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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 ______.

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:

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 \( P \)?

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:

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:

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 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 \( Q \) 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 ________.

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:

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

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 \( \dfrac{x-2}{3}=\dfrac{y+1}{-2}=\dfrac{z+3}{-1} \) 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 \( P \) 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:

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 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 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 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 ____________.

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

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 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:

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 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:

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:

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:

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 __________.

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:

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:

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 \( P_1 : \vec{r} \cdot (2\vec{i} + \vec{j} - 3\vec{k}) = 4 \) 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 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 \( P(-2,-1,1) \) 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 __________.

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:

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 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 __________.

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 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 \( l_1 \) 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 __________.

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JEE Main 2022 27th June Morning Shift

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 ____________.