$A$ particle is projected with a velocity $v$ such that its range on the horizontal plane is twice the greatest height attained by it. The range of the projectile is (where $g$ is acceleration due to gravity)

$A$ graph of magnetic flux $(\phi)$ versus current $(I)$ is drawn for four inductors $A, B, C, D$. Which inductor has the largest value of self-inductance?

$A$ small mass $m$ is attached to a massless string whose other end is fixed at $P$ as shown in the figure. The mass is undergoing circular motion in the $x-y$ plane with center $O$ and constant angular speed $\omega$. If the angular momentum of the system,calculated about $O$ and $P$ is denoted by $\vec L_o$ and $\vec L_p$ respectively,then:

An electron moves with speed $2 \times 10^5 \ m/s$ along the positive $x$-direction in the presence of a magnetic field of induction $B = (\hat{i} + 4\hat{j} - 3\hat{k}) \ T$. The magnitude of the force experienced by the electron in newtons is (Charge on the electron $= 1.6 \times 10^{-19} \ C$)

$A$ gas is allowed to expand in an insulated container against a constant external pressure of $10 \ atm$ from an initial volume $4.2 \ L$ to $5.2 \ L$. The change in internal energy $\Delta E$ of the gas will be $:-$

If $\sin ^{-1}\left(\frac{3}{x}\right)+\sin ^{-1}\left(\frac{4}{x}\right)=\frac{\pi}{2}$,then $x$ is equal to