\begin{array}{l} The\,series\,\sum\limits_{}^{} {a_n b_n } \,\,\,converges\,\,if\,\,a_{n + 1} \le \,a_{n\,} \,for\,\,n\, \ge \,k,\,\,\mathop {\lim }\limits_{x \to \infty } \,\,a_n = - - - - - ,\,\,and\, \\\ |b_k + \,b_{k + 1} + ... + b_n |\,\, \le \,\,M,\,\,for\,\,some\,\,constant\,M. \\\ \end{array}

• A) 1
• B) - 1
• C) None\,of\,these
• D) 0\,

The derivative of the function {x^n} is

• A) {x^{n - 1}}.
• B) n{x^{n - 1}}.
• C) \frac{n}{{{x^{n - 1}}}}.
• D) n{x^n}.

The\,\,\,function\,\,g\left( x \right) = x^2 \,\,is\, - - - - - \,on\,\,[0,\infty )\,.\,

• A) inrea\sin g
• B) decrea\sin g\,\,
• C)
• D)

Investigate\,\,the\,\,\lim it\,\,of\,\,\,\mathop {\lim }\limits_{x \to 0^ - } \,\,\left( {\frac{{\left| x \right|}}{x} + x} \right) = - - - - .

• A) 0
• B) none\,\,of\,\,these
• C) 1
• D) - 1

The function g(x) = {x^2} is

• A) undefined on [0,\infty ).
• B) having negative values on [0,\infty ).
• C) increasing on [0,\infty ).
• D) decreasing on [0,\infty ).

\[ The\,\,\,function\,\,g\left( x \right) = - x^3 \,\,is\,\,decrea\sin g\,\,on\,\, - - - - - .\, \]

• A) \[ \left( { - \infty ,\infty } \right) \]
• B) \[ \left( { - \infty \,,0} \right) \]
• C) \[ [0,\infty )\, \]
• D) \[ none\,\,of\,\,these\, \]

An object in rectilinear motion moves

• A) oscillatory
• B) along straight line
• C) none of these
• D) along curved path

If a particle of mass $2$ units has position vector $\vec{r}=t^{3}\hat{i}+8t\hat{j}-9t\hat{k}$ then find the force acting on the particle.

• A) $12t$
• B) $12t\hat{i}$
• C)
• D)

If a force acts on a particle of constant mass $m$ and $\vec{v}_{1}$, $\vec{v}_{2}$ are velocities at times $t_{1}$ and $t_{2}$, respectively. Then the work done by the force is obtained as

• A) The sum of kinetic energies, i.e., $\frac{1}{2}m\vec{v}_{2}^{2}+\frac{1}{2}m\vec{v}_{1}^{2}$
• B) The change in kinetic energies, i.e., $\frac{1}{2}m\vec{v}_{2}^{2}-\frac{1}{2}m\vec{v}_{1}^{2}$
• C)
• D)

If an object explodes, the center
of mass will

• A) continue its previous motion
• B) come into rest
• C)
• D)

Let 16kgm2 be the momentum of inertia of a system of particles about AB, and 4kg be the total mass of the system then the radius of gyration of the system AB would be _______m.

• A) 2
• B) 8
• C) 4
• D) 1

If a body rotates about some external fixed
point it is called revolution

• A) spin
• B) revolution
• C)
• D)

According to Newton's second law of motion the net force applied on a body is equal to the rate of change of momentum of body.

• A) true
• B) false
• C)
• D)

Which of the following are examples of simple harmonic motion

• A) (c) movement on a slide
• B) (b) object of mass m linked to a spring
• C) (a) movement of the mass attached to a simple pendulum
• D) both (a) and (b) options

A particle moves in a plane in such a way that at any time $t$ its distance from the origin is $r=2t+3t^{2}$ and its angle with positive x-axis is $\theta=\sqrt{t}$. Find the radial and transverse components of velocity.

• A) $2+6t$ and $(\sqrt{t}+\frac{3}{2}t^{\frac{3}{2}})$
• B) $2+6t$ and $\frac{1}{2\sqrt{t}}$
• C)
• D)

Which of the following objects has more inertia?

• A) an object with mass $40kg$
• B) an object with mass $30kg$
• C)
• D)

A car driving along a curved road is an example
of

• A) oscillatory motion
• B) curvilinear motion
• C) rectilinear motion
• D) none of these options

If the total external torque on the system is zero, then its angular momentum will be ______.

• A) zero
• B) increasing
• C) decresing
• D) constant

The product of force and displacement is called

• A) energy
• B) work done
• C)
• D)

Forces that can be expressed in the term of a potential
energy function are called non conservative forces.

• A) true
• B) false
• C)
• D)