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JEE Mains | PYQ

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If the shortest distance between the lines \( \dfrac{x - 4}{1} = \dfrac{y + 1}{2} = \dfrac{z}{-3} \) and \( \dfrac{x - \lambda}{2} = \dfrac{y + 1}{4} = \dfrac{z - 2}{-5} \) is \( \dfrac{6}{\sqrt{5}} \), then the sum of all possible values of \( \lambda \) is:

Let the foot of perpendicular of the point \( P(3,-2,-9) \) on the plane passing through the points \( (-1,-2,-3),(9,3,4),(9,-2,1) \) be \( Q(\alpha,\beta,\gamma) \). Then the distance Q form the origin is:

Let the foot of perpendicular from a point \( P(1,2,-1) \) to the straight line \( L:\ \dfrac{x}{1}=\dfrac{y}{0}=\dfrac{z}{-1} \) be \( N \). Let a line be drawn from \( P \) parallel to the plane \( x+y+2z=0 \) which meets \( L \) at point \( Q \). If \( \alpha \) is the acute angle between the lines \( PN \) and \( PQ \), then \( \cos\alpha \) is equal to ____________.

If the lines \( \dfrac{x-1}{2}=\dfrac{2-y}{-3}=\dfrac{z-3}{\alpha} \) and \( \dfrac{x-4}{5}=\dfrac{y-1}{2}=\dfrac{z}{\beta} \) intersect, then the magnitude of the minimum value of \( 8\alpha\beta \) is __________.

Let the line \(\displaystyle \frac{x-3}{7}=\frac{y-2}{-1}=\frac{z-3}{-4} \) intersect the plane containing the lines \(\displaystyle \frac{x-4}{1}=\frac{y+1}{-2}=\frac{z}{1} \) and \( 4\alpha x-y+5z-7a=0=2x-5y-z-3,\; a\in \mathbb{R} \) at the point \( P(\alpha,\beta,\gamma) \). Then the value of \( \alpha+\beta+\gamma \) is equals ______.

Let the image of the point \( P(0,-5,0) \) in the line \( \dfrac{x-1}{2}=\dfrac{y}{1}=\dfrac{z+1}{-2} \) be the point \( R \) and the image of the point \( Q\!\left(0,\dfrac{-1}{2},0\right) \) in the line \( \dfrac{x-1}{-1}=\dfrac{y+9}{4}=\dfrac{z+1}{1} \) be the point \( S \). Then the square of the area of the parallelogram \( PQRS \) is ________.

Let \( P(\alpha, \beta, \gamma) \) be the point on the line \( \frac{x-1}{2} = \frac{y+1}{-3} = z \) at a distance \( 4\sqrt{14} \) from the point \( (1, -1, 0) \) and nearer to the origin. Then the shortest distance, between the lines \( \frac{x - \alpha}{1} = \frac{y - \beta}{2} = \frac{z - \gamma}{3} \) and \( \frac{x + 5}{2} = \frac{y - 10}{1} = \frac{z - 3}{1} \), is equal to:

Let the shortest distance between the lines \( L:\; \dfrac{x-5}{-2}=\dfrac{y-\lambda}{0}=\dfrac{z+\lambda}{1},\; \lambda \ge 0 \) and \( L_{1}:\; x+1=y-1=4-z \) be \( 2\sqrt{6} \). If \( (\alpha,\beta,\gamma) \) lies on \( L \), then which of the following is NOT possible?

A line with direction ratios \( 2,1,2 \) meets the lines \( x=y+2=z \) and \( x+2=2y=2z \) respectively at the points \( P \) and \( Q \). If the length of the perpendicular from the point \( (1,2,12) \) to the line \( PQ \) is \( l \), then \( l^{2} \) is __________.

If the foot of the perpendicular from the point \( A(-1,4,3) \) on the plane \( P: 2x+my+nz=4 \) is \( \left(-2,\dfrac{7}{2},\dfrac{3}{2}\right) \), then the distance of the point \( A \) from the plane \( P \), measured parallel to a line with direction ratios \( 3,-1,-4 \), is equal to:

For \( a, b \in \mathbb{Z} \) and \( |a-b| \leq 10 \), let the angle between the plane \( P: ax + y - z = b \) and the line \( l: x - 1 = a - y = z + 1 \) be \( \cos^{-1}\!\left(\dfrac{1}{3}\right) \). If the distance of the point \( (6,-6,4) \) from the plane \( P \) is \( 3\sqrt{6} \), then \( a^{4} + b^{2} \) is equal to:

Let the plane \( P \) pass through the intersection of the planes \( 2x+3y-z=2 \) and \( x+2y+3z=6 \), and be perpendicular to the plane \( 2x+y-z+1=0 \). If \( d \) is the distance of \( P \) from the point \( (-7,1,1) \), then \( d^{2} \) is equal to:

Let the lines \(\displaystyle \frac{x-1}{\lambda} = \frac{y-2}{1} = \frac{z-3}{2} \) and \(\displaystyle \frac{x+26}{-2} = \frac{y+18}{3} = \frac{z+28}{\lambda} \) be coplanar and \( P \) be the plane containing these two lines. Then which of the following points does NOT lie on \( P \)?

The shortest distance between the lines x+1=2y=-12z and x=y+2=6z-6 is:

The shortest distance between the lines \( \dfrac{x-4}{4}=\dfrac{y+2}{5}=\dfrac{z+3}{3} \) and \( \dfrac{x-1}{3}=\dfrac{y-3}{4}=\dfrac{z-4}{2} \) is:

If two straight lines whose direction cosines are given by the relations \( l + m - n = 0 \), \( 3l^2 + m^2 + cnl = 0 \) are parallel, then the positive value of \( c \) is:

Let P be the point of intersection of the line \( \dfrac{x+3}{3}=\dfrac{y+2}{1}=\dfrac{1-z}{2} \) and the plane \( x+y+z=2 \). If the distance of the point P from the plane \( 3x-4y+12z=32 \) is \( q \), then \( q \) and \( 2q \) are the roots of the equation.

The distance of the point \( P(3,4,4) \) from the point of intersection of the line joining the points \( Q(3,-4,-5) \) and \( R(2,-3,1) \) and the plane \( 2x+y+z=7 \), is equal to ____________.

If the shortest distance between the straight lines \( 3(x-1)=6(y-2)=2(z-1) \) and \( 4(x-2)=2(y-\lambda)=(z-3), \lambda \in \mathbb{R} \) is \( \dfrac{1}{\sqrt{38}} \), then the integral value of \( \lambda \) is equal to:

If a plane passes through the points \(( -1, k, 0)\), \(( 2, k, -1)\), \(( 1, 1, 2)\) and is parallel to the line \( \dfrac{x-1}{1}=\dfrac{2y+1}{2}=\dfrac{z+1}{-1} \), then the value of \( \dfrac{k^{2}+1}{(k-1)(k-2)} \) is :

If \( A(1,-1,2), \ B(5,7,-6), \ C(3,4,-10) \) and \( D(-1, -4, -2) \) are the vertices of a quadrilateral \( ABCD \), then its area is:

The equation of the planes parallel to the plane \( x - 2y + 2z - 3 = 0 \) which are at unit distance from the point \( (1, 2, 3) \) is \( ax + by + cz + d = 0 \). If \( (b - d) = K(c - a) \), then the positive value of \( K \) is ______________ .

If the shortest distance between the lines \( \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{\lambda} \) and \( \frac{x-2}{1}=\frac{y-4}{4}=\frac{z-5}{5} \) is \( \frac{1}{\sqrt{3}} \), then the sum of all possible values of \( \lambda \) is:

If the image of the point \( P(1, 0, 3) \) in the line joining the points \( A(4, 7, 1) \) and \( B(3, 5, 3) \) is Q\((\alpha, \beta, \gamma)\), then \( \alpha + \beta + \gamma \) is equal to:

Let the plane \( 2x + 3y + z + 20 = 0 \) be rotated through a right angle about its line of intersection with the plane \( x - 3y + 5z = 8 \). If the mirror image of the point \( \left( 2, -\dfrac{1}{2}, 2 \right) \) in the rotated plane is \( B(a,b,c) \), then:

The angle between the straight lines, whose direction cosines are given by the equations \( 2l + 2m - n = 0 \) and \( mn + nl + lm = 0 \), is:

Let the acute angle bisector of the two planes \( x - 2y - 2z + 1 = 0 \) and \( 2x - 3y - 6z + 1 = 0 \) be the plane P. Then which of the following points lies on P?

Let the image of the point \( \left(\dfrac{5}{3},\dfrac{5}{3},\dfrac{8}{3}\right) \) in the plane \( x-2y+z-2=0 \) be \( P \). If the distance of the point \( Q(6,-2,\alpha),\ \alpha>0 \), from \( P \) is \( 13 \), then \( \alpha \) is equal to _____.

If the equation of the plane containing the line \( x+2y+3z-4=0=2x+y-z+5 \) and perpendicular to the plane \( \vec{r}=(\hat{i}-\hat{j})+\lambda(\hat{i}+\hat{j}+\hat{k})+\mu(\hat{i}-2\hat{j}+3\hat{k}) \) is \( ax+by+cz=4 \), then \( (a-b+c) \) is equal to:

If the foot of the perpendicular drawn from \( (1, 9, 7) \) to the line passing through the point \( (3, 2, 1) \) and parallel to the planes \( x + 2y + z = 0 \) and \( 3y - z = 3 \) is \( (\alpha, \beta, \gamma) \), then \( \alpha + \beta + \gamma \) is equal to:

If the equation of the line passing through the point \( \left(0,-\frac{1}{2},0\right) \) and perpendicular to the lines \( \vec{r}=\lambda(\hat{i}+a\hat{j}+b\hat{k}) \) and \( \vec{r}=(\hat{i}-\hat{j}+6\hat{k})+\mu(-b\hat{i}+a\hat{j}+5\hat{k}) \) is \( \frac{x-1}{-2}=\frac{y-4}{d}=\frac{z-c}{-4} \), then \( a+b+c+d \) is equal to:

The line of shortest distance between the lines \(\displaystyle \frac{x-2}{0} = \frac{y-1}{1} = \frac{z}{1} \) and \(\displaystyle \frac{x-3}{2} = \frac{y-5}{2} = \frac{z-1}{1} \) makes an angle of \(\displaystyle \cos^{-1}\!\left(\sqrt{\frac{2}{27}}\right) \) with the plane \( P: ax - y - z = 0, \ (a>0) \). If the image of the point \( (1,1,-5) \) in the plane \( P \) is \( (\alpha,\beta,\gamma) \), then \( \alpha + \beta - \gamma \) is equal to ______.

Let \( \vec a=\hat i+2\hat j+\hat k \) and \( \vec b=2\hat i+7\hat j+3\hat k \). Let \( L_1:\ \vec r=(-\hat i+2\hat j+\hat k)+\lambda \vec a,\, \lambda\in \mathbb{R} \) and \( L_2:\ \vec r=(\hat j+\hat k)+\mu \vec b,\, \mu\in \mathbb{R} \) be two lines. If the line \( L_3 \) passes through the point of intersection of \( L_1 \) and \( L_2 \), and is parallel to \( \vec a+\vec b \), then \( L_3 \) passes through the point:

Let \( Q \) and \( R \) be two points on the line \(\displaystyle \frac{x+1}{2}=\frac{y+2}{3}=\frac{z-1}{2} \) at a distance \( \sqrt{26} \) from the point \( P(4,2,7) \). Then the square of the area of the triangle \( PQR \) is ________.

Let the plane passing through the point \( (-1,0,-2) \) and perpendicular to each of the planes \( 2x+y-z=2 \) and \( x-y-z=3 \) be \( ax+by+cz+8=0 \). Then the value of \( a+b+c \) is equal to:

A line passes through \( A(4,-6,-2) \) and \( B(16,-2,4) \). The point \( P(a,b,c) \), where \( a,b,c \) are non‑negative integers, on the line \( AB \) lies at a distance of \( 21 \) units from the point \( A \). The distance between the points \( P(a,b,c) \) and \( Q(4,-12,3) \) is equal to ________.

If the shortest distance between the lines \( \frac{x-\lambda}{-2}=\frac{y-2}{1}=\frac{z-1}{1} \) and \( \frac{x-\sqrt{3}}{1}=\frac{y-1}{-2}=\frac{z-2}{1} \) is \( 1 \), then the sum of all possible values of \( \lambda \) is:

Let \(d\) be the distance of the point of intersection of the lines \( \dfrac{x+6}{3}=\dfrac{y}{2}=\dfrac{z+1}{1} \) and \( \dfrac{x-7}{4}=\dfrac{y-9}{3}=\dfrac{z-4}{2} \) from the point \( (7,8,9) \). Then \( d^{2}+6 \) is equal to:

Consider the lines \( L_{1} \) and \( L_{2} \) given by:
\( L_{1}:\ \dfrac{x-1}{2}=\dfrac{y-3}{1}=\dfrac{z-2}{2} \)
\( L_{2}:\ \dfrac{x-2}{2}=\dfrac{y-2}{2}=\dfrac{z-3}{3} \)
A line \( L_{3} \) having direction ratios \( 1,-1,-2 \), intersects \( L_{1} \) and \( L_{2} \) at the points \( P \) and \( Q \) respectively. Then the length of line segment \( PQ \) is:

If the lines \( \dfrac{x-k}{1} = \dfrac{y-2}{2} = \dfrac{z-3}{3} \) and \( \dfrac{x+1}{3} = \dfrac{y+2}{2} = \dfrac{z+3}{1} \) are co-planar, then the value of \( k \) is ________.

Let a line passing through the point \( (4,1,0) \) intersect the line \( L_1:\; \dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4} \) at the point \( A(\alpha,\beta,\gamma) \) and the line \( L_2:\; x-6=y=-z+4 \) at the point \( B(a,b,c) \).

If the shortest distance between the lines;

\( L_{1} : \ \vec{r} = (2 + \lambda)\hat{i} + (1 - 3\lambda)\hat{j} + (3 + 4\lambda)\hat{k}, \ \lambda \in \mathbb{R} \)

\( L_{2} : \ \vec{r} = 2(1 + \mu)\hat{i} + 3(1 + \mu)\hat{j} + (5 + \mu)\hat{k}, \ \mu \in \mathbb{R} \)

is \( \dfrac{m}{\sqrt{n}} \), where \( \gcd(m, n) = 1 \), then the value of \( m + n \) equals:

Let the values of \( p \), for which the shortest distance between the lines \( \frac{x+1}{3}=\frac{y}{4}=\frac{z}{5} \) and \( \vec{r}= (p\hat{i}+2\hat{j}+\hat{k})+\lambda(2\hat{i}+3\hat{j}+4\hat{k}) \) is \( \frac{1}{\sqrt{6}} \), be \( a, b \), \( (a<b)\).Then the length of the latus rectum of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) is:

A plane \( E \) is perpendicular to the two planes \( 2x-2y+z=0 \) and \( x-y+2z=4 \), and passes through the point \( P(1,-1,1) \). If the distance of the plane \( E \) from the point \( Q(a,a,2) \) is \( 3\sqrt{2} \), then \( (PQ)^{2} \) is equal to:

Let a triangle \( \triangle PQR \) be such that \( P \) and \( Q \) lie on the line \( \frac{x+3}{8} = \frac{y-4}{2} = \frac{x+1}{2} \) and are at a distance of \( 6 \) units from \( R(1, 2, 3) \). If \( (\alpha, \beta, \gamma) \) is the centroid of \( \triangle PQR \), then \( \alpha + \beta + \gamma \) is equal to:

Let \(a,b\in \mathbb{R}\). If the mirror image of the point \(P(a,6,9)\) with respect to the line \(\dfrac{x-3}{7}=\dfrac{y-2}{5}=\dfrac{z-1}{-9}\) is \(Q(20,b,-a-9)\), then \(|a+b|\) is equal to:

Let \( A(x, y, z) \) be a point in \( xy \)-plane, which is equidistant from three points \( (0, 3, 2) \), \( (2, 0, 3) \) and \( (0, 0, 1) \). Let \( B = (1, 4, -1) \) and \( C = (2, 0, -2) \). Then among the statements: \( (S1) \): \( \triangle ABC \) is an isosceles right angled triangle, and \( (S2) \): the area of \( \triangle ABC \) is \( \dfrac{9\sqrt{2}}{2} \).

Let the foot of perpendicular from the point \( (\lambda,2,3) \) on the line \( \dfrac{x-4}{1}=\dfrac{y-9}{2}=\dfrac{z-5}{1} \) be the point \( (1,\mu,2) \). Then the distance between the lines \( \dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z+4}{6} \) and \( \dfrac{x-\lambda}{2}=\dfrac{y-\mu}{3}=\dfrac{z+5}{6} \) is equal to:

Let the plane \( a x + b y + c z + d = 0 \) bisect the line joining the points \( (4, -3, 1) \) and \( (2, 3, -5) \) at the right angles. If \( a, b, c, d \) are integers, then the minimum value of \( (a^2 + b^2 + c^2 + d^2) \) is ____________.

Let the line passing through the points \( (-1, 2, 1) \) and parallel to the line \( \frac{x-1}{2} = \frac{y+1}{3} = \frac{z}{4} \) intersect the line \( \frac{x+2}{3} = \frac{y-3}{2} = \frac{z-4}{1} \) at the point \( P \). Then the distance of \( P \) from the point \( Q (4, -5, 1) \) is:

If the shortest distance between the lines \( \frac{x-\lambda}{2}=\frac{y-4}{4}=\frac{z-3}{4} \) and \( \frac{x-2}{4}=\frac{y-4}{6}=\frac{z-7}{8} \) is \( \frac{13}{\sqrt{29}} \), then a value of \( \lambda \) is:

Let \( P \) be the foot of the perpendicular from the point \( Q(10,-3,-1) \) on the line \( \dfrac{x-3}{7}=\dfrac{y-2}{-1}=\dfrac{z+1}{-2} \). Then the area of the right-angled triangle \( PQR \), where \( R \) is the point \( (3,-2,1) \) is:

The shortest distance between the lines \( \frac{x-3}{2} = \frac{y-2}{3} = \frac{z-1}{-1} \) and \( \frac{x+3}{2} = \frac{y-6}{1} = \frac{z-5}{3} \), is

If the equation of the plane passing through the point \( (1, 1, 2) \) and perpendicular to the line \( x - 3y + 2z - 1 = 0 = 4x - y + z \) is \( Ax + By + Cz = 1 \), then \( 140(C - B + A) \) is equal to _______.

Let in a \( \triangle ABC \), the length of the side \( AC \) be \( 6 \), the vertex \( B \) be \( (1, 2, 3) \) and the vertices \( A, C \) lie on the line \( \frac{x-6}{3} = \frac{y-7}{2} = \frac{z-7}{-2} \). Then the area (in sq. units) of \( \triangle ABC \) is:

Let $\theta$ be the angle between the planes $P_1 : \vec{r} \cdot (\hat{i} + \hat{j} + 2\hat{k}) = 9$ and $P_2 : \vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 15$. Let $L$ be the line that meets $P_2$ at the point $(4,-2,5)$ and makes an angle $\theta$ with the normal of $P_2$. If $\alpha$ is the angle between $L$ and $P_2$, then $(\tan^2 \theta)(\cot^2 \alpha)$ is equal to _____.

Let \( Q \) be the mirror image of the point \( P(1,2,1) \) with respect to the plane \( x+2y+2z=16 \). Let \( T \) be a plane passing through the point \( Q \) and contains the line \( \vec{r}=-\hat{k}+ \lambda(\hat{i}+\hat{j}+2\hat{k}), \ \lambda \in \mathbb{R} \). Then which of the following points lies on \( T \)?

Let \( \lambda \) be an integer. If the shortest distance between the lines \( x-\lambda = 2y - 1 = -2z \) and \( x = y + 2\lambda = z - \lambda \) is \( \dfrac{\sqrt{7}}{2\sqrt{2}} \), then the value of \( |\lambda| \) is ________.

Let the direction cosines of two lines satisfy the equations: \( 4l + m - n = 0 \) and \( 2mn + 10nl + 3lm = 0 \). Then the cosine of the acute angle between these lines is:

If the shortest distance between the lines \( \frac{x+2}{2} = \frac{y+3}{3} = \frac{z-5}{4} \) and \( \frac{x-3}{1} = \frac{y-2}{-3} = \frac{z+4}{2} \) is \( \frac{38}{3\sqrt{5}} \, k \), and \( \int_{0}^{k} [ x^{2} ]dx = \alpha - \sqrt{\alpha} \), where \( [x] \) denotes the greatest integer function, then \( 6\alpha^{3} \) is equal to ________.

Line \( L_{1} \) passes through the point \( (1,2,3) \) and is parallel to \( z \)-axis. Line \( L_{2} \) passes through the point \( (\lambda,5,6) \) and is parallel to \( y \)-axis. Let for \( \lambda = \lambda_{1}, \lambda_{2}, \lambda_{2} < \lambda_{1} \), the shortest distance between the two lines be \( 3 \). Then the square of the distance of the point \( (\lambda_{1}, \lambda_{2}, 7) \) from the line \( L_{1} \) is:

Let \( \dfrac{x-2}{3}=\dfrac{y+1}{-2}=\dfrac{z+3}{-1} \) lie on the plane \( px-qy+z=5 \), for some \( p,q\in\mathbb{R} \). The shortest distance of the plane from the origin is:

Let a line \( l \) pass through the origin and be perpendicular to the lines \( l_{1}:\ \vec{r}=(\hat{i}-11\hat{j}-7\hat{k})+\lambda(\hat{i}+2\hat{j}+3\hat{k}),\ \lambda\in\mathbb{R} \) and \( l_{2}:\ \vec{r}=(-\hat{i}+\hat{k})+\mu(2\hat{i}+2\hat{j}+\hat{k}),\ \mu\in\mathbb{R} \). If \( P \) is the point of intersection of \( l \) and \( l_{1} \), and \( Q(\alpha,\beta,\gamma) \) is the foot of perpendicular from \( P \) on \( l_{2} \), then \( 9(\alpha+\beta+\gamma) \) is equal to __________ .

If the point of intersection of the lines \( \frac{x+1}{3} = \frac{y+a}{5} = \frac{z+b+1}{7} \) and \( \frac{x-2}{1} = \frac{y-b}{4} = \frac{z-2a}{7} \) lies on \( xy \)-plane, then the value of \( a+b \) is:

If the mirror image of the point \( P(3, 4, 9) \) in the line \( \dfrac{x-1}{3} = \dfrac{y+1}{2} = \dfrac{z-2}{1} \) is \( (\alpha,\beta,\gamma) \), then \( 14(\alpha + \beta + \gamma) \) is:

If the distance of the point \( (a, 2, 5) \) from the image of the point \( (1, 2, 7) \) in the line \( \frac{x}{1} = \frac{y-1}{1} = \frac{z-2}{2} \) is \( 4 \), then the sum of all possible value of \( a \) is equal to:

If the distance of the point \( Q(1,-2,3) \) from the plane \( x+2y-3z+10=0 \) measured parallel to the line \( \dfrac{x-1}{3}=\dfrac{2-y}{m}=\dfrac{z+3}{1} \) is \( \sqrt{\dfrac{{7}}{{2}}} \), then the value of \( |m| \) is equal to ____________.

Let the equation of the plane passing through the line \( x-2y-z-5=0 = x+y+3z-5 \) and parallel to the line \( x+y+2z-7=0 = 2x+3y+z-2 \) be \( ax+by+cz=65 \). Then the distance of the point \( (a,b,c) \) from the plane \( 2x+2y-z+16=0 \) is __________.

The line \( l_{1} \) passes through the point \( (2, 6, 2) \) and is perpendicular to the plane \( 2x + y - 2z = 10 \). Then the shortest distance between the line \( l_{1} \) and the line \( \frac{x+1}{2}=\frac{y+4}{-3}=\frac{z}{2} \) is:

Let \( P \) be the plane containing the straight line \( \displaystyle\frac{x-3}{9}=\frac{y+4}{-1}=\frac{z-7}{-5} \) and perpendicular to the plane containing the straight lines \(\displaystyle \frac{x}{2}=\frac{y}{3}=\frac{z}{5} \) & \(\displaystyle \frac{x}{3}=\frac{y}{7}=\frac{z}{8} \). If \( d \) is the distance of \( P \) from the point \( (2,-5,11) \), then \( d^2 \) is equal to:

The shortest distance between the lines \( \dfrac{x+2}{1}=\dfrac{y}{-2}=\dfrac{z-5}{2} \) and \( \dfrac{x-4}{1}=\dfrac{y-1}{2}=\dfrac{z-3}{0} \) is:

Let the line \( L_1 \) be parallel to the vector \( -3\hat{\text{i}} + 2\hat{\text{j}} + 4\hat{\text{k}} \) and pass through the point \( (2,\, 6,\, 7) \), and the line \( L_2 \) be parallel to the vector \( 2\hat{\text{i}} + \hat{\text{j}} + 3\hat{\text{k}} \) and pass through the point \( (4,\, 3,\, 5) \). If the line \( L_3 \) is parallel to the vector \( -3\hat{\text{i}} + 5\hat{\text{j}} + 16\hat{\text{k}} \) and intersects the lines \( L_1 \) and \( L_2 \) at the points \( C \) and \( D \), respectively, then \( |\overline{CD}|^2 \) is equal to:

If the image of the point \( (4, 4, 3) \) in the line \( \frac{x-1}{2} = \frac{y-2}{1} = \frac{z-1}{3} \) is \( (\alpha, \beta, \gamma) \), then \( \alpha + \beta + \gamma \) is equal to:

Let the image of the point \( P(1, 2, 3) \) in the plane \( 2x - y + z = 9 \) be \( Q \). If the coordinates of the point \( R \) are \( (6, 10, 7) \), then the square of the area of the triangle \( PQR \) is __________.

Let \( P \) be the plane passing through the line \( \dfrac{x-1}{1}=\dfrac{y-2}{-3}=\dfrac{z+5}{7} \) and the point \( (2,4,-3) \). If the image of the point \( (-1,3,4) \) in the plane \( P \) is \( (\alpha,\beta,\gamma) \) then \( \alpha+\beta+\gamma \) is equal to:

Let \( Q \) be the mirror image of the point \( P(1,0,1) \) with respect to the plane \( S:x+y+z=5 \). If a line \( L \) passing through \( (1,-1,-1) \), parallel to the line \( PQ \) meets the plane \( S \) at \( R \), then \( QR^{2} \) is equal to:

Let \( (\alpha, \beta, \gamma) \) be the co-ordinates of the foot of the perpendicular drawn from the point \( (5, 4, 2) \) on the line \( \vec{r} = (\hat{-i} + 3\hat{j} + \hat{k}) + \lambda (2\hat{i} + 3\hat{j} - \hat{k}) \). Then the length of the projection of the vector \( \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \) on the vector \( 6\hat{i} + 2\hat{j} + 3\hat{k} \) is:

If the line \( \frac{2 - x}{3} = \frac{3y - 2}{4\lambda + 1} = 4 - z \) makes a right angle with the line \( \frac{x + 3}{3\mu} = \frac{1 - 2y}{6} = \frac{5 - z}{7} \), then \( 4\lambda + 9\mu \) is equal to:

The square of the distance of the point \( (-2,-8,6) \) from the line \( \dfrac{x-1}{1}=\dfrac{y-1}{2}=\dfrac{z}{-1} \) along the line \( \dfrac{x+5}{1}=\dfrac{y+5}{-1}=\dfrac{z}{2} \) is equal to:

The foot of perpendicular of the point \( (2,0,5) \) on the line \( \frac{x+1}{2}=\frac{y-1}{5}=\frac{z+1}{-1} \) is \( (\alpha,\beta,\gamma) \). Then, which of the following is NOT correct?

Let $Q$ be the foot of the perpendicular from the point $P(7, -2, 13)$ on the plane containing the lines
$ \frac{x+1}{6} = \frac{y-1}{7} = \frac{z-3}{8} \quad \text{and} \quad \frac{x-1}{3} = \frac{y-2}{5} = \frac{z-3}{7}. $
Then $(PQ)^2$, is equal to ________.

The plane, passing through the points \( (0,-1,2) \) and \( (-1,2,1) \) and parallel to the line passing through \( (5,1,-7) \) and \( (1,-1,-1) \), also passes through the point:

Let a line having direction ratios \( 1, -4, 2 \) intersect the lines
\( \dfrac{x-7}{3} = \dfrac{y-1}{-1} = \dfrac{z+2}{1} \) and \( \dfrac{x}{2} = \dfrac{y-7}{3} = \dfrac{z}{1} \) at the point A and B. Then \( (AB)^2 \) is equal to ______________.

Let \( L \) be the line \( \frac{x+1}{2} = \frac{y+1}{3} = \frac{z+3}{6} \) and let \( S \) be the set of all points \( (a, b, c) \) on \( L \), whose distance from the line \( \frac{x+1}{2} = \frac{y+1}{3} = \frac{z-9}{0} \) along the line \( L \) is \( 7 \). Then \( \sum_{(a,b,c)\in S} (a + b + c) \) is equal to:

Let the line \( L \) pass through \( (1, 1, 1) \) and intersect the lines \( \dfrac{x-1}{2} = \dfrac{y+1}{3} = \dfrac{z-1}{4} \) and \( \dfrac{x-3}{1} = \dfrac{y-4}{2} = \dfrac{z}{1} \). Then, which of the following points lies on the line \( L \)?

Let \( ABCD \) be a tetrahedron such that the edges \( AB \), \( AC \) and \( AD \) are mutually perpendicular. Let the areas of the triangles \( ABC \), \( ACD \) and \( ADB \) be \( 5 \), \( 6 \) and \( 7 \) square units respectively. Then the area (in square units) of the \( \triangle BCD \) is equal to:

If \( \left(2\alpha+1,\alpha^2-3\alpha,\frac{\alpha-1}{2}\right) \) is the image of \( (\alpha,2\alpha,1) \) in the line \( \frac{x-2}{3}=\frac{y-1}{2}=\frac{z}{1} \), then the possible value(s) of \( \alpha \) is (are):

Let the point, on the line passing through the points \( P(1, -2, 3) \) and \( Q(5, -4, 7) \), farther from the origin and at a distance of \( 9 \) units from the point \( P \), be \( (\alpha, \beta, \gamma) \). Then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to:

Let the values of \( \lambda \) for which the shortest distance between the lines \( \dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4} \) and \( \dfrac{x-\lambda}{3}=\dfrac{y-4}{4}=\dfrac{z-5}{5} \) is \( \dfrac{1}{\sqrt{6}} \) be \( \lambda_{1} \) and \( \lambda_{2} \). Then the radius of the circle passing through the points \( (0,0) \), \( (\lambda_{1},\lambda_{2}) \) and \( (\lambda_{2},\lambda_{1}) \) is:

The shortest distance between the lines \( \frac{x-3}{4}=\frac{y+7}{-11}=\frac{z-1}{5} \) and \( \frac{x-5}{3}=\frac{y-9}{-6}=\frac{z+2}{1} \) is:

The distance of the point \( (-1,\,2,\,-2) \) from the line of intersection of the planes \( 2x+3y+2z=0 \) and \( x-2y+z=0 \):

Let the image of the point \( P(1, 2, 3) \) in the line \( L:\ \dfrac{x - 6}{3} = \dfrac{y - 1}{2} = \dfrac{z - 2}{3} \) be \( Q \). Let \( R(\alpha, \beta, \gamma) \) be a point that divides internally the line segment \( PQ \) in the ratio \( 1:3 \). Then the value of \( 22(\alpha + \beta + \gamma) \) is equal to \( \_ \_ \_ \_ \_ \_ \).

Let the line \( L : \dfrac{x-1}{2} = \dfrac{y+1}{-1} = \dfrac{z-3}{1} \) intersect the plane \( 2x + y + 3z = 16 \) at the point \( P \). Let the point \( Q \) be the foot of perpendicular from the point \( R(1,-1,-3) \) on the line \( L \). If \( \alpha \) is the area of triangle \( PQR \), then \( \alpha^{2} \) is equal to _____.

Let \( \alpha x + \beta y + yz = 1 \) be the equation of the plane passing through the point \( (3, -2, 5) \) and perpendicular to the line joining the points \( (1, 2, 3) \) and \( (-2, 3, 5) \). Then the value of \( \alpha \beta \gamma \) is equal to ______.

Let the position vectors of three vertices of a triangle be \( 4\vec{p} + \vec{q} - 3\vec{r} \), \( -5\vec{p} + \vec{q} + 2\vec{r} \) and \( 2\vec{p} - \vec{q} + 2\vec{r} \). If the position vectors of the orthocenter and the circumcenter of the triangle are \( \frac{\vec{p} + \vec{q} + \vec{r}}{4} \) and \( \alpha \vec{p} + \beta \vec{q} + \gamma \vec{r} \) respectively, then \( \alpha + 2\beta + 5\gamma \) is equal to:

Consider the lines \( L_{1}: {x-1} = {y-2} = {z} \) and \( L_{2}: {x-2} = {y} = {z-1} \). Let the feet of the perpendiculars from the point \( P(5,1,-3) \) on the lines \( L_{1} \) and \( L_{2} \) be \( Q \) and \( R \) respectively. If the area of the triangle \( PQR \) is \( A \), then \( 4A^{2} \) is equal to:

For the system of linear equations:
\( x - 2y = 1 \)
\( x - y + kz = -2 \)
\( ky + 4z = 6, \; k \in \mathbb{R} \),
Consider the following statements:
(A) The system has unique solution if \( k \neq 2, \, k \neq -2 \).
(B) The system has unique solution if \( k = -2 \).
(C) The system has unique solution if \( k = 2 \).
(D) The system has no-solution if \( k = 2 \).
(E) The system has infinite number of solutions if \( k \neq -2 \).
Which of the following statements are correct?

The distance of the line \(\displaystyle \frac{x - 2}{2} = \frac{y - 6}{3} = \frac{z - 3}{4} \) from the point \( (1, 4, 0) \) along the line \(\displaystyle \frac{x}{1} = \frac{y - 2}{2} = \frac{z + 3}{3} \) is:

Let a plane \( P \) pass through the point \( (3,7,-7) \) and contain the line, \( \dfrac{x-2}{-3}=\dfrac{y-3}{2}=\dfrac{z+2}{1} \). If distance of the plane \( P \) from the origin is \( d \), then \( d^2 \) is equal to \( \underline{\quad} \).

Let the mirror image of the point \( (a, b, c) \) with respect to the plane \( 3x - 4y + 12z + 19 = 0 \) be \( (a - 6, \beta, \gamma) \). If \( a + b + c = 5 \), then \( 7\beta - 9\gamma \) is equal to ____________.

Suppose the line \( \dfrac{x-2}{\alpha}=\dfrac{y-2}{-5}=\dfrac{z+2}{2} \) lies on the plane \( x+3y-2z+\beta=0 \). Then \( (\alpha+\beta) \) is equal to ______.

If the equation of the plane passing through the line of intersection of the planes \( 2x - 7y + 4z - 3 = 0 \), \( 3x - 5y + 4z + 11 = 0 \) and the point \( (-2, 1, 3) \) is \( ax + by + cz - 7 = 0 \), then the value of \( 2a + b + c - 7 \) is _______.

Let the lines \( l_1 : \dfrac{x+5}{3} = \dfrac{y+4}{1} = \dfrac{z-\alpha}{-2} \) and \( l_2 : 3x + 2y + z - 2 = 0 = x - 3y + 2z - 13 \) be coplanar. If the point \( P(a,b,c) \) on \( l_1 \) is nearest to point \( Q(-4,-3,2) \), then \( |a| + |b| + |c| \) is equal to:

If the shortest between the lines: \( \dfrac{x+\sqrt{6}}{2}=\dfrac{y-\sqrt{6}}{3}=\dfrac{z-\sqrt{6}}{4} \) and \( \dfrac{x-\lambda}{3}=\dfrac{y-2\sqrt{6}}{4}=\dfrac{z+2\sqrt{6}}{5} \) is \( 6 \), then the square of sum of all possible values of \( \lambda \) is:

Let the line \( L \) pass through the point \( (0, 1, 2) \), intersect the line \( \dfrac{x-1}{2} = \dfrac{y-2}{3} = \dfrac{z-3}{4} \) and be parallel to the plane \( 2x + y - 3z = 4 \). Then the distance of the point \( P(1, -9, 2) \) from the line \( L \) is:

Let a line \( L_1 \) pass through the origin and be perpendicular to the lines \( L_2:\ \mathbf{\vec r}=(3+t)\mathbf{\hat i}+(2t-1)\mathbf{\hat j}+(2t+4)\mathbf{\hat k} \) and \( L_3:\ \mathbf{\vec r}=(3+2s)\mathbf{\hat i}+(3+2s)\mathbf{\hat j}+(2+s)\mathbf{\hat k},\ t,s\in \mathbb{R} \). If \( (a,b,c),\ a\in \mathbb{Z} \), is the point on \( L_3 \) at a distance of \( \sqrt{17} \) from the point of intersection of \( L_1 \) and \( L_2 \), then \( (a+b+c)^2 \) is equal to \( \_\_\_\_\_ \).

If the square of the shortest distance between the lines \(\displaystyle \frac{x-2}{1} = \frac{y-1}{2} = \frac{z+3}{-3} \) and \(\displaystyle \frac{x+1}{2} = \frac{y+3}{4} = \frac{z+5}{-5} \) is \(\displaystyle \frac{m}{n} \), where \( m, n \) are coprime numbers, then \( m + n \) is equal to:

The shortest distance between the lines \( \dfrac{x+7}{-6}=\dfrac{y-6}{7}=z \) and \( \dfrac{7-x}{2}=y-2=z-6 \) is

One vertex of a rectangular parallelopiped is at the origin \( O \) and the lengths of its edges along \( x, y \) and \( z \) axes are \( 3, 4 \) and \( 5 \) units respectively. Let \( P \) be the vertex \( (3,4,5) \). Then the shortest distance between the diagonal \( OP \) and an edge parallel to \( z \)-axis, not passing through \( O \) or \( P \) is:

Consider the line \( L \) given by the equation \(\displaystyle \frac{x-3}{2} = \frac{y-1}{1} = \frac{z-2}{1} \). Let \( Q \) be the mirror image of the point \( (2, 3, -1) \) with respect to \( L \). Let a plane \( P \) be such that it passes through \( Q \), and the line \( L \) is perpendicular to \( P \). Then which of the following points is on the plane \( P \)?

The sum of all values of \( \alpha \), for which the shortest distance between the lines \( \frac{x+1}{\alpha}=\frac{y-2}{-1}=\frac{z-4}{-\alpha} \) and \( \frac{x}{\alpha}=\frac{y-1}{2}=\frac{z-1}{2\alpha} \) is \( \sqrt{2} \), is:

Let \( L_{1}:\ \vec{r} = (\hat{\imath}-\hat{\jmath}+2\hat{k}) + \lambda(\hat{\imath}-\hat{\jmath}+2\hat{k}),\ \lambda \in \mathbb{R}; \ L_{2}:\ \vec{r} = (j-\hat{k}\,) + \mu(3\hat{\imath}+\hat{\jmath}+p\hat{k}),\ \mu \in \mathbb{R}; \) and \( L_{3}:\ \vec{r} = \delta(\hat{\imath}+m\hat{\jmath}+n\hat{k}),\ \delta \in \mathbb{R} \) be three lines such that \( L_{1} \) is perpendicular to \( L_{2} \) and \( L_{3} \) is perpendicular to both \( L_{1} \) and \( L_{2} \). Then, the point which lies on \( L_{3} \) is:

If the length of the perpendicular drawn from the point \( P(a,4,2), a>0 \) on the line \( \displaystyle\frac{x+1}{2}=\frac{y-3}{3}=\frac{z-1}{-1} \) is \( 2\sqrt{6} \) units and \( Q(\alpha_1,\alpha_2,\alpha_3) \) is the image of the point \( P \) in this line, then \( a + \displaystyle\sum_{i=1}^{3}\alpha_i \) is equal to:

If the line of intersection of the planes \( ax+by=3 \) and \( ax+by+cz=0,\, a>0 \) makes an angle \( 30^\circ \) with the plane \( y-z+2=0 \), then the direction cosines of the line are:

If the shortest distance between the line joining the points \( (1,2,3) \) and \( (2,3,4) \), and the line \( \frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-2}{0} \) is \( \alpha \), then \( 28\alpha^2 \) is equal to ______.

The distance of the point $Q(0, 2, -2)$ from the line passing through the point $P(5, -4, 3)$ and perpendicular to the lines $\vec{r} = (-3\hat{i} + 2\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 5\hat{k}), \lambda \in R$ and $\vec{r} = (\hat{i} - 2\hat{j} + \hat{k}) + \mu(-\hat{i} + 3\hat{j} + 2\hat{k}), \mu \in R$ is:

If \( \lambda_{1}<\lambda_{2} \) are two values of \( \lambda \) such that the angle between the planes \( P_{1}:\ \vec{r}\cdot(3\hat{i}-5\hat{j}+\hat{k})=7 \) and \( P_{2}:\ \vec{r}\cdot(\lambda \hat{i}+\hat{j}-3\hat{k})=9 \) is \( \sin^{-1}\!\left(\dfrac{2\sqrt{6}}{5}\right) \), then the square of the length of perpendicular from the point \( (38\lambda_{1},\,10\lambda_{2},\,2) \) to the plane \( P_{1} \) is \( \_\_\_\_\_ \).

Let the vertices \( Q \) and \( R \) of the triangle \( PQR \) lie on the line \( \dfrac{x+3}{5} = \dfrac{y-1}{2} = \dfrac{z+4}{3} \), \( QR = 5 \) and the coordinates of the point \( P \) be \( (0, 2, 3) \). If the area of the triangle \( PQR \) is \( \dfrac{m}{n} \) then:

For \( \alpha, \beta \in \mathbb{R} \), suppose the system of linear equations
\( x - y + z = 5 \)
\( 2x + 2y + \alpha z = 8 \)
\( 3x - y + 4z = \beta \)
has infinitely many solutions. Then \( \alpha \) and \( \beta \) are the roots of:

Let the mirror image of the point \( (1, 3, a) \) with respect to the plane \( \vec{r}\cdot(2\hat{i} - \hat{j} + \hat{k}) - b = 0 \) be \( (-3, 5, 2) \). Then, the value of \( |\, a + b \,| \) is equal to:

Let \( P(x, y, z) \) be a point in the first octant, whose projection in the \( xy \)-plane is the point \( Q \). Let \( OP = \gamma \); the angle between \( OQ \) and the positive \( x \)-axis be \( \theta \); and the angle between \( OP \) and the positive \( z \)-axis be \( \phi \), where \( O \) is the origin. Then the distance of \( P \) from the \( x \)-axis is:

Let the equation of plane passing through the line of intersection of the planes \( x+2y+az=2 \) and \( x-y+z=3 \) be \( 5x-11y+bz=6a-1 \). For \( c\in \mathbb{Z} \), if the distance of this plane from the point \( (a,-c,c) \) is \( \dfrac{2}{\sqrt{a}} \) then \( \dfrac{a+b}{c} \) is equal to:

The shortest distance between the lines \( \frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5} \) and \( \frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3} \) is:

The lines \( \dfrac{x-2}{2}=\dfrac{y}{-2}=\dfrac{z-7}{16} \) and \( \dfrac{x+3}{4}=\dfrac{y+2}{3}=\dfrac{z+2}{1} \) intersect at the point \( P \). If the distance of \( P \) from the line \( \dfrac{x+1}{2}=\dfrac{y-1}{3}=\dfrac{z-1}{1} \) is \( l \), then \( 14\, l^{2} \) is equal to __________ .

If the mirror image of the point \( (2,4,7) \) in the plane \( 3x-y+4z=2 \) is \( (a,b,c) \), then \( 2a+b+2c \) is equal to:

If the shortest distance between the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $\frac{x}{1} = \frac{y}{\alpha} = \frac{z-5}{1}$ is $\frac{5}{\sqrt{6}}$, then the sum of all possible values of $\alpha$ is:

Let \( N \) be the foot of perpendicular from the point \( P(1,-2,3) \) on the line passing through the points \( (4,5,8) \) and \( (1,-7,5) \). Then the distance of \( N \) from the plane \( 2x-2y+z+5=0 \) is:

Let the lines
\( L_1:\ \vec{r}=\lambda(\hat{i}+2\hat{j}+3\hat{k}),\ \lambda\in\mathbb{R} \) and
\( L_2:\ \vec{r}=(\hat{i}+3\hat{j}+\hat{k})+\mu(\hat{i}+\hat{j}+5\hat{k}),\ \mu\in\mathbb{R} \)
Intersect at the point \( S \). If a plane \( ax+by-z+d=0 \) passes through \( S \) and is parallel to both the lines \( L_1 \) and \( L_2 \), then the value of \( a+b+d \) is equal to ________.

If the system of linear equations.
\( 8x + y + 4z = -2 \)
\( x + y + z = 0 \)
\( \lambda x - 3y = \mu \)
has infinitely many solutions, then the distance of the point \( \left(\lambda, \mu, -\dfrac{1}{2}\right) \) from the plane \( 8x + y + 4z = 0 \) is:

Let \( P \) be the plane passing through the intersection of the planes \( \vec{r} \cdot (\hat{i} + 3\hat{j} - \hat{k}) = 5 \) and \( \vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 3 \), and the point \( (2, 1, -2) \). Let the position vectors of the points \( X \) and \( Y \) be \( \hat{i} - 2\hat{j} + 4\hat{k} \) and \( 5\hat{i} - \hat{j} + 2\hat{k} \) respectively. Then the points

Let \( P \) be the point \( (10, -2, -1) \) and \( Q \) be the foot of the perpendicular drawn from the point \( R(1, 7, 6) \) on the line passing through the points \( (2, -5, 11) \) and \( (-6, 7, -5) \). Then the length of the line segment \( PQ \) is equal to:

The point of intersection \( C \) of the plane \( 8x + y + 2z = 0 \) and the line joining the points \( A(-3, -6, 1) \) and \( B(2, 4, -3) \) divides the line segment \( AB \) internally in the ratio \( k : 1 \). If \( a, b, c \) ( \( |a|, |b|, |c| \) are coprime) are the direction ratios of the perpendicular from the point \( C \) on the line \( \frac{1 - x}{1} = \frac{y + 4}{2} = \frac{z + 2}{3} \), then \( |a + b + c| \) is equal to ______.

Let the points on the plane \( P \) be equidistant from the points \( (-4,2,1) \) and \( (2,-2,3) \). Then the acute angle between the plane \( P \) and the plane \( 2x+y+3z=1 \) is:

Let the line \( L \) intersect the lines \( x - 2 = -y = z - 1 \), \( 2(x + 1) = 2(y - 1) = z + 1 \) and be parallel to the line \( \dfrac{x - 2}{3} = \dfrac{y - 1}{1} = \dfrac{z - 2}{2} \). Then which of the following points lies on \( L \)?

Let two vertices of a triangle \( ABC \) be \( (2, 4, 6) \) and \( (0, -2, -5) \), and its centroid be \( (2, 1, -1) \). If the image of the third vertex in the plane \( x + 2y + 4z = 11 \) is \( (\alpha, \beta, \gamma) \), then \( \alpha\beta + \beta\gamma + \gamma\alpha \) is equal to:

The distance of the point \( (-1, 2, 3) \) from the plane \( \vec r \cdot (\hat i - 2\hat j + 3\hat k) = 10 \) parallel to the line of the shortest distance between the lines \( \vec r = (\hat i - \hat j) + \lambda (2\hat i + \hat k) \) and \( \vec r = (2\hat i - \hat j) + \mu (\hat i - \hat j + \hat k) \) is:

Let \( P(3, 2, 3) \), \( Q(4, 6, 2) \) and \( R(7, 3, 2) \) be the vertices of \( \triangle PQR \). Then, the angle \( \angle QPR \) is:

Let \( P \) be a plane passing through the points \( (1,0,1) \), \( (1,-2,1) \) and \( (0,1,-2) \). Let a vector \( \vec{a}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k} \) be such that \( \vec{a} \) is parallel to the plane \( P \), perpendicular to \( (\hat{i}+2\hat{j}+3\hat{k}) \) and \( \vec{a}\cdot(\hat{i}+\hat{j}+2\hat{k})=2 \), then \( (\alpha-\beta+\gamma)^{2} \) equals _______.

The length of the perpendicular from the point \( (1,-2,5) \) on the line passing through \( (1,2,4) \) and parallel to the line \( x+y-z=0=x-2y+3z-5 \) is:

The line \( L_{1} \) is parallel to the vector \( \vec{a} = -3\hat{i} + 2\hat{j} + 4\hat{k} \) and passes through the point \( (7, 6, 2) \) and the line \( L_{2} \) is parallel to the vector \( \vec{b} = 2\hat{i} + \hat{j} + 3\hat{k} \) and passes through the point \( (5, 3, 4) \). The shortest distance between the lines \( L_{1} \) and \( L_{2} \) is:

The distance of line \( 3y - 2z - 1 = 0 = 3x - z + 4 \) from the point \( (2, -1, 6) \) is:

Let a line \( L \) passing through the point \( P(1, 1, 1) \) be perpendicular to the lines \( \frac{x-4}{4} = \frac{y-1}{1} = \frac{z-1}{1} \) and \( \frac{x-17}{1} = \frac{y-71}{1} = \frac{z}{0} \). Let the lines \( L \) intersect the \( yz \)-plane at the point \( Q \). Another line parallel to \( L \) and passing through the point \( S(1, 0, -1) \) intersects the \( yz \)-plane at the point \( R \). Then the square of the area of the parallelogram \( PQRS \) is equal to _______.

Let the plane $P : \vec{r} \cdot \vec{a} = d$ contain the line of intersection of two planes $\vec{r} \cdot \left(\hat{i} + 3\hat{j} - \hat{k}\right) = 6$ and $\vec{r} \cdot \left(-6\hat{i} + 5\hat{j} - \hat{k}\right) = 7$. If the plane $P$ passes through the point $\left(2, 3, \frac{1}{2}\right)$, then the value of $\frac{|13\vec{a}|^2}{d^2}$ is equal to:

The square of the distance of the image of the point \( (6,1,5) \) in the line \( \dfrac{x-1}{3}=\dfrac{y}{2}=\dfrac{z-2}{4} \), from the origin is_________.

Let \( d \) be the distance between the foot of perpendiculars of the points \( P(1,2,-1) \) and \( Q(2,-1,3) \) on the plane \( -x + y + z = 1 \). Then \( d^2 \) is equal to __________.

Each of the angles \( \beta \) and \( \gamma \) that a given line makes with the positive \( y \)- and \( z \)-axes, respectively, is half of the angle that this line makes with the positive \( x \)-axes. Then the sum of all possible values of the angle \( \beta \) is:

Let \( (\alpha,\beta,\gamma) \) be the mirror image of the point \( (2,3,5) \) in the line \( \dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4} \). Then \( 2\alpha+3\beta+4\gamma \) is equal to:

Let \( P_{1} \) be the plane \( 3x-y-7z=11 \) and \( P_{2} \) be the plane passing through the points \( (2,-1,0) \), \( (2,0,-1) \), and \( (5,1,1) \). If the foot of the perpendicular drawn from the point \( (7,4,-1) \) on the line of intersection of the planes \( P_{1} \) and \( P_{2} \) is \( (\alpha,\beta,\gamma) \), then \( \alpha+\beta+\gamma \) is equal to _______.

Let \( A \) be the point of intersection of the lines \( L_1 : \, \frac{x-7}{1} = \frac{y-5}{0} = \frac{z-3}{-1} \) and \( L_2 : \, \frac{x-1}{3} = \frac{y+3}{4} = \frac{z+7}{5} \).Let \( B \) and \( C \) be the points on the lines \( L_1 \) and \( L_2 \) respectively such that \( AB = AC = \sqrt{15} \). Then the square of the area of the triangle \( ABC \) is:

Let the equation of the plane $P$ containing the line $x+10=\frac{8-y}{2}=z$ be $ax+by+3z=2(a+b)$ and the distance of the plane $P$ from the point $(1, 27, 7)$ be $c$. Then $a^2+b^2+c^2$ is equal to _______.

The vertices \( B \) and \( C \) of a triangle \( ABC \) lie on the line \( \dfrac{x}{1} = \dfrac{1 - y}{-2} = \dfrac{z - 2}{3} \). The coordinates of \( A \) and \( B \) are \( (1, 6, 3) \) and \( (4, 9, \alpha) \) respectively and \( C \) is at a distance of \( 10 \) units from \( B \). The area (in sq. units) of \( \triangle ABC \) is:

If \( d_1 \) is the shortest distance between the lines \( x + 1 = 2y = -12z \) and \( x = y + 2 = 6z - 6 \) and \( d_2 \) is the shortest distance between the lines \( \dfrac{x - 1}{2} = \dfrac{y + 8}{-7} = \dfrac{z - 4}{5} \) and \( \dfrac{x - 1}{2} = \dfrac{y - 2}{1} = \dfrac{z - 6}{-3} \), then the value of \( \dfrac{32\sqrt{3}\, d_1}{d_2} \) is:

The distance of the point \( (-1,9,-16) \) from the plane \( 2x+3y-z=5 \) measured parallel to the line \( \frac{x+4}{3}=\frac{2-y}{4}=\frac{z-3}{12} \) is:

The distance of the point \( P(4,6,-2) \) from the line passing through the point \( (-3,2,3) \) and parallel to a line with direction ratios \( 3, 3, -1 \) is equal to:

The shortest distance between the lines \( \dfrac{x-1}{2}=\dfrac{y+8}{-7}=\dfrac{z-4}{5} \) and \( \dfrac{x-1}{2}=\dfrac{y-2}{1}=\dfrac{z-6}{-3} \) is:

Let the image of the point \( (1,0,7) \) in the line \(\displaystyle \frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3} \) be the point \( (\alpha,\beta,\gamma) \). Then which one of the following points lies on the line passing through \( (\alpha,\beta,\gamma) \) and making angles \(\displaystyle \frac{2\pi}{3} \) and \(\displaystyle \frac{3\pi}{4} \) with \( y \)-axis and \( z \)-axis respectively and an acute angle with \( x \)-axis?

Let the line passing through the points \( P(2,-1,2) \) and \( Q(5,3,4) \) meet the plane \( x-y+z=4 \) at the point \( R \). Then the distance of the point \( R \) from the plane \( x+2y+3z+2=0 \) measured parallel to the line \(\displaystyle \frac{x-7}{2}=\frac{y+3}{2}=\frac{z-2}{1} \) is equal to:

If the foot of the perpendicular from point \( (4, 3, 8) \) on the line \( L_1 : \dfrac{x-a}{1}=\dfrac{y-2}{3}=\dfrac{z-b}{4} \),\(\displaystyle 1 \neq 0\) is \( (3,5,7) \), then the shortest distance between the line \( L_1 \) and line \( L_2 : \dfrac{x-2}{3}=\dfrac{y-4}{4}=\dfrac{z-5}{5} \) is equal to:

If the distances of the point \( (1,2,a) \) from the line \( \dfrac{x-1}{1} = \dfrac{y}{2} = \dfrac{z-1}{1} \) along the lines \( L_{1}: \ \dfrac{x-1}{3} = \dfrac{y-2}{4} = \dfrac{z-a}{b} \) and \( L_{2}: \ \dfrac{x-1}{1} = \dfrac{y-2}{4} = \dfrac{z-a}{c} \) are equal, then \( a+b+c \) is equal to:

The shortest distance between the lines \( \frac{x-2}{3} = \frac{y+1}{2} = \frac{z-6}{2} \) and \( \frac{x-6}{3} = \frac{1-y}{2} = \frac{z+8}{0} \) is equal to ________.

The foot of perpendicular from the origin \( O \) to a plane \( P \) which meets the co-ordinate axes at the points \( A, B, C \) is \( (2,a,4) \), \( a\in \mathbb{N} \). If the volume of the tetrahedron \( OABC \) is \( 144 \,\text{unit}^{3} \), then which of the following points is NOT on \( P \)?

Let the plane \( ax + by + cz = d \) pass through \( (2,3,-5) \) and is perpendicular to the planes \( 2x + y - 5z = 10 \) and \( 3x + 5y - 7z = 12 \). If \( a,b,c,d \) are integers \( d>0 \) and \( \gcd\!\left(|a|,|b|,|c|,d\right)=1 \), then the value of \( a + 7b + c + 20d \) is equal to:

Let the image of the point \( P(1, 2, 6) \) in the plane passing through the points \( A(1, 2, 0) \), \( B(1, 4, 1) \) and \( C(0, 5, 1) \) be \( Q(\alpha, \beta, \gamma) \). Then \( \left( \alpha^{2} + \beta^{2} + \gamma^{2} \right) \) is equal to:

Let the plane containing the line of intersection of the planes \( P1: x + (\lambda + 4)y + z = 1 \) and \( P2: 2x + y + z = 2 \) pass through the points \( (0, 1, 0) \) and \( (1, 0, 1) \). Then the distance of the point \( (2\lambda, \lambda, -\lambda) \) from the plane \( P2 \) is:

The lines \( x = ay - 1 = z - 2 \) and \( x = 3y - 2 = bz - 2 \), \( (ab \neq 0) \) are coplanar, if:

The acute angle between the planes \( P_1 \) and \( P_2 \), when \( P_1 \) and \( P_2 \) are the planes passing through the intersection of the planes \( 5x + 8y + 13z - 29 = 0 \) and \( 8x - 7y + z - 20 = 0 \) and the points \( (2,1,3) \) and \( (0,1,2) \), respectively, is:

Let the point \( A \) be the foot of perpendicular drawn from the point \( P(a,b,0) \) on the line \( \dfrac{x-1}{2}=\dfrac{y-2}{1}=\dfrac{z-\alpha}{3} \). If the midpoint of the line segment \( PA \) is \( \left(0,\dfrac{3}{4},-\dfrac{1}{4}\right) \), then the value of \( a^2+b^2+\alpha^2 \) is equal to:

If the plane \( 2x+y-5z=0 \) is rotated about its line of intersection with the plane \( 3x-y+4z-7=0 \) by an angle of \( \frac{\pi}{2} \), then the plane after the rotation passes through the point:

Let the point \( (+1, \alpha, \beta) \) lie on the line of the shortest distance between the lines \( \dfrac{x + 2}{-3} = \dfrac{y - 2}{4} = \dfrac{z - 5}{2} \) and \( \dfrac{x + 2}{1} = \dfrac{y + 6}{2} = \dfrac{z - 1}{0} \).Then \( (\alpha - \beta)^{2} \) is equal to \( \_\_\_\_\_\_ \).

Let the area of the triangle formed by the lines \( x+2=y-1=z \), \( \frac{x-3}{5}=\frac{y}{-1}=\frac{z-1}{1} \), and \( \frac{x}{-3}=\frac{y-3}{3}=\frac{z-2}{1} \) be \( A \). Then \( A^{2} \) is equal to ______.

Let \( P_1 : \vec{r} \cdot (2\vec{i} + \vec{j} - 3\vec{k}) = 4 \) be a plane. Let \( P_2 \) be another plane which passes through the points \( (2, -3, 2) \), \( (2, -2, -3) \) and \( (1, -4, 2) \). If the direction ratios of the line of intersection of \( P_1 \) and \( P_2 \) be \( 16, \alpha, \beta \), then the value of \( \alpha + \beta \) is equal to _________.

Let the lines $L_1:\vec{r}=\hat{i}+2\hat{j}+3\hat{k}+\lambda(2\hat{i}+3\hat{j}+4\hat{k}),\lambda\in R$ and $L_2:\vec{r}=(4\hat{i}+\hat{j})+\mu(5\hat{i}+2\hat{j}+\hat{k}),\mu\in R$, intersect at the point $R$. Let $P$ and $Q$ be the point lying on lines $L_1$ and $L_2$, respectively, such that $\left|\overrightarrow{PR}\right|=\sqrt{29}$ and $\left|\overrightarrow{PQ}\right|=\sqrt{\frac{47}{3}}$. If the point $P$ lies in the first octant, then $27(QR)^2$ is equal to:

Let a straight line \( L \) pass through the point \( P(2,-1,3) \) and be perpendicular to the lines \(\displaystyle \frac{x-1}{2}=\frac{y+1}{1}=\frac{z-3}{-2} \) and \(\displaystyle \frac{x-3}{1}=\frac{y-2}{3}=\frac{z+2}{4} \). If the line \( L \) intersects the \( yz \)-plane at the point \( Q \), then the distance between the point \( P \) and \( Q \) is:

Let \( (\alpha,\beta,\gamma) \) be the image of the point \( (8,5,7) \) in the line: \( \dfrac{x-1}{2}=\dfrac{y+1}{3}=\dfrac{z-2}{5} \). Then \( \alpha+\beta+\gamma \) is equal to:

Let \( \lambda_1, \lambda_2 \) be the values of \( \lambda \) for which the points \( \left(\dfrac{5}{2},1,\lambda\right) \) and \( (-2,0,1) \) are at equal distance from the plane \( 2x+3y-6z+7=0 \). If \( \lambda_1>\lambda_2 \), then the distance of the point \( (\lambda_1-\lambda_2,\lambda_2,\lambda_1) \) from the line \( \dfrac{x-5}{1}=\dfrac{y-1}{2}=\dfrac{z+7}{2} \) is ________.

Let \( P \) be the plane passing through the point \( (1, 2, 3) \) and the line of intersection of the planes \( \vec{r} \cdot (\hat{i} + \hat{j} + 4\hat{k}) = 16 \) and \( \vec{r} \cdot (-\hat{i} + \hat{j} + \hat{k}) = 6 \). Then which of the following points does NOT lie on \( P \)?

Let the foot of the perpendicular from the point \( (1, 2, 4) \) on the line \( \frac{x+2}{4} = \frac{y-1}{2} = \frac{z+1}{3} \) be \( P \). Then the distance of \( P \) from the plane \( 3x + 4y + 12z + 23 = 0 \) is

Let the image of the point \( P(2,-1,3) \) in the plane \( x+2y-z=0 \) be \( Q \). Then the distance of the plane \( 3x+2y+z+29=0 \) from the point \( Q \) is :

Let the line \( l : x = \dfrac{1-y}{-2} = \dfrac{z-3}{\lambda}, \ \lambda \in \mathbb{R} \) meet the plane \( P : x + 2y + 3z = 4 \) at the point \( (\alpha,\beta,\gamma) \). If the angle between the line \( l \) and the plane \( P \) is \( \cos^{-1}\!\left(\sqrt{\dfrac{5}{14}}\right) \), then \( \alpha + 2\beta + 6\gamma \) is equal to __________.

Let the co-ordinates of one vertex of \( \triangle ABC \) be \( A(0, 2, \alpha) \) and the other two vertices lie on the line \( \dfrac{x+\alpha}{5} = \dfrac{y-1}{2} = \dfrac{z+4}{3} \). For \( \alpha \in \) \(\mathbb{Z}\) , if the area of \( \triangle ABC \) is \( 21 \) sq. units and the line segment \( BC \) has length \( 2\sqrt{21} \) units, then \( \alpha^{2} \) is equal to ________.

Let \( P(-2,-1,1) \) and \( Q\!\left( \dfrac{56}{17}, \dfrac{43}{17}, \dfrac{111}{17} \right) \) be the vertices of the rhombus \( PRQS \). If the direction ratios of the diagonal \( RS \) are \( \alpha, -1, \beta \), where both \( \alpha \) and \( \beta \) are integers of minimum absolute values, then \( \alpha^{2} + \beta^{2} \) is equal to __________.

Let \( (\alpha, \beta, \gamma) \) be the image of the point \( P (2, 3, 5) \) in the plane \( 2x + y - 3z = 6 \). Then \( \alpha + \beta + \gamma \) is equal to:

If the lines \( \dfrac{x-1}{1}=\dfrac{y-2}{2}=\dfrac{z+3}{1} \) and \( \dfrac{x-a}{2}=\dfrac{y+2}{3}=\dfrac{z-3}{1} \) intersect at the point \( P \), then the distance of the point \( P \) from the plane \( z=a \) is:

Let a line pass through two distinct points \( P(-2,-1,3) \) and \( Q \), and be parallel to the vector \( 3\hat{i} + 2\hat{j} + 2\hat{k} \). If the distance of the point \( Q \) from the point \( R(1,3,3) \) is \( 5 \), then the square of the area of \( \triangle PQR \) is equal to:

Let \( P \) be the plane, passing through the point \( (1,-1,-5) \) and perpendicular to the line joining the points \( (4,1,-3) \) and \( (2,4,3) \). Then the distance of \( P \) from the point \( (3,-2,2) \) is:

Let \( Q \) and \( R \) be the feet of perpendiculars from the point \( P(a,a,a) \) on the lines \( x=y, \ z=1 \) and \( x=-y, \ z=-1 \) respectively. If \( \angle QPR \) is a right angle, then \( 12a^2 \) is equal to:

If the image of the point \( P(1, 2, a) \) in the line \( \frac{x-6}{3} = \frac{y-7}{2} = \frac{7 - z}{2} \) is \( Q(5, b, c) \), then \( a^{2} + b^{2} + c^{2} \) is equal to:

Let \( L_1 : \frac{x-1}{1} = \frac{y-2}{-1} = \frac{z-1}{2} \) and \( L_2 : \frac{x+1}{-1} = \frac{y-2}{2} = \frac{z}{1} \) be two lines. Let \( L_3 \) be a line passing through the point \( (\alpha, \beta, \gamma) \) and be perpendicular to both \( L_1 \) and \( L_2 \). If \( L_3 \) intersects \( L_1 \), then \( |\,5\alpha - 11\beta - 8\gamma\,| \) equals:

The equation of the plane which contains the y-axis and passes through the point \( (1, 2, 3) \) is;

The distance of the point \( (1, -2, 3) \) from the plane \( x - y + z = 5 \) measured parallel to a line, whose direction ratios are \( 2, 3, -6 \), is:

Let \( P \) be the point of intersection of the lines \( \dfrac{x-2}{1} = \dfrac{y-4}{5} = \dfrac{z-2}{1} \) and \( \dfrac{x-3}{2} = \dfrac{y-2}{3} = \dfrac{z-3}{2} \). Then, the shortest distance of \( P \) from the line \( 4x = 2y = z \) is:

A plane \( P \) is parallel to two lines whose direction ratios are \( -2, 1, -3 \) and \( -1, 2, -2 \) and it contains the point \( (2, 2, -2) \). Let \( P \) intersect the co-ordinate axes at the points \( A, B, C \) making the intercepts \( \alpha, \beta, \gamma \). If \( V \) is the volume of the tetrahedron \( OABC \), where \( O \) is the origin, and \( p = \alpha + \beta + \gamma \), then the ordered pair \( (V, \ p) \) is equal to:

If the two lines \( l_1 : \dfrac{x-2}{3} = \dfrac{y+1}{-2}, \; z = 2 \) and \( l_2 : \dfrac{x-1}{1} = \dfrac{2y+3}{\alpha} = \dfrac{z+5}{2} \) are perpendicular, then an angle between the lines \( l_2 \) and \( l_3 : \dfrac{1-x}{3} = \dfrac{2y-1}{-4} = \dfrac{z}{4} \) is:

Let the equation of the plane, that passes through the point \( (1,4,-3) \) and contains the line of intersection of the planes \( 3x-2y+4z-7=0 \) and \( x+5y-2z+9=0 \), be \( \alpha x + \beta y + \gamma z + 3 = 0 \), then \( \alpha + \beta + \gamma \) is equal to:

Consider a line \( L \) passing through the points \( P(1,2,1) \) and \( Q(2,1,-1) \). If the mirror image of the point \( A(2,2,2) \) in the line \( L \) is \( (\alpha,\beta,\gamma) \), then \( \alpha+\beta+6\gamma \) is equal to ________________.

If \( (x,y,z) \) be an arbitrary point lying on a plane \( P \) which passes through the points \( (42,0,0) \), \( (0,42,0) \) and \( (0,0,42) \), then the value of the expression \(\displaystyle 3 + \frac{x-11}{(y-19)^{2}(z-12)^{2}} + \frac{y-19}{(x-11)^{2}(z-12)^{2}} + \frac{z-12}{(x-11)^{2}(y-19)^{2}} + \frac{x+y+z}{14(x-11)(y-19)(z-12)} \) is equal to:

The plane \( 2x - y + z = 4 \) intersects the line segment joining the points \( A(a, -2, 4) \) and \( B(2, b, -3) \) at the point \( C \) in the ratio \( 2 : 1 \) and the distance of the point \( C \) from the origin is \( \sqrt{5} \). If \( ab < 0 \) and \( P \) is the point \( (a - b, 2b - a) \) then \( CP^2 \) is equal to:

Let a line \( L \) be perpendicular to both the lines \( L_{1} : \, \dfrac{x+1}{3} = \dfrac{y+3}{5} = \dfrac{z+5}{7} \) and \( L_{2} : \, \dfrac{x-2}{1} = \dfrac{y-4}{4} = \dfrac{z-6}{7} \). If \( \theta \) is the acute angle between the lines \( L \) and \( L_{3} : \, \dfrac{x-8/7}{2} = \dfrac{y-4/7}{1} = \dfrac{z}{2} \), then \( \theta \) is equal to:

Let a line \( L \) pass through the point \( P(2,3,1) \) and be parallel to the line \( x + 3y - 2z - 2 = 0 = x - y + 2z \). If the distance of \( L \) from the point \( (5,3,8) \) is \( \alpha \), then \( 3\alpha^{2} \) is equal to __________.

If two distinct points \( Q, R \) lie on the line of intersection of the planes \( -x + 2y - z = 0 \) and \( 3x - 5y + 2z = 0 \) and \( PQ = PR = \sqrt{18} \) where the point \( P \) is \( (1,-2,3) \), then the area of the triangle \( PQR \) is equal to:

Let the foot of perpendicular from the point \( A(4, 3, 1) \) on the plane \( P : x - y + 2z + 3 = 0 \) be \( N \). If \( B(5, \alpha, \beta), \ \alpha, \beta \in \mathbb{Z} \) is a point on plane \( P \) such that the area of the triangle \( ABN \) is \( 3\sqrt{2} \), then \( \alpha^{2} + \beta^{2} + \alpha\beta \) is equal to ________.

Let \( P(\alpha,\beta,\gamma) \) be the image of the point \( Q(1,6,4) \) in the line \(\displaystyle \frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3} \). Then \( 2\alpha+\beta+\gamma \) equal to ________.

Let a line with direction ratios \( a,-4a,-7 \) be perpendicular to the lines with direction ratios \( 3,-1,2b \) and \( b,a,-2 \). If the point of intersection of the line \(\displaystyle \frac{x+1}{a^2+b^2}=\frac{y-2}{a^2-b^2}=\frac{z}{1} \) and the plane \( x-y+z=0 \) is \( (\alpha,\beta,\gamma) \), then \( \alpha+\beta+\gamma \) is equal to __________.

Let a unit vector $\widehat{OP}$ make angles $\alpha, \beta, \gamma$ with the positive directions of the co-ordinate axes $OX, OY, OZ$ respectively, where $\beta \in \left(0, \frac{\pi}{2}\right)$. If $\widehat{OP}$ is perpendicular to the plane through points $(1, 2, 3)$, $(2, 3, 4)$ and $(1, 5, 7)$, then which one of the following is true?

Let \( P \) and \( Q \) be the points on the line \( \dfrac{x+3}{8} = \dfrac{y-4}{2} = \dfrac{z+1}{2} \) which are at a distance of \( 6 \) units from the point \( R(1, 2, 3) \). If the centroid of the triangle \( PQR \) is \( (\alpha,\beta,\gamma) \), then \( \alpha^{2} + \beta^{2} + \gamma^{2} \) is:

Let \( O \) be the origin, and \( M \) and \( N \) be the points on the lines \( \dfrac{x-5}{4}=\dfrac{y-4}{1}=\dfrac{z-5}{3} \) and \( \dfrac{x+8}{12}=\dfrac{y+2}{5}=\dfrac{z-11}{9} \) respectively such that \( MN \) is the shortest distance between the given lines. Then \( \overrightarrow{OM}\cdot\overrightarrow{ON} \) is equal to ________.

Let the plane \( P: 8x + \alpha_1 y + \alpha_2 z + 12 = 0 \) be parallel to the line \( L : \frac{x+2}{2} = \frac{y-3}{3} = \frac{z+4}{5} \). If the intercept of \( P \) on the \( y \)-axis is \( 1 \), then the distance between \( P \) and \( L \) is:

Let \( P \) be the foot of the perpendicular from the point \( (1, 2, 2) \) on the line \( L : \dfrac{x-1}{1}=\dfrac{y+1}{-1}=\dfrac{z-2}{2} \).
Let the line \( \vec{r}=(\hat{-i}+\hat{j}-2\hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k}),\, \lambda\in\mathbb{R} \), intersect the line \( L \) at \( Q \). Then \( 2(PQ)^2 \) is equal to:

Let \( P \) be the plane passing through the points \( (5,3,0) \), \( (13,3,-2) \) and \( (1,6,2) \). For \( \alpha \in \mathbb{N} \); if the distances of the points \( A(3,4,\alpha) \) and \( B(2,\alpha,a) \) from the plane \( P \) are \( 2 \) and \( 3 \) respectively, then the positive value of \( a \) is:

Consider the line \( L \) passing through the points \( (1, 2, 3) \) and \( (2, 3, 5) \). The distance of the point \(\displaystyle \left( \frac{11}{3}, \frac{11}{3}, \frac{19}{3} \right) \) from the line \( L \) along the line \(\displaystyle \frac{3x-11}{2} = \frac{3y-11}{1} = \frac{3z-19}{2} \) is equal to

The foot of the perpendicular from a point on the circle \( x^2 + y^2 = 1, \, z = 0 \) to the plane \( 2x + 3y + z = 6 \) lies on which one of the following curves?

If the plane \( P \) passes through the intersection of two mutually perpendicular planes \( 2x + ky - 5z = 1 \) and \( 3kx - ky + z = 5, \; k < 3 \) and intercepts a unit length on positive \( x \)-axis, then the intercept made by the plane \( P \) on the \( y \)-axis is:

The distance of the point \( (7,-3,-4) \) from the plane passing through the points \( (2,-3,1) \), \( (-1,1,-2) \) and \( (3,-4,2) \) is:

The square of the distance of the point \(\displaystyle \left(\frac{15}{7},\frac{32}{7},7\right) \) from the line \(\displaystyle \frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7} \) in direction of the vector \( \hat{i}+4\hat{j}+7\hat{k} \) is:

The largest value of \( a \), for which the perpendicular distance of the plane containing the lines \( \vec{r} = (\hat{i}+\hat{j}) + \lambda(\hat{i}+a\hat{j}-\hat{k}) \) and \( \vec{r} = (\hat{i}+\hat{j}) + \mu(-\hat{i}+\hat{j}-a\hat{k}) \) from the point \( (2,1,4) \) is \( \sqrt{3} \), is ______.

Let a line passing through the point \( (-1, 2, 3) \) intersects the lines \( L_1 : \dfrac{x-1}{3}=\dfrac{y-2}{2}=\dfrac{z+1}{-2} \) at \( M(\alpha,\beta,\gamma) \) and \( L_2 : \dfrac{x+2}{-3}=\dfrac{y-2}{-2}=\dfrac{z-1}{4} \) at \( N(a,b,c) \). Then, the value of \( \dfrac{(\alpha+\beta+\gamma)^2}{(a+b+c)^2} \) equals _______.

A line with direction ratios \( 1,-1,2 \) intersects the lines \( \dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z+1}{3} \) and \( \dfrac{x+1}{-1}=\dfrac{y-2}{1}=\dfrac{z}{4} \) at the points \( P \) and \( Q \), respectively. If the length of the line segment \( PQ \) is \( a \), then \( 225a^{2} \) is equal to:

Let \( S \) be the set of all values of \( \lambda \), for which the shortest distance between the lines \( \dfrac{x-\lambda}{0}=\dfrac{y-3}{4}=\dfrac{z+6}{1} \) and \( \dfrac{x+\lambda}{3}=\dfrac{y}{-4}=\dfrac{z-6}{0} \) is \( 13 \). Then \( 8\left|\sum\limits_{\lambda\in S}\lambda\right| \) is equal to:

If the shortest distance between the lines \( \vec{r}_{1}=\alpha\hat{i}+2\hat{j}+2\hat{k}+\lambda(\hat{i}-2\hat{j}+2\hat{k}),\, \lambda\in\mathbb{R},\, \alpha>0 \) and \( \vec{r}_{2}=-4\hat{i}-\hat{k}+\mu(3\hat{i}-2\hat{j}-2\hat{k}),\, \mu\in\mathbb{R} \) is \( 9 \), then \( \alpha \) is equal to ______.

If the equation of plane passing through the mirror image of a point \( (2, 3, 1) \) with respect to line \( \dfrac{x + 1}{2} = \dfrac{y - 3}{1} = \dfrac{z - 2}{-1} \) and containing the line \( \dfrac{x - 2}{3} = \dfrac{1 - y}{2} = \dfrac{z + 1}{1} \) is \( \alpha x + \beta y + \gamma z = 24 \), then \( \alpha + \beta + \gamma \) is equal to:

Let the plane \( x + 3y - 2z + 6 = 0 \) meet the co-ordinate axes at the points \( A, B, C \). If the orthocenter of the triangle \( ABC \) is \( \left(\alpha, \beta, \dfrac{6}{7}\right) \), then \( 98(\alpha+\beta)^2 \) is equal to____.

If equation of the plane that contains the point \( (-2, 3, 5) \) and is perpendicular to each of the planes \( 2x + 4y + 5z = 8 \) and \( 3x - 2y + 3z = 5 \) is \( \alpha x + \beta y + \gamma z + 97 = 0 \) then \( \alpha + \beta + \gamma = \)

Let the plane \( P : 4x - y + z = 10 \) be rotated by an angle \( \dfrac{\pi}{2} \) about its line of intersection with plane \( x + y - z = 4 \). If \( \alpha \) is the distance of the points \( (2, 3, -4) \) from the new position of the plane \( P \), then \( 35\alpha \) is equal to:

A plane \( P \) contains the line \( x+2y+3z+1=0 = x-y-z-6 \), and is perpendicular to the plane \( -2x+y+z+8=0 \). Then which of the following points lies on \( P \)?

Let the line of the shortest distance between the lines \( L_1 : \vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}) \) and \( L_2 : \vec{r} = (4\hat{i} + 5\hat{j} + 6\hat{k}) + \mu(\hat{i} + \hat{j} - \hat{k}) \) intersect \( L_1 \) and \( L_2 \) at \( P \) and \( Q \) respectively. If \( (\alpha, \beta, \gamma) \) is the mid point of the line segment \( PQ \), then \( 2(\alpha + \beta + \gamma) \) is equal to ______.

The perpendicular distance, of the line \(\displaystyle \frac{x-1}{2}=\frac{y+2}{-1}=\frac{z+3}{2} \) from the point \( P(2,-10,1) \), is:

Let \( L_{1} : \dfrac{x-1}{2} = \dfrac{y-2}{3} = \dfrac{z-3}{4} \) and \( L_{2} : \dfrac{x-2}{3} = \dfrac{y-4}{4} = \dfrac{z-5}{5} \) be two lines. Then which of the following points lies on the line of the shortest distance between \( L_{1} \) and \( L_{2} \)?

If the equation of the plane passing through the line of intersection of the planes \( 2x - y + z = 3 \), \( 4x - 3y + 5z + 9 = 0 \) and parallel to the line \( \dfrac{x+1}{-2} = \dfrac{y+3}{4} = \dfrac{z-2}{5} \) is \( ax + by + cz + 6 = 0 \), then \( a + b + c \) is equal to:

A vector \( \vec{v} \) in the first octant is inclined to the x-axis at \( 60^\circ \), to the y-axis at \( 45^\circ \) and to the z-axis at an acute angle. If a plane passing through the points \( (\sqrt{2},-1,1) \) and \( (a,b,c) \), is normal to \( \vec{v} \), then:

Let a line \( L \) passing through the point \( (1,1,1) \) be perpendicular to both the vectors \( 2\hat{i}+2\hat{j}+\hat{k} \) and \( \hat{i}+2\hat{j}+2\hat{k} \). If \( P(a,b,c) \) is the foot of perpendicular from the origin on the line \( L \), then the value of \( 34(a+b+c) \) is:

Let the line \( L \) be the projection of the line \( \frac{x-1}{2} = \frac{y-3}{1} = \frac{z-4}{2} \) in the plane \( x - 2y - z = 3 \). If \( d \) is the distance of the point \( (0, 0, 6) \) from \( L \), then \( d^2 \) is equal to ______.

Equations of a plane at a distance \( \sqrt{\dfrac{2}{21}} \) from the origin, which contains the line of intersection of the planes \( x - y - z - 1 = 0 \) and \( 2x + y - 3z + 4 = 0 \), is:

The distance of the point \( (7, 10, 11) \) from the line \( \frac{x-4}{1} = \frac{y-4}{0} = \frac{z-2}{3} \) along the line \( \frac{x-9}{2} = \frac{y-13}{3} = \frac{z-17}{6} \) is:

Let \( l_1 \) be the line in \( xy \)-plane with \( x \) and \( y \) intercepts \( \frac{1}{8} \) and \( \frac{1}{4\sqrt{2}} \) respectively, and \( l_2 \) be the line in \( zx \)-plane with \( x \) and \( z \) intercepts \( -\frac{1}{8} \) and \( -\frac{1}{6\sqrt{3}} \) respectively. If \( d \) is the shortest distance between the line \( l_1 \) and \( l_2 \), then \( d^{-2} \) is equal to __________.

The equation of the plane passing through the line of intersection of the planes \( \vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1 \) and \( \vec{r} \cdot (2\hat{i} + 3\hat{j} - \hat{k}) + 4 = 0 \) and parallel to the \( x \)-axis is:

The shortest distance, between lines \( L_1 \) and \( L_2 \), where \( L_1:\; \dfrac{x-1}{2}=\dfrac{y+1}{-3}=\dfrac{z+4}{2} \) and \( L_2 \) s the line, passing through the line, passing through the points \( A=(-4,4,3), B=(-1,6,3) \) and perpendicular to the line \( \dfrac{x-3}{-2}=\dfrac{y}{3}=\dfrac{z-1}{1} \) is:

The shortest distance between the lines \( \frac{x-5}{1}=\frac{y-2}{2}=\frac{z-4}{-3} \) and \( \frac{x+3}{1}=\frac{y+5}{4}=\frac{z-1}{-5} \) is:

Let \( PQR \) be a triangle with \( R(-1, 4, 2) \). Suppose \( M(2, 1, 2) \) is the mid point of \( PQ \). The distance of the centroid of \( \Delta PQR \) from the point of intersection of the lines \( \frac{x-2}{0} = \frac{y}{2} = \frac{z+3}{-1} \) and \( \frac{x-1}{1} = \frac{y+3}{-3} = \frac{z+1}{1} \) is:

The distance, of the point \( (7, -2, 11) \) from the line \( \dfrac{x-6}{1} = \dfrac{y-4}{0} = \dfrac{z-8}{3} \) along the line \( \dfrac{x-5}{2} = \dfrac{y-1}{-3} = \dfrac{z-5}{6} \), is:

Let \( (\alpha, \beta, \gamma) \) be the foot of perpendicular from the point \( (1, 2, 3) \) on the line \( \frac{x + 3}{5} = \frac{y - 1}{2} = \frac{z + 4}{3} \). Then \( 19(\alpha + \beta + \gamma) \) is equal to:

Let \( S \) be the mirror image of the point \( Q(1,\,3,\,4) \) with respect to the plane \( 2x-y+z+3=0 \) and let \( R(3,\,5,\gamma) \) be a point of this plane. Then the square of the length of the line segment \( SR \) is ______.

Let the image of the point \( P(1,6,a) \) in the line \( L : \, \dfrac{x}{1} = \dfrac{y-1}{2} = \dfrac{z-a+1}{b}, \, b>0 \), be \( \left(\dfrac{a}{3},0,a+c\right) \). If \( S(\alpha,\beta,\gamma), \, \alpha>0 \), is the point on \( L \) such that the distance of \( S \) from the foot of perpendicular from the point \( P \) on \( L \) is \( 2\sqrt{14} \), then \( \alpha + \beta + \gamma \) is equal to:

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JEE Main 2023 6th April Morning Shift

Let the image of the point \( P(1, 2, 3) \) in the plane \( 2x - y + z = 9 \) be \( Q \). If the coordinates of the point \( R \) are \( (6, 10, 7) \), then the square of the area of the triangle \( PQR \) is __________.