A particle is moving in a straight line such that its velocity is increasing at \(5\,ms ^{-1}\) per meter. The acceleration of the particle is \(ms ^{-2}\) at a point where its velocity is \(20\,ms ^{-1}\).

The one division of main scale of vernier callipers reads \(1\,mm\) and \(10\) divisions of Vernier scale is equal to the \(9\) divisions on main scale. When the two jaws of the instrument touch each other the \(zero\) of the Vernier lies to the right of \(zero\) of the main scale and its fourth division coincides with a main scale division. When a spherical bob is tightly placed between the two jaws, the \(zero\) of the Vernier scale lies in between \(4.1\,cm\) and \(4.2\,cm\) and \(6^{\text {th }}\) Vernier division coincides with a main scale division. The diameter of the bob will be \(.............10^{-2}\,cm\)

Cross-section view of a prism is the equilateral triangle \({ABC}\) in the figure. The minimum deviation is observed using this prism when the angle of incidence is equal to the prism angle. The time taken by light to travel from midpoint of \({BC}\) to \({A}\) is \(.....\times 10^{-10}\, {s}\). \((\) Given, speed of light in vacuum \(=3 \times 10^{8} \,{m} / {s}\) and \(\left.\cos 30^{\circ}=\frac{\sqrt{3}}{2}\right)\)

A sinusoidal voltage of peak value \(250\, V\) is applied to a series \(LCR\) circuit, in which \(R =8 \Omega, L =24\, mH\) and \(C =60 \mu F\). The value of power dissipated at resonant condition is \('x'\, kW\). The value of \(x\) to the nearest integer is .............

In Young's double slit experiment, carried out with light of wavelength \(5000\) \(A\), the distance between the slits is \(0.3 \mathrm{~mm}\) and the screen is at \(200 \mathrm{~cm}\) from the slits. The central maximum is at \(x=0 \mathrm{~cm}\). The value of \(\mathrm{x}\) for third maxima is _______ \(\mathrm{mm}\).

For a constant collector-emitter voltage of \(8\,V\), the collector current of a transistor reached to the value of \(6\,mA\) from \(4\,mA\), whereas base current changed from \(20\,\mu A\) to \(25\,\mu\,A\) value. If transistor is in active state, small signal current gain (current amplification factor) will be.

Wires \({W}_{1}\) and \({W}_{2}\) are made of same material having the breaking stress of \(1.25 \times 10^{9} \,{N} / {m}^{2}\) \({W}_{1}\) and \({W}_{2}\) have cross-sectional area of \(8 \times 10^{-7}\, {m}^{2}\) and \(4 \times 10^{-7}\, {m}^{2}\), respectively. Masses of \(20 \,{kg}\) and \(10\, {kg}\) hang from them as shown in the figure. The maximum mass that can be placed in the pan without breaking the wires is \(.....{kg}\) (Use \(\left.{g}=10\, {m} / {s}^{2}\right)\)

A solid metallic cube having total surface area \(24\;m ^{2}\) is uniformly heated. If its temperature is increased by \(10\,^{\circ} C\), calculate the increase in volume of the cube \(\left(\right.\) Given \(\left.: \alpha=5.0 \times 10^{-4}{ }^{\circ} C ^{-1}\right)\)

An \(npn\) transistor operates as a common emitter amplifier with a power gain of \(10^{6} .\) The input circuit resistance is \(100\, \Omega\) and the output load resistance is \(10\, K \Omega\). The common emitter current gain ' \(\beta\) ' will be............. (Round off to the Nearest Integer)

A force \(\mathrm{f}=\mathrm{x}^2 \mathrm{y} \hat{\mathrm{i}}+\mathrm{y}^2 \hat{\mathrm{j}}\) acts on a particle in a plane \(\mathrm{x}+\mathrm{y}=10\). The work done by this force during a displacement from \((0,0)\) to \((4 \mathrm{~m}, 2 \mathrm{~m})\) is Joule (round off to the nearest integer)