$A$ $2 \ m$ long rod of radius $1 \ cm$ which is fixed from one end is given a twist of $0.8 \ radians$. The shear strain developed will be

$A$ linear charge having linear charge density $\lambda$ penetrates a cube diagonally and then it penetrates a sphere diametrically as shown. What will be the ratio of flux coming out of the cube and the sphere?

If three points $A, B$ and $C$ have position vectors $(1, x, 3)$,$(3, 4, 7)$ and $(y, -2, -5)$ respectively and if they are collinear,then $(x, y) = $

If $a, b$ and $c$ are the roots of $x^3+4x+1=0$,then $\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}=$

Suppose a pure $Si$ crystal has $5 \times 10^{28} \text{ atoms } m^{-3}$. It is doped by $1 \text{ ppm}$ concentration of pentavalent $As$. Calculate the number of electrons and holes. Given that $n_{i} = 1.5 \times 10^{16} \text{ m}^{-3}$.

$A$ block of mass $m$ lying on a rough horizontal plane is acted upon by a horizontal force $P$ and another force $Q$ inclined at an angle $\theta$ to the vertical. The minimum value of the coefficient of friction between the block and the surface for which the block will remain in equilibrium is: