The isotope ${}_5{}^{12}B$ having a mass 12.014 u undergoes $\beta$ - decay to ${}_6{}^{12}C . {}_6{}^{12}C$ has an excited state of the nucleus $\left({}_6{}^{12}{C}^{*}\right)$ at 4.041 MeV above its ground state. If ${}_5{}^{12}B$ decays to ${}_6{}^{12}{C}^{*},$ the maximum kinetic energy of the $\beta$ - particle in units of MeV is ( $1\mu =931.5 Me\frac{V}{c^2},$ where c is the speed of light in vacuum).

A container of fixed volume has a mixture of one mole of hydrogen and one mole of helium in equilibrium at temperature $\mathrm{T}$ . Assuming the gases are ideal, the correct statement(s) is (are)

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A small block of mass \(M\) moves on a frictionless surface of an inclined plane, as shown in figure. The angle of the incline suddenly changes from \(60^{\circ}\) to \(30^{\circ}\) at point \(B\). The block is initially at rest at \(A\). Assume that collisions between the block and the incline are totally inelastic \(\left(g=10 \mathrm{~m} / \mathrm{s}^2\right)\)

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If collision between the block and the incline is completely elastic, then the vertical (upward) component of the velocity of the block at point \(B\), immediately after it strikes the second incline is

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A uniform thin cylindrical disk of mass \(M\) and radius \(R\) is attached to two identical massless springs of spring constant \(k\) which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance \(d\) from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is \(L\). The disk is initially at its equilibrium position with its centre of mass \((C M)\) at a distance Lfrom the wall. The disk rolls without slipping with velocity \(\mathbf{v}_0=v_0 \hat{\mathbf{i}}\) The coefficient of friction is \(\mu\).
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The maximum value of \(v_0\) for which the disk will roll without slipping is

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The key feature of Bohr's theory of spectrum of hydrogen atom is the quantization of angular momentum when an electron is revolving around a proton. We will extend this to a general rotational motion to find quantized rotational energy of a diatomic molecule assuming it to be rigid. The rule to be applied is Bohr's quantization condition.
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A diatomic molecule has moment of inertia I. By Bohr's quantization condition its rotational energy in the \(n\)th level ( \(n=0\) is not allowed) is

During Searle's experiment, zero of the Vernier scale lies between $3.20\times {10}^{-2}\mathrm{ }\mathrm{m}$ and $3.25\times {10}^{-2}\mathrm{ }\mathrm{m}$ of the main scale. The ${20}^{\mathrm{t}\mathrm{h}}$ division of the Vernier scale exactly coincides with one of the main scale divisions. When an additional load of 2 kg is applied to the wire, the zero of the Vernier scale still lies between $3.20\times {10}^{-2\mathrm{ }}\mathrm{m}$ and $3.25\times {10}^{-2}\mathrm{ }\mathrm{m}$ of the main scale but now the ${45}^{\mathrm{t}\mathrm{h}}$ division of Vernier scale coincides with one of the main scale divisions. The length of the thin metallic wire is 2 m and its cross-sectional area is $8\times {10}^{-7}\mathrm{ }{\mathrm{m}}^2$ . The least count of the Vernier scale is $1.0\times \mathrm{ }{10}^{-5}\mathrm{ }\mathrm{m}$ . The maximum percentage error in the Young's modulus of the wire is

Two batteries of different emfs and different internal resistances are connected as shown. The voltage across \(A B\) in volt is

The figure shows the \(p-V\) plot of an ideal gas taken through a cycle \(A B C D A\). The part \(A B C\) is a semi-circle and \(C D A\) is half of an ellipse. Then,

In the following reaction, compound $\mathrm{Q}$ is obtained from compound $\mathrm{P}$ via an ionic intermediate.

What is the degree of unsaturation of $\mathrm{Q}$?

Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with the increase in temperature. If at room temperature, \(100 \mathrm{~W}, 60 \mathrm{~W}\) and \(40 \mathrm{~W}\) bulbs have filament resistances \(R_{100}, R_{60}\) and \(R_{40}\), respectively, the relation between these resistances is

For all \(x>0\), let \(y_1(x), y_2(x)\), and \(y_3(x)\) be the functions satisfying
\(\begin{aligned} & \frac{d y_1}{d x}-(\sin x)^2 y_1=0, y_1(1)=5 \\ & \frac{d y_2}{d x}-(\cos x)^2 y_2=0, y_2(1)=\frac{1}{3} \\ & \frac{d y_3}{d x}-\left(\frac{2-x^3}{x^3}\right) y_3=0, y_3(1)=\frac{3}{5 e}\end{aligned}\)
respectively. Then \(\lim _{x \rightarrow 0^{+}} \frac{y_1(x) y_2(x) y_3(x)+2 x}{e^{3 x} \sin x}\) is equal to _____

Let \(a_0, a_1, \ldots, a_{23}\) be real numbers such that
\(\left(1+\frac{2}{5} x\right)^{23}=\sum_{i=0}^{23} a_i x^i\)
for every real number \(x\). let \(a_r\) be the largest among the numbers \(a_j\) for \(0 \leq j \leq 23\).
The the value of \(r\) is ______

A solid sphere is in pure rolling motion on an inclined surface having inclination \(\theta\).