KCET 2024

The angle between the line $x+y=3$ and the line joining the points $(1,1)$ and $(-3,4)$ is

The sine of the angle between the straight line $\frac{x-2}{3}=\frac{y-3}{4}=\frac{4-z}{-5}$ are the plane $2 x-2 y+z=5$ is

The distance between the two planes $2 x+3 y+4 z=4$ and $4 x+6 y+8 z=12$ is

If lines $\frac{x-1}{-3}=\frac{y-2}{2 k}=\frac{z-3}{2}$ and $\frac{x-1}{3 k}=\frac{y-5}{1}=\frac{z-6}{-5}$ are mutually perpendicular, then $k$ is equal to

The equation $x y=0$ in three-dimensional space represents

The plane containing the point $(3,2,0)$ and the line $\frac{x-3}{1}=\frac{y-6}{5}=\frac{z-4}{4}$ is

The volume of the parallelopiped whose co terminous edges are $\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+\hat{\mathbf{j}}$ is

The vectors $\mathbf{A B}=3 \hat{\mathbf{i}}+4 \hat{\mathbf{k}}$ and $\mathbf{A C}=5 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ are the sides of a $\triangle A B C$, The length of the median through $A$ is

Let $\mathbf{a}$ and $\mathbf{b}$ be two unit vectors and $\theta$ is the angle between them. Then, $\mathbf{a}+\mathbf{b}$ is a unit vector, if

If $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are three non-coplanar vectors and $p, q$ and $r$ are vectors defined by $\mathbf{p}=\frac{\mathbf{a} \times \mathbf{c}}{[\mathbf{a b c}]}, \mathbf{q}=\frac{\mathbf{c} \times \mathbf{a}}{[\mathbf...