For an ideal gas the instantaneous change in pressure \('{p} '\) with volume \('v'\) is given by the equation \(\frac{{dp}}{{dv}}=-{ap} .\) If \({p}={p}_{0}\) at \({v}=0\) is the given boundary condition, then the maximum temperature one mole of gas can attain is : (Here \({R}\) is the gas constant)

The speed of a wave produced in water is given by \(v=\lambda^a g^b \rho^c\). Where \(\lambda\), g and \(\rho\) are wavelength of wave, acceleration due to gravity and density of water respectively. The values of \(a , b\) and \(c\) respectively, are

For a transverse wave travelling along a straight line, the distance between two peaks (crests) is \(5 \,m ,\) while the distance between one crest and one trough is \(1.5 \,m\) The possible wavelengths (in \(m\) ) of the waves are

When two soap bubbles of radii \(a\) and \(b ( b > a )\) coalesce, the radius of curvature of common surface is

A wire of length \(L\) and radius \(r\) is clamped rigidly at one end. When the other end of the wire is pulled by a force \(f\), its length increases by \(l\). Another wire of same material of length \(2 L\) and radius \(2 r\) is pulled by a force \(2 f\). Then the increase in its length will be

A body of mass 1 kg is suspended with the help of two strings making angles as shown in figure. Magnitude of tensions \(T_1\) and \(T_2\), respectively, are (in N) :

A toy-car, blowing its horn, is moving with a steady speed of \(5\, m/s\) , away from a wall. An observer, towards whom the toy car is moving, is able to hear \(5\, beats\) per second. If the velocity of sound in air is \(340\, m/s\), the frequency of the horn of the toy car is close to  ... \(Hz\)

If the area (in sq. units) bounded by the parabola \(y^2 =4\lambda x\) and the line \(y = \lambda x\), \(\lambda  > 0\), is \(\frac{1}{9}\), then \(\lambda \) is equal to

The acceleration due to gravity is found upto an accuracy of \(4 \,\%\) on a planet. The energy supplied to a simple pendulum to known mass ' \({m}\) ' to undertake oscillations of time period \(T\) is being estimated. If time period is measured to an accuracy of \(3\, \%\), the accuracy to which \({E}\) is known as \(..........\,\%\)

Let quadratic curve passing through the point \((-1,0)\) and touching the line \(y=x\) at \((1,1)\) be \(y=\) \(f(x)\). Then the \(x\)-intercept of the normal to the curve at the point \((\alpha, \alpha+1)\) in the first quadrant is \(..........\).

A star has \(100 \%\) helium composition. It starts to convert three \({ }^4 \mathrm{He}\) into one \({ }^{12} \mathrm{C}\) via triple alpha process as \({ }^4 \mathrm{He}+{ }^4 \mathrm{He}+{ }^4 \mathrm{He} \rightarrow{ }^{12} \mathrm{C}+\mathrm{Q}\). The mass of the star is \(2.0 \times 10^{32} \mathrm{~kg}\) and it generates energy at the rate of \(5.808 \times 10^{30} \mathrm{~W}\). The rate of converting these \({ }^4 \mathrm{He}\) to \({ }^{12} \mathrm{C}\) is \(\mathrm{n} \times 10^{42} \mathrm{~s}^{-1}\), where \(\mathrm{n}\) is _______. [Take, mass of \({ }^4 \mathrm{He}=4.0026 \mathrm{u}\), mass of \({ }^{12} \mathrm{C}=12 \mathrm{u}\)]

The function of time representing a simple harmonic motion with a period of \(\frac{\pi}{\omega}\) is :

The integral \(\int_{-1}^{\frac{3}{2}}\left(\left|\pi^2 x \sin (\pi x)\right|\right) d x\) is equal to: