The vector equation of the plane through the line of intersection of the planes $x + y + z = 1{\rm{ }}\,{\rm{and}}\,{\rm{ 2}}x + 3y + 4z = 5$ which is perpendicular to the plane $x - y + z = 0{\rm{ }}$ is

$r \times (\hat i + \hat k) + 2 = 0$

$r \times (\hat i - \hat k) + 2 = 0$

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JEE • MATHEMATICS • 3D PLANES

3d planes practice · Medium

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 1-30 of 59 questions

1 Medium

1 of 59

Analyzing +4 | -1

If the planes

x-c y-b z=0, c x-y+a z=0

and

b x+a y-z=0

pass through a straight line, then the value of

a^2+b^2+c^2+2 a b c

is:

View answer and explanation

Correct option: D

The correct answer is D (1) because if the planes pass through a straight line, the determinant of the coefficients of x, y, and z must be zero, leading to the given expression equaling 1.

2 Medium

2 of 59

Analyzing +4 | -1

Let the plane

2 x-3 y+9 z=0

P

, the equation of line passing through

(0,1,-1)

and lying-in plane

4 x-2 y+7 z+9=0

& parallel to

P

is:

View answer and explanation

Correct option: B

\quad \frac{x}{-3}=\frac{y-1}{22}=\frac{z+1}{8}

Option B (

\frac{x}{-3}=\frac{y-1}{22}=\frac{z+1}{8}

) is correct because it represents the line passing through the given point and parallel to the plane P, lying in the plane

4x - 2y + 7z + 9 = 0

3 Medium

3 of 59

Application +4 | -1

Distance of point

(-2,3,4)

from the line

\frac{x+2}{3}=\frac{2 y+3}{4}=\frac{3 x+4}{5}

measured parallel to the plane

8 x+6 y-9 z+1=0

View answer and explanation

Correct option: B

\frac{\sqrt{401}}{2}

Option B (

\frac{\sqrt{401}}{2}

) is correct because it represents the distance calculated by finding the intersection point of the line and a line parallel to the plane.

4 Medium

4 of 59

Application +4 | -1

A variable plane is at a constant distance '

r

' from the origin and meets the axes in

A, B, C

then, the locus of the centroid of the tetrahedron

{OABC}

is-

View answer and explanation

Correct option: A

\frac{1}{{x}^2}+\frac{1}{{y}^2}+\frac{1}{{z}^2}=\frac{16}{{r}^2}

Option A (

\frac{1}{{x}^2}+\frac{1}{{y}^2}+\frac{1}{{z}^2}=\frac{16}{{r}^2}

) is correct because it represents the locus of the centroid of the tetrahedron, derived from the equation of the plane and the centroid coordinates.

5 Medium

5 of 59

Application +4 | -1

If the lines

\frac{x-1}{1}=\frac{y-2}{2}=\frac{z+3}{1}

and

\frac{x-a}{2}=\frac{y+2}{3}=\frac{z-3}{1}

intersect at the point

P

, then the distance of the point

P

from the plane

z=a

View answer and explanation

Correct option: A

Option A (28) is correct because after finding the intersection point P and the value of a, the distance from P to the plane z=a is calculated correctly.

6 Medium

6 of 59

Application +4 | -1

If the greatest and the least resultants of two forces are

{P}

and

{Q}

respectively, then the resultant of these forces, when acting at right angles, will be

View answer and explanation

Correct option: D

\sqrt{\frac{\left({P}^2+{Q}^2\right)}{2}}

The correct answer is D (

\sqrt{\frac{\left({P}^2+{Q}^2\right)}{2}}

) because it correctly calculates the resultant of two forces acting at right angles, given their greatest and least resultants.

7 Medium

7 of 59

Application +4 | -1

A ray of light passing through the point

A(1,2,3)

, strikes the plane

x+y+z=12

B

and on reflection passes through point

{C}(3,5,9)

. Coordinates of point

{B}

is-

View answer and explanation

Correct option: B

(-7,0,19)

Option B ((-7,0,19)) is correct because it is the point of intersection of the reflected ray BC with the plane x+y+z=12, calculated using the image of point A.

8 Medium

8 of 59

Application +4 | -1

Let the image of the point

P(2,-1,3)

in the plane

x+2 y-z=0

{Q}

. Then the distance of the plane

3 x+2 y+z+29=0

from the point

Q

View answer and explanation

Correct option: D

3 \sqrt{14}

Option D (

3 \sqrt{14}

) is correct because it represents the distance of the plane from the image point Q, calculated using the formula for the distance between a point and a plane.

9 Medium

9 of 59

Application +4 | -1

If the equation of the plane that contains the point

(-2,3,5)

and is perpendicular to each of the planes

2 x+4 y+5 z=8

and

3 x-2 y+3 z=5

\alpha x+\beta y+\gamma z+97=0

then

\alpha+\beta+\gamma=

View answer and explanation

Correct option: A

Option A (15) is correct because it is the sum of the coefficients of x, y, and z in the plane's equation, determined by using the point-normal form and the given perpendicularity conditions.

10 Medium

10 of 59

Application +4 | -1

The equation of a plane passing through the line of intersection of the planes

x+2 y+3 z=2

and

x-y+z=3

and at a distance

\frac{2}{\sqrt{3}}

from the point

(3,1,-1)

View answer and explanation

Correct option: A

5 x-11 y+z=17

Option A (5x - 11y + z = 17) is correct because it represents a plane passing through the intersection of the given planes and satisfies the distance condition from the given point.

11 Medium

11 of 59

Application +4 | -1

Find the foot of perpendicular from the point

(4,6,2)

on the line

\vec{r}=2(\hat{i}+\hat{j}+\hat{k})+t(3 \hat{i}+2 \hat{j}+\hat{k})

View answer and explanation

Correct option: C

(5,4,3)

Option C is correct because the foot of the perpendicular is found by finding a point on the line such that the direction vector from the given point to that point is perpendicular to the line's direction vector.

12 Medium

12 of 59

Application +4 | -1

P, Q, R, S

are four coplanar points on the sides

A B, B C

C D, D A

of a skew quadrilateral, then

\frac{A P}{P B} \cdot \frac{B Q}{Q C} \cdot \frac{C R}{R D} \cdot \frac{D S}{S A}

equals

View answer and explanation

Correct option: A

Option A (1) is correct because the product of the ratios of segments on the sides of a skew quadrilateral with coplanar points is 1.

13 Medium

13 of 59

Application +4 | -1

The line joining the points

(1,1,2)

and

(3,-2,1)

meets the plane

3 x+2 y+z=6

at the point

View answer and explanation

Correct option: B

(3,-2,1)

Option B (

(3,-2,1)

) is correct because substituting the parametric form of the line into the plane equation yields r=1, resulting in the point (3,-2,1).

14 Medium

14 of 59

Application +4 | -1

P(0,1,0)

and

Q(0,0,1)

are two points, then the projection of

\mathrm{PQ}

on the plane

x+y+z=3

View answer and explanation

Correct option: C

\sqrt{2}

Option C (

\sqrt{{2}}

) is correct because the projection of PQ on the plane is equal to the length of PQ since PQ is parallel to the plane.

15 Medium

15 of 59

Application +4 | -1

4 x+4 y-k z=0

is the equation of the plane through the origin that contains the line

\frac{x-1}{2}=\frac{y+1}{3}=\frac{z}{4}

, then

k=

View answer and explanation

Correct option: C

Option C (5) is correct because substituting the direction ratios of the line into the plane equation and solving for k yields k=5.

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16 Medium

16 of 59

Application +4 | -1

The minimum distance of the point

(1,1,1)

from the plane

x+y+z=1

measured perpendicular to the line

\frac{x-x_1}{1}=\frac{y-y_1}{2}=\frac{z-z_1}{3}

View answer and explanation

Correct option: A

\sqrt{\frac{28}{3}}

Option D (

\frac{{4 \sqrt{{7}}}}{{3}}

) is correct because it represents the minimum distance calculated by projecting the vector connecting the point to the plane onto a vector perpendicular to the given line.

17 Medium

17 of 59

Application +4 | -1

A plane passing through the point

\left( {3,1,1} \right)

contains two lines whose direction ratios are

1, - 2,2

and

2,3, - 1

respectively. If this plane also passes through the point

\left( {\alpha , - 3,\,\,5} \right),

then

\alpha

is equal to

View answer and explanation

Correct option: A

Option A (5) is correct because it is the value of

\alpha

that satisfies the equation of the plane containing the given lines and passing through the point (

\alpha

, -3, 5).

18 Medium

18 of 59

Application +4 | -1

The distance of the point

P(-1,-2,4)

from the plane

x+y+z=7

measured parallel to

z

-axis is

View answer and explanation

Correct option: D

Option D (6) is correct because it represents the distance calculated by finding the intersection point of the line parallel to the z-axis and the plane, then determining the z-coordinate difference.

19 Medium

19 of 59

Analyzing +4 | -1

The plane containing the line

\frac{x-3}{2}=\frac{y-b}{4}=\frac{z-3}{3}

passes through the points

(a, 1,2),(2,1,4),(2,3,5)

, then

3 a+5 b

is equal to

View answer and explanation

Correct option: C

-16

Option C (-16) is correct because substituting the points into the plane equation allows us to solve for a and b, and then calculate 3a + 5b.

20 Medium

20 of 59

Application +4 | -1

A plane passing through the points (0, -1, 0) and (0, 0, 1) and making an angle

\frac{\pi }{4}

with the plane

y - z + 5 = 0,

also passes through the point

View answer and explanation

Correct option: A

\left( {\sqrt 2 ,\,1,\,4} \right)

Option A (

\left( {\sqrt 2 ,\,1,\,4} \right)

) is correct because, after finding the equation of the plane using the given points and angle condition, this point satisfies the equation.

