The vector equation of a plane passing through the line of intersection of the planes $\mathbf{r} \cdot(\hat{\mathbf{i}}-2 \hat{\mathbf{k}})=3, \mathbf{r} \cdot(2 \hat{\mathbf{j}}+\hat{\mathbf{k}})=5$ and the point $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ is

The points $A(-1,2,3), B(2,-3,1)$ and $C(3,1,-2)$

The directions cosines of the line making angles $\frac{\pi}{4}, \frac{\pi}{3}$ and $\theta\left(0<\theta<\frac{\pi}{2}\right)$ respectively with $X, Y$ and $Z$ axes are

If the equation of the plane passing through the point $(3,2,5)$ and perpendicular to the planes $2 x-3 y+5 z=7$ and $5 x+2 y-3 z=11$ is $x+b y+c z+d=0$, then $2 b+3 c+d=$

The circumradius of the triangle formed by the points $(2,-1,1),(1,-3,-5)$ and $(3,-4,-4)$ is

Let $A(2,3,5), B(-1,3,2)$ and $C(\lambda, 5, \mu)$ be the vertices of $\triangle A B C$. If the median through the vertex $A$ is equally inclined to the coordinate axes, then

Equation of the plane passing through the origin and perpendicular to the planes $x+2 y-z=1$ and $3 x-4 y+z=5$ is

30. Line $L_1$ passes through the point $\hat{\mathbf{i}}+\hat{\mathbf{j}}$ and $\hat{\mathbf{k}}-\hat{\mathbf{i}}$. Line $L_2$ passes through the point $\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and is parallel to the vector $\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$. If $x \hat{\mathbf{i}}+y \hat{\mathbf{j}}+z \hat{\mathbf{k}}$ is the point of intersection of the lines $L_1$ and $L_2$, then $(y-x)=$

The point in the $X Y$ - plane which is equidistant from the points $A(2,0,3), B(0,3,2)$ and $C(0,0,1)$ has the coordinates

If the direction ratio of two lines $L_1$ and $L_2$ are given by $(1,-2,2)$ and $(-2,3,-6)$ respectively, then the direction ratios of the line which is perpendicular to the linesh and $L_2$ are

If the image of the point $A(1,1,1)$ with respect to the plane $4 x+2 y+4 z+1=0$ is $B(\alpha, \beta, \gamma)$, then $\alpha+\beta+\gamma=$

Assertion (A) For the lines $\mathbf{r}=\mathbf{a}+t \mathbf{b}$ and $\mathbf{r}=\mathbf{p}+s \mathbf{q}$, if $(\mathbf{a}-\mathbf{p}) \cdot(\mathbf{b} \times \mathbf{q}) \neq 0$, then the two lines are coplanar.

Reason $(\mathrm{R})|(\mathbf{a}-\mathbf{p}) \cdot(\mathbf{b} \times \mathbf{q})|$ is $|\mathbf{b} \times \mathbf{q}|$ times the shortest distance between the lines $\mathbf{r}=\mathbf{a}+t \mathbf{b}$ and $\mathbf{r}=\mathbf{p}+s \mathbf{q}$.

The locus of a point at which the line joining the points $(-3,1,2),(1,-2,4)$ subtends a right angle, is

If $A(1,2,3), B(2,3,-1), C(3,-1,-2)$ are the vertices of a $\triangle A B C$, then the direction ratios of the bisector of $\angle A B C$ are

Let $A=(2,0,-1), B=(1,-2,0), C=(1,2,-1)$ and $D=(0,-1,-2)$ be four points.

If $\theta$ is the acute angle between the plane determined by $A, B, C$ and the plane determined by $A, C, D$, then $\tan \theta=$

If $A(0,1,2), B(2,-1,3)$ and $C(1,-3,1)$ are the vertices of a triangle, then the distance between its circumcentre and orthocentre is

If the direction cosines of two lines satisfy the equations $l-2 m+n=0, l m+10 m n-2 n l=0$ and $\theta$ is the angle between the lines, then $\cos \theta=$

If $(2,-1,3)$ is the foot of the perpendicular drawn from the origin $(0,0,0)$ to a plane, then the equation of that plane is

If $A(2,-1,1), B(2,5,1)$ and $C(0,-2,3)$ are the vertices of a triangle. If $D$ is the point of intersection of the side $B C$ and the internal angular bisector of angle $A$, then $A D=$

A plane $\pi$ given by $a x+b y+11 z+d=0$ is perpendicular to the planes $2 x-3 y+z=4$, $3 x+y-z=5$ and the perpendicular distance from the origin to the plane $\pi$ is $\sqrt{6}$ units. If all the intercepts made by the plane $\pi$ on the coordinate axes are positive, then $d=$

For a positive real number $p$, if the perpendicular distance from a point $-\hat{\mathbf{i}}+p \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ to the plane $\mathbf{r} \cdot(2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}})=7$ is 6 units, then $p=$

If $Q(\alpha, \beta, \gamma)$ is the harmonic conjugate of the point $P(0,-7,1)$ with respect to the line segment joining the points $(2,-5,3)$ and $(-1,-8,0)$, then $\alpha-\beta+\gamma=$

On a line with direction cosines $l, m, n, A\left(x_1, y_1, z_1\right)$ is a fixed point. If $B=\left(x_1+4 k l, y_1+4 k m, z_1+4 k n\right)$ and $C=\left(x_1+k l, y_1+k m, z_1+k n\right)(k>0)$, then the ratio in which the point $B$ divides the line segment joining $A$ and $C$ is

If the line of intersection of the planes $2 x+3 y+z=1$ and $x+3 y+2 z=2$ makes an angle $\alpha$ with the positive $X$-axis, then $\cos \alpha=$

$\hat{\mathbf{i}}-2 \hat{\mathbf{j}}$ is a point on the line parallel to the vector $2 \hat{\mathbf{i}}+\hat{\mathbf{k}}$. If $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$ is a point on the plane parallel to the vectors $2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+2 \hat{\mathbf{k}}$, then the point of intersection of the line and the plane is

Angle between a diagonal of a cube and a diagonal of its face which are coterminus is

A plane $\pi$ is passing through the points $A(1,-2,3)$ and $B(6,4,5)$. If the plane $\pi$ is perpendicular the plane $3 x-y+z=2$, then the perpendicular distance from $(0,0,0)$ to the plane $\pi$ is

The point of intersection of the lines represented by $\mathbf{r}=(\hat{\mathbf{i}}-6 \hat{\mathbf{j}}+2 \hat{\mathbf{k}})+\mathbf{t}(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}})$ and $\mathbf{r}=(4 \hat{\mathbf{j}}+\hat{\mathbf{k}})+\mathbf{s}(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}})$ is

If the four points $(6,2,4),(1,3,5),(1,-2,3)$ and $(6, k, 2)$ are coplanar, then $k=$

$G(1,0,1)$ is the centroid of the $\triangle A B C$. If $A=(1,-4,2)$ and $B=(3,1,0)$, then $A G^2+C G^2=$

If the sum of the distances of the point $(3,4, \alpha), \alpha \in R$ from $X$-axis, $Y$-axis and $Z$-axis is minimum, then $\sec \alpha=$

If the equation of the plane passing through the point $(2,-1,3)$ and perpendicular to each of the planes $3 x-2 y+z=8$ and $x+y+z=6$ is $l x+m y+n z=1$, then $4 m+2 n-3 l=$

If $\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}, \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}$ are the vertices of a tetrahedron, then its volume is

If a line $L$ makes angles $\frac{\pi}{3}$ and $\frac{\pi}{4}$ with $Y$-axis and $Z$-axis respectively, then the angle between $L$ and another line having direction ratio $1,1,1$ is

If P divides the line segment joining the points $A(1,2,-1)$ and $B(-1,0,1)$ externally in the ratio 1 : 2 and $Q=(1,3,-1)$, then $PQ=$

If the direction cosines of a line are $\left(\frac{a}{\sqrt{83}}, \frac{5}{\sqrt{83}}, \frac{c}{\sqrt{83}}\right)$ and $c-a=4$, then $ca=$

Let the plane $\pi$ pass through the point (1, 0, 1) and perpendicular to the planes $2x + 3y - z = 2$ and $x - y + 2z = 1$. Let the equation of the plane passing through the point (11, 7, 5) and parallel to the plane $\pi$ be $ax + by - z - d = 0$. Then, ${a \over b} + {b \over d} = $

$D, E, F$ are respectively the points on the sides $B C, C A$ and $A B$ of a $\triangle A B C$ dividing them in the ratio $2: 3,1: 2,3: 1$ internally. The lines $\mathbf{B E}$ and $\mathbf{C F}$ intersect on the line $\mathbf{A D}$ at $P$. If $\mathbf{A P}=x_1 \cdot \mathbf{A} \mathbf{B}+y_1 \cdot \mathbf{A C}$, then $x_1+y_1=$

If the equation of the plane passing through the point $A(-2,1,3)$ and perpendicular to the vector $3 \hat{i}+\hat{j}+5 \hat{k}$ is $a x+b y+c z+d=0$, then $\frac{a+b}{c+d}=$

