If \(t =\sqrt{ x }+4\), then \(\left(\frac{ dx }{ dt }\right)_{ t =4}\) is.

A body of mass \(2\, kg\) moves under a force of \((2 \hat{ i }+3 \hat{ j }+5 \hat{ k }) \,N\). It starts from rest and was at the origin initially. After \(4 \,s\), its new coordinates are \((8, b, 20)\). The value of \(b\) is ........ . (Round off to the Nearest Integer)

The current required to be passed through a solenoid of \(15\,cm\) length and 60 turns in order to demagnetise a bar magnet of magnetic intensity \(2.4 \times 10^3\,Am ^{-1}\) is \(.........A\).

A solid sphere of mass ' \(m\) ' and radius ' \(r\) ' is allowed to roll without slipping from the highest point of an inclined plane of length ' \(L\) ' and makes an angle \(30^{\circ}\) with the horizontal. The speed of the particle at the bottom of the plane is \(v_1\). If the angle of inclination is increased to \(45^{\circ}\) while keeping \(L\) constant. Then the new speed of the sphere at the bottom of the plane is \(v_2\). The ratio \(v_1^2: v_2^2\) is:

Two radioactive substances \(A\) and \(B\) have decay constants \(5\lambda \) and \(\lambda \) respectively. At \(t = 0\), a sample has the same number of the two nuclei. The time taken for the ratio of the number of nuclei to become \((\frac {1}{e})^2\) will be

The first overtone frequency of an open organ pipe is equal to the fundamental frequency of a closed organ pipe. If the length of the closed organ pipe is \(20 \,cm\). The length of the open organ pipe is........\(cm\).

The fractional compression \(\left(\frac{\Delta \mathrm{V}}{\mathrm{V}}\right)\) of water at the depth of 2.5 km below the sea level is ______ \(\%\). Given, the Bulk modulus of water \(=2 \times 10^9 \mathrm{~N} \mathrm{~m}^{-2}\), density of water \(=10^3\) \(\mathrm{kg} \mathrm{m}^{-3}\), acceleration due to gravity \(=\mathrm{g}=10 \mathrm{~m} \mathrm{~s}^{-2}\).

In Bohr's atomic model, the electron is assumed to revolve in a circular orbit of radius \(0.5 \times 10^{-10}\,m\). If the speed of electron is \(2.2 \times 10^{6} \;{m} / {s}\), then the current associated with the electron will be \(....\,\times 10^{-2} \;{mA}\). [Take \(\pi\) as \(\left.\frac{22}{7}\right]\)

If speed \(V,\) area \(A\) and force \(F\) are chosen as fundamental units, then the dimension of Young's modulus will be :

A square loop of side \(10 \mathrm{~cm}\) and resistance \(0.7\  \Omega\) is placed vertically in east-west plane. A uniform magnetic field of \(0.20 \mathrm{~T}\) is set up across the plane in north east direction. The magnetic field is decreased to zero in \(1 \mathrm{~s}\) at a steady rate. Then, magnitude of induced emf is \(\sqrt{\mathrm{x}} \times 10^{-3} \mathrm{~V}\). The value of \(x\) is _______.

In the given figure the blocks \(A , B\) and C weigh \(4 kg, 6 kg\) and 8 kg respectively. The co-efficient of sliding friction between any two surfaces is 0.5 . The force \(\overrightarrow{ F }\) required to slide the block C with constant speed is _________ N . (Used \(g =10 m / s ^2\) )

The \(V-I\) characteristic of a diode is shown in the figure. The ratio of forward to reverse bias resistance is

A radioactive material decays by simultaneous emissions of two particles with half lives of \(1400\, years\) and \(700\, years\) respectively. What will be the time after which one third of the material remains? (Take In \(3=1.1\) ) (In \(years\))