3D Geometry PYQs - Last 10 Years

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MathematicsAlgebra2017-2026

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Practice 346 JEE Main Mathematics questions from 3D Geometry. Use the year-wise and type-wise breakdown to prioritize recent PYQs, then continue into the question list below.

346

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Mathematics / Algebra

2017-2026

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228

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2022-2026

346

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2017-2026

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Question Types

346PYQs

MCQ75.7%

INTEGER24.3%

Difficulty Mix

#1 Medium315

#2 Hard20

#3 Easy11

228 in last 5 years346 in last 10 years

Last 10 Years 3D Geometry Questions

Showing 50 of 346 filtered questions.

Let the foot of perpendicular from a point P(1, 2, \(-\)1) to the straight line \(L:{x \over 1} = {y \over 0} = {z \over { - 1}}\) be N. Let a line be drawn from P parallel to the plane x + y + 2z = 0 which meets L at point Q. If \(\alpha\)...

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

INTEGER+4 / -12021

Let \(\alpha\) be the angle between the lines whose direction cosines satisfy the equations l + m \(-\) n = 0 and l2 + m2 \(-\) n2 = 0. Then the value of sin4\(\alpha\) + cos4\(\alpha\) is :

The equation of the line through the point (0, 1, 2) and perpendicular to the line \({{x - 1} \over 2} = {{y + 1} \over 3} = {{z - 1} \over { - 2}}\) 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 \(\overrightarrow {OP}\) on this plane is of length :

A line 'l' passing through origin is perpendicular to the lines\({l_1}:\overrightarrow r = (3 + t)\widehat i + ( - 1 + 2t)\widehat j + (4 + 2t)\widehat k\)$${l_2}:\overrightarrow r = (3 + 2s)\widehat i + (3 + 2s)\widehat j + (2 + s)\wideh...

INTEGER+4 / -12021

The distance of the point (1, 1, 9) from the point of intersection of the line
\({{x - 3} \over 1} = {{y - 4} \over 2} = {{z - 5} \over 2}\)
and the plane x + y + z = 17 is :

The equation of the plane passing through the point (1, 2, -3) and perpendicular to the
planes 3x + y - 2z = 5 and 2x - 5y - z = 7, is :

The vector equation of the plane passing through the intersection of the planes \(\overrightarrow r .\left( {\widehat i + \widehat j + \widehat k} \right) = 1\) and \(\overrightarrow r .\left( {\widehat i - 2\widehat j} \right) = - 2\), an...

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

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 \({{\sqrt 7 } \over {2\sqrt 2 }}\), then the value of | \(\lambda\) | is ...

INTEGER+4 / -12021

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

Let L be the line of intersection of planes \(\overrightarrow r .(\widehat i - \widehat j + 2\widehat k) = 2\) and \(\overrightarrow r .(2\widehat i + \widehat j - \widehat k) = 2\). If \(P(\alpha ,\beta ,\gamma )\) is the foot of perpendic...

Let P be a plane passing through the points (1, 0, 1), (1, \(-\)2, 1) and (0, 1, \(-\)2). Let a vector \(\overrightarrow a = \alpha \widehat i + \beta \widehat j + \gamma \widehat k\) be such that \(\overrightarrow a\) is parallel to the ...

INTEGER+4 / -12021

Consider the line L given by the equation \({{x - 3} \over 2} = {{y - 1} \over 1} = {{z - 2} \over 1}\). 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...

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

The distance of line \(3y - 2z - 1 = 0 = 3x - z + 4\) from the point (2, \(-\)1, 6) 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 ax + by + cz + 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 (a2 + b2 + c2 + d2) is _________.

INTEGER+4 / -12021

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 :

INTEGER+4 / -12021

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

INTEGER+4 / -12021

Let P be a plane containing the line \({{x - 1} \over 3} = {{y + 6} \over 4} = {{z + 5} \over 2}\) and parallel to the line \({{x - 1} \over 4} = {{y - 2} \over { - 3}} = {{z + 5} \over 7}\). If the point (1, \(-\)1, \(\alpha\)) lies on the...

INTEGER+4 / -12021

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 __...

INTEGER+4 / -12021

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

Let P be an arbitrary point having sum of the squares of the distances from the planes x + y + z = 0, lx \(-\) nz = 0 and x \(-\) 2y + z = 0, equal to 9. If the locus of the point P is x2 + y2 + z2 = 9, then the value of l \(-\) n is equal ...

INTEGER+4 / -12021

If the equation of plane passing through the mirror image of a point (2, 3, 1) with respect to line \({{x + 1} \over 2} = {{y - 3} \over 1} = {{z + 2} \over { - 1}}\) and containing the line $${{x - 2} \over 3} = {{1 - y} \over 2} = {{z + 1...

