List of top Mathematics Questions on Three Dimensional Geometry asked in JEE Main

The acute angle between the planes P₁ and P₂, when P₁ and P₂ 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 π/2, then the plane after the rotation passes through the point:

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
$\frac{x-2}{3} = \frac{y+1}{-2} = \frac{z+3}{-1}$
lie on the plane px – qy + z = 5, for some p, q ∈ ℝ. The shortest distance of the plane from the origin 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
$\vec{r}=−\hat{k}+λ(\hat{i}+\hat{j}+2\hat{k}), λ ∈ R.$
Then, which of the following points lies on T?

Let the locus of the centre $(\alpha, \beta), $ $\beta>0$, of the circle which touches the circle $x ^2+( y -1)^2=1$ externally and also touches the x-axis be L Then the area bounded by L and the line y=4 is :

Let the plane
$P : \stackrel{→}{r} . \stackrel{→}{a} = d$
contain the line of intersection of two planes
$\stackrel{→}{r} . ( \hat{i} + 3\hat{j} - \hat{k} ) = 6$
and
$\stackrel{→}{r} . ( -6\hat{i} + 5\hat{j} - \hat{k} ) = 7$
. If the plane P passes through the point (2, 3, 1/2),
then the value of $\frac{| 13a→|² }{d²}$ 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, β, γ). If a + b + c = 5, then 7β – 9γ is equal to __________.

If the two lines
$l1:\frac{(x−2)}{3}=\frac{(y+1)}{−2},z=2 $
and
$ l2:\frac{(x−1)}{1}=\frac{(2y+3)}{α}=\frac{(z+5)}{2} $
are perpendicular, then an angle between the lines l₂and
$l3:\frac{(1−x)}{3}=\frac{(2y−1)}{−4}=\frac{z}{4} $
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
$(2,−\frac{1}{2},2) $
in the rotated plane is B( a, b, c),then

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

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 line
$\frac{x - 3}{7} = \frac{y - 2}{-1} = \frac{z - 3}{-4}$
intersect the plane containing the lines
$\frac{x - 4}{1} = \frac{y + 1}{-2} = \frac{z}{1}$ and $4ax-y+5z-7a = 0 = 2x-5y-z-3, a∈R$
at the point P(α, β, γ). Then the value of α + β + γ equals _____.

A hall has a square floor of dimension 10 m $\times$ 10 m (see the figure) and vertical walls. If the angle GPH between the diagonals AG and BH is $\cos^{-1}\frac{1}{5}$, then the height of the hall (in meters) is :

A plane passes through the points A(1, 2, 3), B(2, 3, 1) and C(2, 4, 2). If O is the origin and P is (2, -1, 1), then the projection of $\vec{OP}$ on this plane is of length :

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 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 foot of perpendicular from a point $P(1, 2, -1)$ to the straight line $L: \frac{x}{1} = \frac{y}{0} = \frac{z}{-1}$ 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 ________.

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

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 _________.

The square of the distance of the point of intersection of the line $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z+1}{6}$ and the plane $2x - y + z = 6$ from the point $(-1, -1, 2)$ is _____________

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