If the slope of one of the pair of lines represented by $2x^2 + 3xy + Ky^2 = 0$ is $2$,then the angle between the pair of lines is

Three forces acting on a body are shown in the figure. To have the resultant force only along the $y-$ direction,the magnitude of the minimum additional force needed is.........$N$.

Let $\bar{a} = 3 \hat{i} + 2 \hat{j} + x \hat{k}$ and $\bar{b} = \hat{i} - \hat{j} + \hat{k}$,for some real $x$. Then $|\bar{a} \times \bar{b}| = r$ is possible,if

The magnetic flux through a circuit of resistance $R$ changes by an amount $\Delta \phi$ in time $\Delta t$. Then the total quantity of electric charge $Q$ passing during this time through any point of the circuit is given by:

Two blocks of masses $1 ~kg$ and $2 ~kg$ are connected by a metal wire going over a smooth pulley as shown in the figure. The breaking stress of the metal is $2 \times 10^9 ~N/m^2$. What should be the minimum radius of the wire used if it is not to break? Take $g = 10 ~m/s^2$.

In the expansion of $(1+x)^n$,the coefficients of $p$-th and $(p+1)$-th terms are respectively $p$ and $q$. Then $p+q=$