If $x$-coordinate of a point $P$ on the line joining the points $Q(2,2,1)$ and $R(5,2,-2)$ is 4, then the $y$-coordinate of $P=$

If $(2,3, c)$ are the direction ratios of a ray passing through the point $C(5, q, 1)$ and also the mid-point of the line segment joining the points $A(p,-4,2)$ and $B(3,2,-4)$, then $c \cdot(p+7 q)=$

If the equation of the plane which is at a distance of $1 / 3$ units from the origin and perpendicular to a line whose directional ratios are $(1,2,2)$ is $x+p y+q z+r=0$, then $\sqrt{p^2+q^2+r^2}=$

The point of intersection of the lines $\mathbf{r}=2 \mathbf{b}+t(6 \mathbf{c}-\mathbf{a})$ and $\mathbf{r}=\mathbf{a}+s(\mathbf{b}-3 \mathbf{c})$ is

If the point $(a, 8,-2)$ divides the line segment joining the points $(1,4,6)$ and $(5,2,10)$ in the ratio $m: n$, then $\frac{2 m}{n}-\frac{a}{3}=$

If $(a, b, c)$ are the direction ratios of a line joining the points $(4,3,-5)$ and $(-2,1,-8)$, then the point $P(a, 3 b, 2 c)$ lies on the plane

The $x$-intercept of a plane $\pi$ passing through the point $(1,1,1)$ is $\frac{5}{2}$ and the perpendicular distance from the origin to the plane $\pi$ is $\frac{5}{7}$. If the $y$-intercept of the plane $\pi$ is negative and the $z$-intercept is positive, then its $y$-intercept is

The equation of the plane passing through $3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}$ and parallel to the vectors $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ is

The direction cosines of the line joining the points $(-2,4,-5)$ and $(1,2,3)$ are

The points (2, 3, 4), ($-$1, $-$2, 1) and (5, 8, 7) are

The sum of intercepts of the plane $4 x+3 y+2 z=2$ on the coordinate axes is

If the lines, $\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{\lambda}$ and $\frac{x-2}{3}=\frac{y-3}{2}=\frac{z-2}{3}$ are coplanar, then $\sin ^{-1}(\sin \lambda)+\cos ^{-1}(\cos \lambda)$ is equal to

The line passing through $(1,1,-1)$ and parallel to the vector $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ meets the line $\frac{x-3}{-1}=\frac{y+2}{5}=\frac{z-2}{-4}$ at $A$ and the plane $2 x-y+2 z+7=0$ at $B$. Then $A B$ is equal to

If the vertices of the triangles are (1, 2, 3), (2, 3, 1), (3, 1, 2) and if H, G, S and I respectively denote its orthocentre, centroid, circumcentre and incentre, then H + G + S + I is equal to

A(2, 3, 4), B(4, 5, 7), C(2, $-$6, 3) and D(4, $-$4, k) are four points. If the line AB is parallel to CD, then k is equal to

If the direction cosines of two lines are $\left( {{2 \over 3},{2 \over 3},{1 \over 3}} \right)$ and $\left( {{5 \over {13}},{{12} \over {13}},0} \right)$, then identify the direction ratios of a line which is bisecting one o the angle between them.

$X$ intercept of the plane containing the line of intersection of the planes $x-2 y+z+2=0$ and $3 x-y-z+1=0$ and also passing through $(1,1,1)$ is

Let $L_1$ (resp, $L_2$ ) be the line passing through $2 \hat{\mathbf{i}}-\hat{\mathbf{k}}$ (resp. $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}})$ and parallel to $3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ ( resp. $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ ). Then the shortest distance between the lines $L_1$ and $L_2$ is equal to

If the points (2, 4, $-$1), (3, 6, $-$1) and (4, 5, $-$1) are three consecutive vertices of a parallelogram, then its fourth vertex is

$A(-1,2-3), B(5,0,-6)$ and $C(0,4,-1)$ are the vertices of a $\triangle A B C$. The direction cosines of internal bisector of $\angle B A C$ are

If the projections of the line segment AB on xy, yz and zx planes are $\sqrt{15},\sqrt{46},7$ respectively, then the projection of AB on Y-axis is

Find the equation of the plane passing through the point $(2,1,3)$ and perpendicular to the planes $x-2 y+2 z+3=0$ and $3 x-2 y+4 z-4=0$.

The ratio in which the YZ-plane divides the line joining (2, 4, 5) and (3, 5, $-$4) is

The direction cosines of a line which makes equal angles with the coordinate axes are

Let $O$ be the origin and $P$ be a point which is at a distance of 3 units from the origin. If the direction ratios of $\overline{O P}$ are $(1,-2,-2)$, then the coordinates of $P$ are