KCET 2023

A line passes through \((2,2)\) and is perpendicular to the line \(3 x+y=3\). Its \(y\)-intercept is

If a line makes an angle of \(\frac{\pi}{3}\) with each \(X\) and \(Y\) axis, then the acute angle made by \(\mathrm{Z}\)-axis is

The length of perpendicular drawn from the point \((3,-1,11)\) to the line \(\frac{x}{2}=\frac{y-2}{3}=\frac{z-3}{4}\) is

The equation of the plane through the points \((2,1,0),(3,2,-2)\) and \((3,1,7)\) is

The point of intersection of the line \(x+1=\frac{y+3}{3}=\frac{-z+2}{2}\) with the plane \(3 x+4 y+5 z=10\) is

If \((2,3,-1)\) is the foot of the perpendicular from \((4,2,1)\) to a plane, then the equation of the plane is

\(|\mathbf{a} \times \mathbf{b}|^2+|\mathbf{a} \cdot \mathbf{b}|^2=144\) and \(|\mathbf{a}|=4\), then \(|\mathbf{b}|\) is equal to

If \(\mathbf{a}+2 \mathbf{b}+3 \mathbf{c}=0\) and \((\mathbf{a} \times \mathbf{b})+(\mathbf{b} \times \mathbf{c})+(\mathbf{c} \times \mathbf{a})=\lambda(\mathbf{b} \times \mathbf{c})\), then the value of \(\lambda\) is equal to

If \(|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|\), then

The component of \(\hat{\mathbf{i}}\) in the direction of the vector \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) is