A boy thrown a ball into air at \(45^{\circ}\) from the horizontal to land it on a roof of a building of height H . If the ball attains maximum height in 2 s and lands on the building in 3 s after launch, then value of H is _________ m .
\(\left( g =10 m / s ^2\right)\)

A particle \(P\) is formed due to a completely inelastic collision of particles \(x\) and \(y\) having de -Broglie wavelengths \({\lambda _x}\) and \({\lambda _y}\) respectively. If \(x\) and \(y\) were moving in opposite directions, then the de -Broglie wavelength of \(P\) is

A ring is hung on a nail. It can oscillate, without slipping or sliding \((i)\) in its plane with a time period \(T_{1}\) and, \((ii)\) back and forth in a direction perpendicular to its plane, with a period \(T _{2}\). the ratio \(\frac{ T _{1}}{ T _{2}}\) will be 

A travelling harmonic wave is represented by the equation \(y(x, t) = 10^{-3}\,sin\,(50t + 2x)\), where \(x\) and \(y\) are in meter and \(t\) is in seconds. Which of the following is a correct statement about the wave?

Equivalent resistance of the following network is _______ \(\Omega\).

In photoelectric effect, the stopping potential \(\left(\mathrm{V}_0\right) \mathrm{v} / \mathrm{s}\) frequency \((\nu)\) curve is plotted.
( h is the Planck's constant and \(\phi_0\) is work function of metal )
(A) \(\mathrm{V}_0 \mathrm{v} / \mathrm{s} \nu\) is linear.
(B) The slope of \(\mathrm{V}_0 \mathrm{v} / \mathrm{s} \nu\) curve \(=\frac{\phi_0}{\mathrm{~h}}\)
(C) h constant is related to the slope of \(\mathrm{V}_0 \mathrm{v} / \mathrm{s} \nu\) line.
(D) The value of electric charge of electron is not required to determine \(h\) using the \(V_0 \mathrm{v} / \mathrm{s} \nu\) curve.
(E) The work function can be estimated without knowing the value of \(h\).
Choose the correct answer from the options given below :

The magnetic intensity at the centre of a long current carrying solenoid is found to be \(1.6 \times 10^3 Am ^{-1}\). If the number of turns is 8 per cm, then the current flowing through the solenoid is \(................\,A\)

In a collinear collision, a particle with an initial speed \(v_0\) strikes a stationary particle of the same mass. If the final total kinetic energy is \(50\%\) greater than the original kinetic energy, the magnitude of the relative velocity between the two particles, after collision, is:

Match List\(-I\) with List\(-II.\)

List\(-I\)	List\(-II\)
\((a)\) Torque	\((i)\) \({MLT}^{-1}\)
\((b)\) Impulse	\((ii)\) \({MT}^{-2}\)
\((c)\) Tension	\((iii)\) \({ML}^{2} {T}^{-2}\)
\((d)\) Surface Tension	\((iv)\) \({MI} {T}^{-2}\)

Choose the most appropriate answer from the option given below :

The figure shows two solid discs with radius \({R}\) and \({r}\) respectively. If mass per unit area is same for both, what is the ratio of \({MI}\) of bigger disc around axis \({AB}\) (Which is \(\perp\) to the plane of the disc and passing through its centre) of MI of smaller disc around one of its diameters lying on its plane? Given \('M'\) is the mass of the larger disc. (\(MI\) stands for moment of inertia)

A \(10\, \Omega, 20 \; mH\) coil carrying constant current is connected to a battery of \(20 \; V\) through a switch is opened current becomes zero in \(100 \; \mu s\). The average emf induced in the coil is \(\dots \; V\).

If \(2 \tan ^2 \theta-5 \sec \theta=1\) has exactly \(7\) solutions in the interval \(\left[0, \frac{n \pi}{2}\right]\), for the least value of \(n \in N\) then \(\sum_{\mathrm{k}=1}^{\mathrm{n}} \frac{\mathrm{k}}{2^{\mathrm{k}}}\) is equal to :

If \(\left(\sin ^{-1} x\right)^{2}-\left(\cos ^{-1} x\right)^{2}=a ; 0\,<\,x\,<\,1, a \neq 0\), then the value of \(2 \mathrm{x}^{2}-1\) is :