Three Dimensional Geometry - Algebra - Mathematics Previous Year Questions

Let $A$ be a point having position vector $\hat{\mathbf{i}}-3 \hat{\mathbf{j}}$ and $\mathbf{r}=(\hat{\mathbf{i}}-3 \hat{\mathbf{j}})+t(\hat{\mathbf{j}}-2 \hat{\mathbf{k}})$ be a line. If $P$ is a point on this line and is at a minimum distance from the plane $\mathbf{r} .(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}})=0$, then the equation of the plane through $P$ and perpendicular to $A P$, is

If $L$ is a line common to the planes $3 x+4 y+7 z=1$, $x-y+z=5$, then the direction ratios of the line $L$ are

If the points $(1,1, \lambda)$ and $(-3,0,1)$ are equidistant from the plane $3 x+4 y-12 z+13=0$, then the values of $\lambda$ are

If the equation of the plane passing through the points $(2,1,2),(1,2,1)$ and perpendicular to the plane $2 x-y+2 z=1$ is $a x+b y+c z+d=0$, then $\frac{a+b}{c+d}=$

If the direction cosines of two lines satisfy the equation $2 l+m-n=0, l^2-2 m^2+n^2=0$ and $\theta$ is the angle between the lines, then $\cos \theta=$

If $A(0,3,4), B(1,5,6), C(-2,0,-2)$ are the vertices of a $\triangle A B C$ and the bisector of angle $A$ meets the side $B C$ at $D$, then $A D=$

The shortest distance between the lines

$$\begin{aligned} & \mathbf{r}=(3 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}+2 \hat{\mathbf{k}})+t(4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-\hat{\mathbf{k}}) \text { and } \\ & \mathbf{r}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-4 \hat{\mathbf{k}})+s(6 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}) \text { is } \end{aligned}$$

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