The number of values of ' $k$ ' for which the points $(-4,9, k),(-1,6, k),(0,7,10)$ from right-angled isosceles triangle is

A line makes angles $60^{\circ}, 45^{\circ}, \theta$ with positive $X, Y, Z$ axes respectively. If $\theta$ is an acute angle, then $\tan \theta=$

If the foot of the perpendicular drawn from the point $(2,0,-3)$ to the plane $\pi$ is $(1,-2,0)$ and the equation of the plane $\pi$ is $a x+b y-3 z+d=0$, then $a+b+d=$

Let $\pi_1$ be the plane determined by the vectors $\hat{\mathbf{i}}+\hat{\mathbf{j}}$. $\hat{\mathbf{i}}+\hat{\mathbf{k}}$ and $\pi_2$ be the plane determined by the vectors $\hat{\mathbf{j}}-\hat{\mathbf{k}}, \hat{\mathbf{k}}-\hat{\mathbf{i}}$. Let $\mathbf{a}$ be a non-zero vector parallel to the line of intersection of the planes $\pi_1$ and $\pi_2$. If $\mathbf{b}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$, then the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ is