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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:

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 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?

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:

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:

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.

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 :

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:

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 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 $\theta$ 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 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 __________ .

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:

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 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:

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?

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 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 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:

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:

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 ______.

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 \( \_\_\_\_\_ \).

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 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:

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:

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 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 \( P_{1} \) 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 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 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 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:

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 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:

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 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 \( (\alpha, \beta, \gamma) \) 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 \( P \) 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:

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 \( L \) 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 __________.

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 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 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 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:

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:

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:

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:

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:

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:

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JEE Main 2023 13th April Evening Shift

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: