The radius of an atomic nucleus of mass number $64$ is $4.8$ fermi. Then the mass number of another atomic nucleus of radius $6$ fermi is

The dispersive powers of the materials of two lenses forming an achromatic combination are in the ratio of \(4: 3\). Effective focal length of the two lenses is \(+60 \mathrm{~cm}\) then the focal lengths of the lenses should be

A block of mass \(8 \mathrm{~kg}\) is suspended by a rope of length \(3 \mathrm{~m}\) from the ceiling. A force of \(40 \mathrm{~N}\) is applied horizontally to the block. Then the angle that the rope makes with the vertical in equilibrium is
(acceleration due to gravity \(=10 \mathrm{~m} \mathrm{~s}^{-2}\), neglect the mass of the rope)

A $500 \mathrm{kg}$ car takes a round turn of radius $50 \mathrm{m}$ with a velocity of $36\mathrm{kmph}$. The centripetal force acting on the car is

An electron having kinetic energy of \(100 \mathrm{eV}\) circulates in a path of radius \(10 \mathrm{~cm}\) in a magnetic field. The magnitude of magnetic field \(|\mathbf{B}|\) is approximately [Mass of electron \(=0.5 \mathrm{MeV} \mathrm{c}^{-2}\), where \(\mathrm{c}\) is the velocity of light].

A body is projected at an angle of \(60^{\circ}\) with the horizontal. If the initial kinetic energy of the body is X, then its kinetic energy at the highest point is

A force $\overrightarrow{F_1}$ of magnitude $4 \mathrm{N}$ acts on an object of mass $1 \mathrm{kg}$ at origin in a direction $30°$ above the positive $x$- axis. A second force $\overrightarrow{F_2}$ of magnitude $4 \mathrm{N}$ acts on the same object in the direction of the positive $y$- axis. The magnitude of the acceleration of the object is nearly

Two persons \(A\) and \(B\) throw a pair of dice alternately until one of them gets the sum of the numbers appeared on the dice as 4 and the person who gets this result first is declared as the winner. If A starts the game: then the probability that \(B\) wins the game is

Match the item given in Column-I (method used for determining colligative property) and Column-II (for the corresponding colligative property) and find the correct order.
\(\begin{array}{|c|c|c|}
\hline & \text { Column I } & \text { Column II } \\
\hline \text { A. } & \text { Beckmann method } & \text { 1. Osmotic pressure } \\
\hline \text { B. } & \begin{array}{l}
\text { Ostwald-Walker method }
\end{array} & \text { 2. Elevation in B.P } \\
\hline \text { C. } & \begin{array}{l}
\text { Berkeley-Hartley method }
\end{array} & \text { 3. Depression in F.P. } \\
\hline \text { D. } & \text { Landsberger method } & \begin{array}{l}
\text { 4. Relative lowering of vapour pressure }
\end{array} \\
\hline
\end{array}\)

If a function \(f\) is defined by :
\(\begin{aligned}
f(x) & =0, \text { when } x=1, \\
& =x^3-1, \text { when } 1 < x < \infty,
\end{aligned}\)
\(=x-1\), when \(-\infty < x < 1\), then at \(x=1, f\) is

Assertion (A) : For an endothermic dissolution process, increase in temperature increases the solubility in a nearly saturated solution.
Reason (R) : In saturated solution dynamic equilibrium exists between dissolved solute and undissolved solute.

\(\int \frac{3 x^9+7 x^8}{\left(x^2+2 x+5 x^8\right)^2} d x=\)

If \(\hat{i}-2 \hat{j}+3 \hat{k}, 2 \hat{i}+3 \hat{j}+\hat{k},-3 \hat{i}-\hat{j}-2 \hat{k}\) are the position vectors of three points A, B, C respectively, then \(\mathrm{A}, \mathrm{B}, \mathrm{C}\)