Electric potential at any point is $V = - 5x + 3y + \sqrt {15} z$,then the magnitude of the electric field is

The central fringe of the interference pattern produced by light of wavelength $6000 \, \mathring{A}$ is found to shift to the position of the $4^{th}$ bright fringe after a glass plate of refractive index $1.5$ is introduced. The thickness of the glass plate would be ........ $\mu m$.

If the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$ intersect,then the value of $k$ is

If $a, b, c$ are unit vectors and the maximum value of $|a-b|^2+|b-c|^2+|c-a|^2$ is $k$,then $k(2|a|^2+3|b|^2-4|c|^2)=$

The value of the integral $\int_0^4 \frac{dx}{1+x^2}$ obtained by using the Trapezoidal rule with $h=1$ is

If $y=a e^x+b e^{-x}+c$,where $a, b, c$ are parameters,then $y^{\prime \prime \prime}$ is equal to