The recoil speed of a hydrogen atom after it emits a photon in going from \(n =5\) state to \(n =1\) state will be ..... \(m/s\)

A ball is projected from the ground with a speed \(15 \,ms ^{-1}\) at an angle \(\theta\) with horizontal so that its range and maximum height are equal, then \(tan\,\theta\) will be equal to 

A Carnot engine whose heat \(\operatorname{sinks}\) at \(27\,^{\circ} C\), has an efficiency of \(25 \%\). By how many degrees should the temperature of the source be changed to increase the efficiency by \(100 \%\) of the original efficiency \(?\)

Identify the logic operation performed by the given circuit.

A test particle is moving in a circular orbit in the gravitational field produced by a mass density \(\rho \left( r \right) = \frac{K}{{{r^2}}}\). Identify the correct relation between the radius \(R\) of the particle’s orbit and its period \(T\)

A thermodynamic system is taken from an original state \(D\) to an intermediate state \(E\) by the linear process shown in the figure. Its volume is then reduced to the original volume from \(E\) to \(F\) by an isobaric process. The total work done by the gas from \(D\) to \(E\) to \(F\) will be \(......J\)

A student measured the length of a rod and wrote it as \(3.50\;cm\). Which instrument did he use to measure it?

The equation \(x^2-4 x+[x]+3=x[x]\), where \([x]\) denotes the greatest integer function, has:

As shown in the figure, a plane mirror is fixed at a height of \(50\,cm\) from the bottom of tank containing water \(\left(\mu=\frac{4}{3}\right)\). The height of water in the tank is \(8\,cm\). A small bulb is placed at the bottom of the water tank. The distance of image of the bulb formed by mirror from the bottom of the tank is \(......cm\)

A rectangular loop of sides \(12 \mathrm{~cm}\) and \(5 \mathrm{~cm}\), with its sides parallel to the \(x\)-axis and \(y\)-axis respectively moves with a velocity of \(5 \mathrm{~cm} / \mathrm{s}\) in the positive \(\mathrm{x}\) axis direction, in a space containing a variable magnetic field in the positive \(z\) direction. The field has a gradient of \(10^{-3} \mathrm{~T} / \mathrm{cm}\) along the negative \(\mathrm{x}\) direction and it is decreasing with time at the rate of \(10^{-5} \mathrm{~T} / \mathrm{s}\). If the resistance of the loop is \(6 \mathrm{~m} \Omega\), the power dissipated by the loop as heat is _______ \(\times 10^{-9} \mathrm{~W}\).

Let \(y=y(x)\) be a solution curve of the differential equation, \(\left(1-x^2 y^2\right) d x=y d x+x d y\).If the line \(x =1\) intersects the curve \(y = y ( x )\) at \(y =2\) and the line \(x =2\) intersects the curve \(y=y(x)\) at \(y=\alpha\), then a value of \(\alpha\) is

Let \(f(x) = \int \left(\dfrac{16x + 24}{x^2 + 2x - 15}\right) dx\). If \(f(4) = 14 \log_e(3)\) and \(f(7) = \log_e(2^{\alpha} \cdot 3^{\beta})\), \(\alpha, \beta \in \mathbb{N}\), then \(\alpha + \beta\) is equal to:

Let the solution curve of the differential equation \(xdy-ydx=\sqrt{x^{2}+y^{2}}dx, x>0, y(1)=0\), be \(y=y(x)\). Then \(y(3)\) is equal to