The internal energy \((U),\) pressure \((P)\) and volume \(( V )\) of an ideal gas are related as \(U =\) \(3 P V+4\). The gas is :-

For a thin symmetric prism made of glass (refractive index \(1.5\)), the ratio of incident angle and minimum deviation will be _______.

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) :


Three identical spheres of same mass undergo one dimensional motion as shown in figure with initial velocities \(v_{\mathrm{A}}=5 \mathrm{~m} / \mathrm{s}, v_{\mathrm{B}}=2 \mathrm{~m} / \mathrm{s}, v_{\mathrm{C}}=4 \mathrm{~m} / \mathrm{s}\). If we wait sufficiently long for elastic collision to happen, then \(v_{\mathrm{A}}=4 \mathrm{~m} / \mathrm{s}, v_{\mathrm{B}}=2 \mathrm{~m} / \mathrm{s}\), \(v_{\mathrm{C}}=5 \mathrm{~m} / \mathrm{s}\) will be the final velocities.

Reason (R): In an elastic collision between identical masses, two objects exchange their velocities.
In the light of the above statements, choose the correct answer from the options given below :

Given below are two statements: Statement \(I:\) For a planet, if the ratio of mass of the planet to its radius increases, the escape velocity from the planet also increases. Statement \(II:\) Escape velocity is independent of the radius of the planet. In the light of above statements, choose the most appropriate answer from the options given below on :

Suppose there is a uniform circular disc of mass M kg and radius r m shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis A of the disc is given by \(\frac{ x }{256} Mr ^2\). The value of x is ___________.

Two wires \(A\) and \(B\) are made up of the same material and have the same mass. Wire A has radius of \(2.0 \mathrm{~mm}\) and wire \(B\) has radius of \(4.0 \mathrm{~mm}\). The resistance of wire B is \(2 \Omega\). The resistance of wire \(\mathrm{A}\) is _______ \(\Omega\).

The \(P-V\) diagram of a diatomic ideal gas system going under cyclic process as shown in figure. The work done during an adiabatic process \(CD\) is (use \(\gamma=1.4\)) (in \(J\))

If \(a, b, c\) are sides of a scalene triangle, then the value of \(\left| \begin{array}{*{20}{c}}
a&b&c\\
b&c&a\\
c&a&b
\end{array} \right|\) is

If the area of the region bounded by \(16x^2 - 9y^2 = 144\) and \(8x - 3y = 24\) is A, then \(3(A + 6 \log_e(3))\) is equal to _______.

Let \(\left\{a_{n}\right\}_{n-1}^{\infty}\) be a sequence such that \(a_{1}=1, a_{2}=1\) and \(a_{n+2}=2 a_{n+1}+a_{n}\) for all \(n \geq 1 .\) Then tha value of \(47 \sum_{n=1}^{\infty} \frac{a_{n}}{2^{3 n}}\) is equal to \(.....\)

Two polarisers \(P_1\) and \(P_2\) are placed in such a way that the intensity of the transmitted light will be zero. A third polariser \(P_3\) is inserted in between \(P_1\) and \(\mathrm{P}_2\), at the particular angle between \(\mathrm{P}_2\) and \(\mathrm{P}_3\). The transmitted intensity of the light passing the through all three polarisers is maximum. The angle between the polarisers \(\mathrm{P}_2\) and \(\mathrm{P}_3\) is :

Let \( \vec{a}=2\hat{i}-5\hat{j}+5\hat{k} \) and \( \vec{b}=\hat{i}-\hat{j}+3\hat{k}. \) If \( \vec{c} \) is a vector such that
\( 2(\vec{a}\times\vec{c})+3(\vec{b}\times\vec{c})=\vec{0} \) and \( (\vec{a}-\vec{b})\cdot\vec{c}=-97, \) then \( |\vec{c}\times \hat{k}|^{2} \) is equal to

Let \(f ( x )=2 x ^{ n }+\lambda, \lambda \in R , n \in N\), and \(f (4)=133\), \(f(5)=255\). Then the sum of all the positive integer divisors of \(( f (3)- f (2))\) is