Images of same size are formed by a convex lens when an object is placed either at 20 cm or 10 cm distance from the lens. The focal length of the lens is

A $20 \mathrm{V}$ AC  is applied to a circuit consisting of a resistor and a coil with negligible resistance. If the voltage across the resistor is $12 \mathrm{V}$, the voltage across the coil is

A body is projected at an angle of \(45^{\circ}\) from a point on the ground at a distance of \(30 \mathrm{~m}\) from the foot of a vertical pole of height \(20 \mathrm{~m}\). The body just crosses the top of the pole and strikes the ground at a distance \(s\) from the foot of the pole on the other side of the pole. Then, \(s\)

The speed of a particle changes from \(\sqrt{5} \mathrm{~ms}^{-1}\) to \(2 \sqrt{5} \mathrm{~ms}^{-1}\) in a time \(t\). If the magnitude of change in its velocity is \(5 \mathrm{~ms}^{-1}\), the angle between the initial and final velocities of the particle is

A block of mass \(90 \mathrm{~kg}\) is suspended by three strings \(A, B\) and \(C\) as shown in figure. Tensions in the strings \(A, B\) and \(C\) respectively are \(\left(g=10 \mathrm{~ms}^{-2}, \sin 37^{\circ}=0.6, \cos 37^{\circ}=0.8\right)\)

A sphere of surface area $4 {\mathrm{m}}^2$ at temperature $400 \mathrm{K}$ and having emissivity $0.5$ is located in an environment of temperature $200 \mathrm{K}$. The net rate of energy exchange of the sphere is (Stefan Boltzmann constant $\sigma =5.67\times {10}^{-8} \mathrm{W} {\mathrm{m}}^{-2} {\mathrm{K}}^4$)

The efficiency of a Carnot's engine is \(100 \%\) only when

The probability distribution of a random variable \(X\) is given by
\begin{array}{l|r|r|r|r|c}
\hlineX=X & 0 & 1 & 2 & 3 & 4 \\
\hlineP(X=x) & 0.4 & 0.3 & 0.1 & 0.1 & 0.1 \\
\hline
\end{array}
The variance of \(X\) is

Two balls of charge \(q_1\) and \(q_2\) initially have a velocity of the same magnitude and direction. After a uniform electric field has been applied during a certain time, the direction of the velocity of the first ball changes by \(60^{\circ}\), and the velocity magnitude is reduced by half. The direction of the velocity of the second ball changes thereby by \(90^{\circ}\).
In what proportion will the velocity of the second ball change? Determine the magnitude of the charge-to-mass ratio for the second ball if it is equal to \(k_1\) for the first ball. The electrostatic interaction between the balls should be neglected.

The value of \(\left|\begin{array}{cc}\log _5 729 & \log _3 5 \\ \log _5 27 & \log _9 25\end{array}\right|\) \(\times\left|\begin{array}{ll}\log _3 5 & \log _{27} 5 \\ \log _5 9 & \log _5 9\end{array}\right|\) is

An ideal gas is taken around ABCA as shown in the \(\mathrm{P}-\mathrm{V}\) diagram. The work done during a cycle is

RMS velocity of one mole of an ideal gas was measured at different temperatures. A graph of \(\left(\mathrm{u}_{\mathrm{rms}}\right)^2\) (on y -axis) and \(\mathrm{T}(\mathrm{K})\) (on x -axis) gave straight line passing through the origin and its slope is \(249 \mathrm{~m}^2 \mathrm{~s}^{-2} \mathrm{~K}^{-1}\). What is the molar mass (in \(\left.\mathrm{kg} \mathrm{mol}^{-1}\right)\) of ideal gas? \(\left(\mathrm{R}=8.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right)\)

A one-rupee coin, a two rupee coin, a five rupee coin and a ten rupee coin are tossed simultaneously. Then, the expected value of the sum of the values of coins that show heads up is