The maximum height reached by a projectile is \(64 \mathrm{~m}\). If the initial velocity is halved, the new maximum height of the projectile is _______ \(\mathrm{m}\).

An excited \(He^+\) ion emits two photons in succession, with wavelengths \(108.5\, nm\) and \(30.4\, nm\), in making a transition to ground state. The quantum number \(n\), corresponding to its initial excited state is \(n=........\) (for photon of wavelength \(\lambda \), energy \(E = \frac{{1240\,eV}}{{\lambda \,(in\,nm)}}\))

To find the focal length of a convex mirror, a student records the following data

Object Pin	Convex Lens	Convex Mirror	Image Pin
\(22.2\,cm\)	\(32.2\,cm\)	\(45.8\,cm\)	\(71.2\,cm\)

The focal length of the convex lens is \(f_1\)  and that of mirror is \(f_2\). Then taking index correction to be negligibly small, \(f_1\)  and  \(f_2\) are close to

For a given transistor amplifier circuit in \(CE\) configuration \(V_{C C}=1 V , R_c=1 k \Omega, R_b=100 k \Omega\) and \(\beta=100\). Value of base current \(I_b\) is

A uniform thin rod \(AB\) of length \(L\) has linear mass density \(\mu \left( x \right) = a + \frac{{bx}}{L}\) , where \(x\) is measured from \(A\). If the \(CM\) of the rod lies at a distance of \(\left( {\frac{7}{12}} \right)L\) from \(A\), then \(a\) and \(b\) are related as

A thermodynamic system is taken through the cyclic process ABC as shown in the figure (P-V graph : A(2, 100), B(5, 300), C(5, 100). The total work done by the system during the cycle ABC is ___________ J.

In meter bridge experiment for measuring unknown resistance ' \(S\) ', the null point is obtained at a distance \(30 cm\) from the left side as shown at point \(D\). If \(R\) is \(5.6 k \Omega\), then the value of unknown resistance ' \(S\) ' will be \(\Omega .\)

Assume there are two identical simple pendulum Clocks\(-1\) is placed on the earth and Clock\(-2\) is placed on a space station located at a height \(h\) above the earth surface. Clock\(-1\) and Clock\(-2\) operate at time periods \(4\,s\) and \(6\,s\) respectively. Then the value of \(h\) is \(....km\) (consider radius of earth \(R _{ E }=6400\,km\) and \(g\) on earth \(10\,m / s ^{2}\) )

A series \(LCR\) circuit driven by \(300\, {V}\) at a frequency of \(50 \,{Hz}\) contains a resistance \({R}=3 \,{k} \Omega\), an inductor of inductive reactance \(X_{L}=250 \,\pi \Omega\) and an unknown capacitor. The value of capacitance to maximize the average power should be : (Take \(\pi^{2}=10\) ) (in \(\mu {F}\))

A circular ring and a solid sphere having same radius roll down on an inclined plane from rest without slipping. The ratio of their velocities when reached at the bottom of the plane is \(\sqrt{\frac{x}{5}}\) where \(\mathrm{x}=\) _____

A square frame of side \(10\, cm\) and a long straight wire carrying current \(1\, A\) are in the plate of the paper. Starting from close to the wire, the frame moves towards the right with a constant speed of \(10\, ms^{-1}\) (see figure). The \(e.m. f\) induced at the time the left arm of the frame is at \(x\, = 10\, cm\) from the wire is .....\(\mu V\)

A cube of side \('a'\) has point charges \(+Q\) located at each of its vertices except at the origin where the charge is \(- Q\). The electric field at the centre of cube is

Three infinitely long wires with linear charge density \(\lambda\) are placed along the \(x-a x i s, y\)-axis and \(z-\) axis respectively. Which of the following denotes an equipotential surface?