Mathematics Mock Test - 9

If $k$ is a scalar and $I$ is a unit matrix of order $3$, then $\text{adj}(kI)=$

Find the angle between the pair of lines $\frac{x-5}{3}=\frac{y+2}{5}=\frac{z+2}{4}$ and $\frac{x-1}{1}=\frac{y-3}{1}=\frac{z-3}{2}$.

Find the number of ways of picking $4$ cards from a set of $52$ playing cards if all four cards are of the same suit?

If $\vec{a}=2\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}=-\hat{i}+2\hat{j}+\hat{k}$ and $\vec{c}=3\hat{i}+\hat{j}$ are such that $\vec{a}+\gamma \vec{b}$ is perpendicular to $\vec{c}$, then determine the value of $\gamma$?

Find the Cartesian equation of the plane which passes through the point $(5,2,-4)$ and is perpendicular to the line with direction ratios $2,3,-1$?