\(200 \mathrm{~cc}\) of an ideal gas \((\gamma=1.5)\) expands adiabatically. If the rms speed of the gas molecules becomes half of the initial value, the final volume of the gas is

When the temperature difference between the source and sink increases, the efficiency of the heat engine

The maximum possible height of a mountain on earth is approximately
(elastic limit of mountain rock \(=30 \times 10^7 \mathrm{Nm}^{-2}\), average density of mountain rock \(=3 \times 10^3 \mathrm{kgm}^{-3}, \mathrm{~g}=10 \mathrm{~ms}^{-2}\) )

An object requires \(500 \mathrm{~N}\) force to be pulled up on a \(30^{\circ}\) frictionless smooth inclined plane at a constant speed. The weight of the object is

A conductor of resistance \(3 \Omega\) is stretched uniformly till its length is doubled. The wire is now bent in the form of an equilateral triangle. The effective resistance between the ends of any side of the triangle in ohms is

Pure silicon at 300 K has equal electron and hole concentration of \(1.5 \times 10^{16} \mathrm{~m}^{-3}\). If the hole concentration increases to \(3 \times 10^{22} \mathrm{~m}^{-3}\), then electron concentration in the silicon is

Truth table for the given circuit is

\(\int \sin ^4 x \cos ^4 x d x=\)

Assertion: Transition metals and their complexes show catalytic activity. Reason: The activation energy of a reaction is lowered by the catalyst.

Which of the following are disproportionation reactions?
\((\mathrm{g}=\sigma, l=(ద, \mathrm{aq}=\text { జe, } \mathrm{s}=\text { ఘ) }\)
(A) \(2 \mathrm{NO}_{2(\mathrm{~g})}+2 \mathrm{NaOH}_{(\mathrm{aq})} \rightarrow \mathrm{NaNO}_{2(\mathrm{aq})}+\mathrm{NaNO}_{3(\mathrm{aq})}+\mathrm{H}_2 \mathrm{O}_{(l)}\)
(B) \(\mathrm{Cl}_{2(\mathrm{~g})}+2 \mathrm{NaOH}_{(\mathrm{aq})} \rightarrow \mathrm{NaClO}_{(\mathrm{aq})}+\mathrm{NaCl}_{(\mathrm{aq})}+\mathrm{H}_2 \mathrm{O}_{(\mathrm{l})}\)
(C) \(3 \mathrm{ClO}^{-} \rightarrow 2 \mathrm{Cl}^{-}+\mathrm{ClO}_3^{-}\)
(D) \(3 \mathrm{Mg}_{(\mathrm{s})}+\mathrm{N}_{2(\mathrm{~g})} \rightarrow \mathrm{Mg}_3 \mathrm{~N}_{2(\mathrm{~s})}\)

Let $\alpha$ and $\beta \left(\alpha <\beta \right)$ are roots of $18x^2-9\pi x+{\pi }^2=0, f\left(x\right)=x^2, g\left(x\right)=\cos x$. Then ${\int }_{\alpha }^{\beta }x\left(gof\left(x\right)\right)dx=$

If the origin is shifted to a point P by the translation of axes to remove the \(y\)-term from the equation \(x^2-y^2+2 y-1\) \(=0\). then the transformed equation of it is

Number of solutions of the trigonometric equation \(2 \tan 2 \theta-\cot 2 \theta+1=0\) lying in the interval \([0, \pi]\) is