A small particle moves to position \(5 \hat{i}-2 \hat{j}+\hat{k}\) from its initial position \(2 \hat{i}+3 \hat{j}-4 \hat{k}\) under the action of force \(5 \hat{i}+2 \hat{j}+7 \hat{k} N\). The value of work done will be \(............J\).

A trransistor is connected in common emitter circuit configuration, the collector supply voltage is \(10 \,{V}\) and the voltage drop across a resistor of \(1000\, \Omega\) in the collector circuit is \(0.6\, {V}\). If the current gain factor \((\beta)\) is \(24\) , then the base currect is \(....\,\mu\) \(A.\) (Round off to the Nearest Integer)

A magnetic needle of magnetic moment \(6.7 \times 10^{-2} \) \(Am^2\) and moment of inertia \(7.5 \times  10^{-6}\) \( kgm^2\) is performing simple harmonic oscillations in a magnetic field of \(0.01\ T\). Time taken for \(10\) complete oscillations is.....\(s\)

Consider a solid sphere of radius \(\mathrm{R}\) and mass density \(\rho(\mathrm{r})=\rho_{0}\left(1-\frac{\mathrm{r}^{2}}{\mathrm{R}^{2}}\right),  0<\mathrm{r} \leq \mathrm{R} .\) The minimum density of a liquid in which it will float is

Inductance of a coil with \(10^4\) turns is 10 mH and it is connected to a dc source of 10 V with internal resistance of \(10 \Omega\). The energy density in the inductor when the current reaches \(\left(\frac{1}{ e }\right)\) of its maximum value is \(\alpha \pi \times \frac{1}{ e ^2} J / m ^3\). The value of \(\alpha\) is __________ . \(\left(\mu_0=4 \pi \times 10^{-7} Tm / A \right)\).

A bullet of \(10\, {g}\), moving with velocity \(v\), collides head-on with the stationary bob of a pendulum and recoils with velocity \(100 \, {m} / {s}\). The length of the pendulum is \(0.5\, {m}\) and mass of the bob is \(1\, {kg}\). The minimum value of \(v=\) \(....{m} / {s}\) so that the pendulum describes a circle. (Assume the string to be inextensible and \({g}=10\, {m} / {s}^{2}\) )

The velocity of a particle executing simple harmonic motion along \(x\)-axis is described as \(v^2 = 50 - x^2\), where \(x\) represents displacement. If the time period of motion is \(\dfrac{x}{7}\) s, the value of \(x\) is _____.

A radioactive nucleus \(n_2\) has 3 times the decay constant as compared to the decay constant of another radioactive nucleus \(n_1\). If initial number of both nuclei are the same, what is the ratio of number of nuclei of \(n_2\) to the number of nuclei of \(n_1\), after one half-life of \(n_1\) ?

A telescope with objective diameter \(R\) is used to observe a distant star emitting light of wavelength \(500\) nm, at a resolution of \(5 \times 10^{-7}\) radian. The value of \(R\) is _____ cm.

A wheel initially at rest is subjected to a uniform angular acceleration about its axis. In the first \(2\text{ s}\) it rotates through an angle \(\theta_1\) and in the next \(2\text{ s}\) it rotates through an angle \(\theta_2\). The ratio \(\dfrac{\theta_2}{\theta_1}\) is _______.

Two wires of same length and radius are joined end to end and loaded. The Young's modulii of the materials of the two wires are \(Y_{1}\) and \(Y_{2}\). The combination behaves as a single wire then its Young's modulus is: