A monoatomic gas \(\left(\gamma=\frac{5}{3}\right)\) at a pressure of \(4 \mathrm{~atm}\) is compressed adiabatically so that its temperature rises from \(27^{\circ} \mathrm{C}\) to \(327^{\circ} \mathrm{C}\). The pressure of the gas in its final state is

The density (in \(\mathrm{kg} \mathrm{m}^{-3}\) ) of nuclear matter is of the order of

A transverse wave is described by the equation \(y=y_0 \sin 2 \pi\left(v t-\frac{x}{\lambda}\right)\). The maximum particle velocity is equal to four times the wave velocity if \(\lambda=\)

If the average translational kinetic energy of a molecule in a gas is equal to the kinetic energy of an electron accelerating from rest through \(10 \mathrm{~V}\), then the temperature of the gas molecule is
\[
\left(\text { Boltzmann constant }=1.38 \times 10^{-23} \mathrm{JK}^{-1}\right)
\]

The potential energy of a particle of mass 10 g as a function of displacement \(x\) is \(\left(50 x^2+100\right) \mathrm{J}\). The frequency of oscillation is

A wire of length 0.5 m and area of cross-section \(4 \times 10^{-6} \mathrm{~m}^2\) at a temperature of \(100^{\circ} \mathrm{C}\) is suspended vertically by fixing its upper end to the ceiling. The wire is then cooled to \(0^{\circ} \mathrm{C}\), but is prevented from contracting, by attaching a mass at the lower end. If the mass of the wire is negligible, then the value of the mass attached to the wire is
[Young's modulus of material of the wire \(=10^{11} \mathrm{~N} \mathrm{~m}^{-2}\); coefficient of linear expansion of the material of the wire \(=10^{-5} \mathrm{~K}^{-1}\) and acceleration due to gravity \(\left.=10 \mathrm{~m} \mathrm{~s}^{-2}\right]\)

A motor is used to deliver water at a certain rate through a given horizontal pipe. To deliver \(n\)-times the water through the same pipe in the same time the power of the motor must be increased as follows :

Maximum number of electrons theoretically possible for an orbit with principal quantum number \(n=6\) in an atom is

If $l, m\left(l<m\right)$ are roots of $a{x}^2+bx+c=0$, then $\underset{x\to a}{\lim }\frac{\left|a{x}^2+bx+c\right|}{a{x}^2+bx+c}=$

Suppose that the circle \(x^2+y^2+2 g x+2 f y+c=0\) has its centre on \(2 x+3 y-7=0\) and cuts the circles \(x^2+y^2-4 x-6 y+11=0\) and \(x^2+y^2-10 x-4 y+21=0\) orthogonally. Then \(5 g-10 f+3 c=\)

Two trains, which are moving along different tracks in opposite directions are put on the same track by mistake. On noticing the mistake, when the trains are \(300 \mathrm{~m}\) apart the drivers start slowing down the trains. The graphs given below show decrease in their velocities as function of time. The separation between the trains when both have stopped is

The sum of the distinct values of \(x\) for which the matrix \(A=\left[\begin{array}{lll}1 & 1 & x \\ 1 & x & 1 \\ x & 1 & 1\end{array}\right]\) has no inverse, is

In which of the following reactions of \(\mathrm{H}_2 \mathrm{O}_2\) acts as an oxidising agent (either in acidic, alkaline or neutral medium)?
(i) \(2 \mathrm{Fe}^{2+}+\mathrm{H}_2 \mathrm{O}_2 \rightarrow\)
(ii) \(2 \mathrm{MnO}_4^{-}+6 \mathrm{H}^{+}+5 \mathrm{H}_2 \mathrm{O}_2 \rightarrow\)
(iii) \(\mathrm{I}_2+\mathrm{H}_2 \mathrm{O}_2+2 \mathrm{OH}^{-} \rightarrow\)
(iv) \(\mathrm{Mn}^{2+}+\mathrm{H}_2 \mathrm{O}_2 \rightarrow\)