The work done by a force $\vec F = (-6x^3 \hat i) \, N$,in displacing a particle from $x = 4 \, m$ to $x = -2 \, m$ is ................ $J$.

If the system of equations $ax + y + z = 0$,$x + by + z = 0$,and $x + y + cz = 0$ $(a, b, c \neq 1)$ has a non-trivial solution,then the value of $\frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c}$ is:

Two rods of same material have same length and area. The heat $\Delta Q$ flows through them for $12 \, \text{minutes}$ when they are joined in series. If now both the rods are joined in parallel,then the same amount of heat $\Delta Q$ will flow in ........ $\text{minutes}$.

$\begin{aligned} & f(x, y)=2(x-y)^2-x^4-y^4 \\ & \left|\left(f_{x x} f_{y y}-f_{x y}^2\right)\right|_{(0,0)} \end{aligned}$

The ratio of the longest wavelengths corresponding to the Lyman and Balmer series in the hydrogen spectrum is

The magnetic needle of a vibration magnetometer makes $12$ oscillations per minute in the horizontal component of the earth's magnetic field. When an external short bar magnet is placed at some distance along the axis of the needle in the same line,it makes $15$ oscillations per minute. If the poles of the bar magnet are interchanged,the number of oscillations it makes per minute is