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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 ____________.

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:

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 ______________ .

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 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:

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 \(a,b\in \mathbb{R}\). 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 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 \( \lambda \) 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 ________.

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

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?

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 \( \underline{\quad} \).

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 _______.

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 \( P \)?

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

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:

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

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 \( P \)?

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 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:

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:

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:

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 \( L \) 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 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:

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 ______.

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JEE Main 2021 27th August Morning Shift

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: