal position, its instantaneous angular acceleration will be (1) $\frac { g } { 2l }$ (2) $\frac { 7g } { 3l }$ (3) $\frac { g } { 3l }$ (4) $\frac { g } { 13l }$
A particle executes simple harmonic motion with an amplitude of $5 cm$. When the particle is at $4 cm$ from the mean position, the magnitude of its velocity in SI units is equal to that of its acceleration. Then, its periodic time in seconds is: (1) $\frac { 8 \pi } { 3 }$ (2) $\frac { 3 } { 8 } \pi$ (3) $\frac { 4 \pi } { 3 }$ (4) $\frac { 7 } { 3 } \pi$
A cylindrical plastic bottle of negligible mass is filled with 310 ml of water and left floating in a pond with still water. If pressed downward slightly and released, it starts performing simple harmonic motion at angular frequency $\omega$. If the radius of the bottle is 2.5 cm then $\omega$ is close to: (density of water $= 10 ^ { 3 } \mathrm {~kg} / \mathrm { m } ^ { 3 }$) (1) $5.00 \mathrm { rad } \mathrm { sec } ^ { - 1 }$ (2) $2.50 \mathrm { rad } \mathrm { sec } ^ { - 1 }$ (3) $7.9 \mathrm { rad } \mathrm { sec } ^ { - 1 }$ (4) $3.75 \mathrm { rad } \mathrm { sec } ^ { - 1 }$
The value of $\lambda$ such that sum of the squares of the roots of the quadratic equation, $x ^ { 2 } + ( 3 - \lambda ) x + 2 = \lambda$ has the least value is: (1) 2 (2) $\frac { 4 } { 9 }$ (3) $\frac { 15 } { 8 }$ (4) 1
The positive value of $\lambda$ for which the co-efficient of $x ^ { 2 }$ in the expansion $x ^ { 2 } \left( \sqrt { x } + \frac { \lambda } { x ^ { 2 } } \right) ^ { 10 }$ is 720, is (1) $\sqrt { 5 }$ (2) 3 (3) 4 (4) $2 \sqrt { 2 }$