21 Medium

21 of 59

Application +4 | -1

The equation of the planes passing through the line of intersection of the planes

3 x-y-4 z=0

and

x+3 y+6=0

whose distance from the origin is 1 , are

View answer and explanation

Correct option: A

\begin{aligned} & x-2 y-2 z-3=0, \\ & 2 x+y-2 z+3=0 \end{aligned}

Option A is correct because it represents the equations of the planes that satisfy the given conditions of passing through the line of intersection and being at a distance of 1 from the origin.

22 Medium

22 of 59

Application +4 | -1

Let

P

be the point of intersection of the line

\frac{x+3}{3}=\frac{y+2}{1}=\frac{1-z}{2}

and the plane

x+y+z=2

. If the distance of the point

P

from the plane

3 x-4 y+

12 z=32

q

, then

q

and

2 q

are the roots of the equation

View answer and explanation

Correct option: B

x^2-18 x+72=0

Option B (

x^2-18 x+72=0

) is correct because it involves finding the point of intersection of a line and a plane, then finding the distance of that point from another plane, and finally forming a quadratic equation with the given roots.

23 Medium

23 of 59

Application +4 | -1

Angle between the straight line

\frac{x-1}{2}=\frac{y+3}{-1}=\frac{z-5}{2}

and the plane

4 x-2 y+4 z=9

View answer and explanation

Correct option: B

90^{\circ}

Option B (2) is correct. Since the point (-1, 2) lies outside the circle, there are two tangents that can be drawn from the point to the circle.

24 Medium

24 of 59

Application +4 | -1

Let

P

be the plane passing through the line

\frac{x-1}{1}=\frac{y-2}{-3}=\frac{z+5}{7}

and the point

(2,4,-3)

. If the image of the point

(-1,3,4)

in the plane

P

(\alpha, \beta, \gamma)

, then

\alpha+\beta+\gamma

is equal to

View answer and explanation

Correct option: A

Option A (10) is correct because it is the sum of the coordinates of the image point after reflection in the plane.

25 Medium

25 of 59

Analyzing +4 | -1

The value of '

\lambda

' for which the lines

3 x+2 y+z-5=0=x+y-2 z-3

and

2 x-y-\lambda z=0=7 x+10 y-8 z

are perpendicular to each other is

View answer and explanation

Correct option: C

Option C (1) is correct because the lines are perpendicular, so the dot product of their direction ratios is zero, leading to lambda = 1.

26 Medium

26 of 59

Application +4 | -1

A plane meets the coordinate axes at

A,B,C

such that the centroid of the triangle

ABC

is at

{G} (a, b, c).

The equation of the plane

{ABC}

is-

View answer and explanation

Correct option: C

\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=3

The correct answer is

\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=3

because it represents the equation of a plane whose intercepts are three times the coordinates of the centroid.

27 Medium

27 of 59

Application +4 | -1

The position vectors of points

\mathbf{a}

and

\mathbf{b}

are

\hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}}

and

3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}

respectively. The equation of a plane is r.

(5 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-7 \hat{\mathbf{k}})+9=0

. The points a and

\mathbf{b}

View answer and explanation

Correct option: C

are on the opposite side of the plane

Option C (are on the opposite side of the plane) is correct because substituting the position vectors into the plane's equation yields opposite signs.

28 Medium

28 of 59

Analyzing +4 | -1

If a plane passes through the points

(-1, k, 0)

(2, k,-1),(1,1,2)

and is parallel to the line

\frac{x-1}{1}=\frac{2 y+1}{2}=\frac{z+1}{-1}

, then the value of

\frac{k^2+1}{(k-1)(k-2)}

View answer and explanation

Correct option: A

\frac{13}{6}

Option A (

\frac{{13}}{{6}}

) is correct because it is obtained by finding the normal vector to the plane, using the parallel line condition to solve for k, and then substituting k into the expression.

29 Medium

29 of 59

Application +4 | -1

The vector equation of the plane through the line of intersection of the planes

x + y + z = 1\mathrm{ }\,\mathrm{and}\,\mathrm{ 2}x + 3y + 4z = 5

which is perpendicular to the plane

x - y + z = 0\mathrm{ }

View answer and explanation

Correct option: D

r \cdot (\hat i - \hat k) + 2 = 0

Option D (

r \cdot (\hat i - \hat k) + 2 = 0

) is correct because it represents the equation of a plane that passes through the intersection of the given planes and is perpendicular to the plane

x - y + z = 0

30 Medium

30 of 59

Application +4 | -1

Equation of the plane that contains the lines

\mathbf{r}=(\hat{\mathbf{i}}+\hat{\mathbf{j}})+\lambda(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}})

and,

\mathbf{r}=(\hat{\mathbf{i}}+\hat{\mathbf{j}})+\mu(-\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}})

View answer and explanation

Correct option: C

\mathbf{r} \cdot(-\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})=0

Option C (

\mathbf{r} \cdot(-\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})=0

) is correct because the normal vector to the plane is found by taking the cross product of the direction vectors of the lines, and the plane passes through the given point.

Revision notes and key takeaways

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Key Takeaways

✓ This set contains 59 single-correct-answer MCQs on 3D PLANES.
✓ Try each question before opening its answer. Read the explanation where one is available.
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