Let the position vectors of two points P and Q be 3\(\widehat i\) \(-\) \(\widehat j\) + 2\(\widehat k\) and \(\widehat i\) + 2\(\widehat j\) \(-\) 4\(\widehat k\), respectively. Let R and S be two points such that the direction ratios of l...

If for a > 0, the feet of perpendiculars from the points A(a, \(-\)2a, 3) and B(0, 4, 5) on the plane lx + my + nz = 0 are points C(0, \(-\)a, \(-\)1) and D respectively, then the length of line segment CD is equal to :

Let P be a plane lx + my + nz = 0 containing the line, \({{1 - x} \over 1} = {{y + 4} \over 2} = {{z + 2} \over 3}\). If plane P divides the line segment AB joining points A(\(-\)3, \(-\)6, 1) and B(2, 4, \(-\)3) in ratio k : 1 then the val...

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

INTEGER+4 / -12021

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 $$3 + {{x - 11} \over {{{(y - 19)}^2}{{(z - 12)}^2}}} + {{y - 19} \over {{{(x - 11...

If the foot of the perpendicular from point (4, 3, 8) on the line \({L_1}:{{x - a} \over l} = {{y - 2} \over 3} = {{z - b} \over 4}\), l \(\ne\) 0 is (3, 5, 7), then the shortest distance between the line L1 and line $${L_2}:{{x - 2} \over ...

The projection of the line segment joining the
points (1, –1, 3) and (2, –4, 11) on the line
joining the points (–1, 2, 3) and (3, –2, 10)
is ____________.

INTEGER+4 / -02020

If the distance between the plane,
23x – 10y – 2z + 48 = 0 and the plane
containing the lines
\({{x + 1} \over 2} = {{y - 3} \over 4} = {{z + 1} \over 3}\) and
$${{x + 3} \over 2} = {{y + 2} \over 6} = {{z - 1} \over \lambda }\left( {\lambd...

INTEGER+4 / -02020

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

The mirror image of the point (1, 2, 3) in a plane
is \(\left( { - {7 \over 3}, - {4 \over 3}, - {1 \over 3}} \right)\). Which of the following
points lies on this plane ?

Let P be a plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R be any point
(2, 1, 6). Then the image of R in the plane P is :

If the foot of the perpendicular drawn from the point (1, 0, 3) on a line passing through (\(\alpha\), 7, 1)
is
\(\left( {{5 \over 3},{7 \over 3},{{17} \over 3}} \right)\), then \(\alpha\) is equal to______.

INTEGER+4 / -02020

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

A plane P meets the coordinate axes at A, B
and C respectively. The centroid of \(\Delta\)ABC is
given to be (1, 1, 2). Then the equation of the
line through this centroid and perpendicular to
the plane P is :

If (a, b, c) is the image of the point (1, 2, -3) in the line \({{x + 1} \over 2} = {{y - 3} \over { - 2}} = {z \over { - 1}}\), then a + b + c is :

If for some \(\alpha\) \(\in\) R, the lines
L1 : \({{x + 1} \over 2} = {{y - 2} \over { - 1}} = {{z - 1} \over 1}\) and
L2 : \({{x + 2} \over \alpha } = {{y + 1} \over {5 - \alpha }} = {{z + 1} \over 1}\) are coplanar,
then the line L2
...

If the equation of a plane P, passing through the intersection of the planes, x + 4y - z + 7 = 0
and 3x + y + 5z = 8 is ax + by + 6z = 15 for some a, b \(\in\) R, then the distance of the point
(3, 2, -1) from the plane P is...........

INTEGER+4 / -02020

The distance of the point (1, –2, 3) from the plane x – y + z = 5 measured parallel to the line \({x \over 2} = {y \over 3} = {z \over { - 6}}\) is :

The foot of the perpendicular drawn from the
point (4, 2, 3) to the line joining the points
(1, –2, 3) and (1, 1, 0) lies on the plane :

The plane which bisects the line joining, the
points (4, –2, 3) and (2, 4, –1) at right angles
also passes through the point :

Let a plane P contain two lines
\(\overrightarrow r = \widehat i + \lambda \left( {\widehat i + \widehat j} \right)\), \(\lambda \in R\) and
\(\overrightarrow r = - \widehat j + \mu \left( {\widehat j - \widehat k} \right)\), $$\mu \in...

INTEGER+4 / -02020

The plane passing through the points (1, 2, 1),
(2, 1, 2) and parallel to the line, 2x = 3y, z = 1
also passes through the point :

A plane passing through the point (3, 1, 1)
contains two lines whose direction ratios are 1,
–2, 2 and 2, 3, –1 respectively. If this plane also
passes through the point (\(\alpha\), –3, 5), then
\(\alpha\) is
equal to:

The equation of the line passing through (–4, 3, 1), parallel
to the plane x + 2y – z – 5 = 0 and intersecting
the line \({{x + 1} \over { - 3}} = {{y - 3} \over 2} = {{z - 2} \over { - 1}}\) is :