A thin uniform bar of length \(L\) and mass \(8\,m\) lies on a smooth horizontal table. Two point masses \(m\) and \(2\,m\) moving in the same horizontal plane from opposite sides of the bar with speeds \(2v\) and \(v\) respectively. The masses stick to the bar after collision at a distance \(\frac{L}{3}\) and \(\frac{L}{6}\) respectively from the centre of the bar. If the bar starts rotating about its center of mass as a result of collision, the angular speed of the bar will be

In the reported figure, a capacitor is formed by placing a compound dielectric between the plates of parallel plate capacitor. The expression for the capacity of the said capacitor will be (Given area of plate \(=A\) )

A \(0.5 \,kg\) block moving at a speed of \(12 \,ms ^{-1}\) compresses a spring through a distance \(30\, cm\) when its speed is halved. The spring constant of the spring will be \(Nm ^{-1}\).

Three containers \(\mathrm{C}_{1}, \mathrm{C}_{2}\) and \(\mathrm{C}_{3}\) have water at different temperatures. The table below shows the final temperature \(T\) when different amounts of water (given in litres) are taken from each containers and mixed (assume no loss of heat during the process) \(\begin{array}{|c|c|c|c|}\hline \mathrm{C_{1 }} & {\mathrm{C}_{2}} & {\mathrm{C}_{3}} & {\mathrm{T}} \\ \hline {1 l} & {2 l} & {-} & {60^{\circ} \mathrm{C}} \\ \hline {-} & {1 l} & {2 l} & {30^{\circ} \mathrm{C}} \\ \hline  {2 l} & {-} & {1 l} & {60^{\circ} \mathrm{C}} \\ \hline  {1 l} & {1 l} & {1 l} & {\theta} \\ \hline\end{array}\) The value of \(\theta\) (in \(^{\circ} \mathrm{C}\) to the nearest integer) is

In a Young's double slit experiment, the slits are separated by 0.2 mm. If the slits separation is increased to 0.4 mm , the percentage change of the fringe width is _______.

One mole of a diatomic ideal gas undergoes a cyclic process \(ABC\) as shown in figure. The process \(BC\) is adiabatic. The temperatures at \(A, B\) and \(C\) are \(400\ K, 800\ K \) and \(600\ K\) respectively. Choose the correct statement

A particle of mass \(m\) moves on a straight line with its velocity increasing with distance according to the equation \(\mathrm{v}=\alpha \sqrt{\mathrm{x}}\), where \(\alpha\) is a constant. The total work done by all the forces applied on the particle during its displacement from \(\mathrm{x}=0\) to \(\mathrm{x}=\mathrm{d}\), will be _______.

Suppose you have taken a dilute solution of oleic acid in such a way that its concentration becomes \(0.01 \,cm ^{3}\) of oleic acid per \(cm ^{3}\) of the solution. Then you make a thin film of this solution (monomolecular thickness) of area \(4\, cm ^{2}\) by considering \(100\) spherical drops of radius \(\left(\frac{3}{40 \pi}\right)^{\frac{1}{3}} \times 10^{-3}\, cm .\) Then the thickness of oleic acid layer will be \(x \times 10^{-14} \,m\). Where \(x\) is ...... .

The mean and standard deviation of \(40\) observations are \(30\) and \(5\) respectively. It was noticed that two of these observations \(12\) and \(10\) were wrongly recorded. If \(\sigma\) is the standard deviation of the data after omitting the two wrong observations from the data, then \(38 \sigma^{2}\) is equal to\(.........\)

Two circles in the first quadrant of radii \(r_1\) and \(r_2\) touch the coordinate axes. Each of them cuts off an intercept of \(2\) units with the line \(x+y=2\). Then \(r_1^2+r_2^2-r_1 r_2\) is equal to \(...........\)

\(\mathop {\lim }\limits_{x \to \pi /4} \frac{{{{\cot }^3}\,x - \tan \,x}}{{\cos \left( {x + \pi /4} \right)}}\) is

The radius of circle, having minimum area, which touches the cruve  \(y = 4 - {x^2}\) and the lines \(y = \left| x \right|\) is :

All the pairs \((x, y)\) that satisfy the inequality \({2^{\sqrt {{{\sin }^2}{\kern 1pt} x - 2\sin {\kern 1pt} x + 5} }}.\frac{1}{{{4^{{{\sin }^2}\,y}}}} \leq 1\) also Satisfy the equation