# MCQ Questions for Class 12 Maths Chapter 11 Three Dimensional Geometry with Answers

Question 1.

The direction cosines of the y-axis are

(a) (6, 0, 0)

(b) (1, 0, 0)

(c) (0, 1, 0)

(d) (0, 0, 1)

## Answer

Answer: (c) (0, 1, 0)

Question 2.

The direction ratios of the line joining the points (x, y, z) and (x_{2}, y_{2}, z_{1}) are

(a) x_{1} + x_{2}, y_{1} + y_{2}, z_{1} + z_{2}

(b) \(\sqrt{(x_1 – x_2)^2 + (y_1 – y_2)^2 + (z_1 + z_2)^2}\)

(c) \(\frac{x_1+x_2}{2}\), \(\frac{y_1+y_2}{2}\), \(\frac{z_1+z_2}{2}\)

(d) x_{2} – x_{1}, y_{2} – y_{1}, z_{2} – z_{1}

## Answer

Answer: (d) x_{2} – x_{1}, y_{2} – y_{1}, z_{2} – z_{1}

Question 3.

The coordinates of the midpoints of the line segment joining the points (2, 3, 4) and (8, -3, 8) are

(a) (10, 0, 12)

(b) (5, 6, 0)

(c) (6, 5, 0)

(d) (5, 0, 6)

## Answer

Answer: (d) (5, 0, 6)

Question 4.

If the planes a_{1}x + b, y + c, z + d_{1} = 0 and a_{2}x + b, y + c_{2}z + d_{2} = 0 are perpendicular to each other then

(a) \(\frac{a_1}{a_2}\) = \(\frac{b_1}{b_2}\) = \(\frac{c_1}{c_2}\)

(b) \(\frac{a_1}{a_2}\) + \(\frac{b_1}{b_2}\), \(\frac{c_1}{c_2}\)

(c) a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0

(d) a\(_{1}^{2}\)a\(_{2}^{2}\) + b\(_{1}^{2}\)b\(_{2}^{2}\) + c\(_{1}^{2}\)c\(_{2}^{2}\) = 0

## Answer

Answer: (c) a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0

Question 5.

The distance of the plane 2x – 3y + 6z + 7 = 0 from the point (2, -3, -1) is

(a) 4

(b) 3

(c) 2

(d) \(\frac{1}{5}\)

## Answer

Answer: (c) 2

Question 6.

The direction cosines of the normal to the plane 2x – 3y – 6z – 3 = 0 are

(a) \(\frac{2}{7}\), \(\frac{-3}{7}\), \(\frac{-6}{7}\)

(b) \(\frac{2}{7}\), \(\frac{3}{7}\), \(\frac{6}{7}\)

(c) \(\frac{-2}{7}\), \(\frac{-3}{7}\), \(\frac{-6}{7}\)

(d) None of these

## Answer

Answer: (a) \(\frac{2}{7}\), \(\frac{-3}{7}\), \(\frac{-6}{7}\)

Question 7.

If 2x + 5y – 6z + 3 = 0 be the equation of the plane, then the equation of any plane parallel to the given plane is

(a) 3x + 5y – 6z + 3 = 0

(b) 2x – 5y – 6z + 3 = 0

(c) 2x + 5y – 6z + k = 0

(d) None of these

## Answer

Answer: (c) 2x + 5y – 6z + k = 0

Question 8.

(2, – 3, – 1) 2x – 3y + 6z + 7 = 0

(a) 4

(b) 3

(c) 2

(d) \(\frac{1}{5}\)

## Answer

Answer: (c) 2

Question 9.

The length of the ⊥^{er} from the point (0, – 1, 3) to the plane 2x + y – 2z + 1 = 0 is

(a) 0

(b) 2√3

(c) \(\frac{2}{3}\)

(d) 2

## Answer

Answer: (d) 2

Question 10.

The shortest distance between the lines \(\vec{r}\) = \(\vec{a}\) + k\(\vec{b}\) and r = \(\vec{a}\) + l\(\vec{c}\) is (\(\vec{b}\) and \(\vec{c}\) are non-collinear)

(a) 0

(b) |\(\vec{b}\).\(\vec{c}\)|

(c) \(\frac{|\vec{b}×\vec{c}|}{|\vec {a}|}\)

(d) \(\frac{|\vec{b}.\vec{c}|}{|\vec {a}|}\)

## Answer

Answer: (a) 0

Question 11.

The equation xy = 0 in three dimensional space is represented by

(a) a plane

(b) two plane are right angles

(c) a pair of parallel planes

(d) a pair of st. line

## Answer

Answer: (b) two plane are right angles

Question 12.

The direction cosines of any normal to the xy plane are

(a) 1, 0 ,0

(b) 0, 1, 0

(c) 1, 1, 0

(d) 1, 1, 0

## Answer

Answer: (d) 1, 1, 0

Question 13.

How many lines through the origin in make equal angles with the coordinate axis?

(a) 1

(b) 4

(c) 8

(d) 2

## Answer

Answer: (c) 8

Question 14.

The direction cosines of the line joining (1, -1, 1) and (-1, 1, 1) are

(a) 2, -2, 0

(b) 1, -1, 0

(c) \(\frac{1}{√2}\), – \(\frac{1}{√2}\)

(d) None of these

## Answer

Answer: (c) \(\frac{1}{√2}\), – \(\frac{1}{√2}\)

Question 15.

The equation x² – x – 2 = 0 in three dimensional space is represented by

(a) A pair of parallel planes

(b) A pair of straight lines

(c) A pair of perpendicular plane

(d) None of these

## Answer

Answer: (a) A pair of parallel planes

Question 16.

The distance of the point (-3, 4, 5) from the origin

(a) 50

(b) 5√2

(c) 6

(d) None of these

## Answer

Answer: (b) 5√2

Question 17.

If a line makes angles Q_{1}, Q_{21} and Q_{3} respectively with the coordinate axis then the value of cos² Q_{1} + cos² Q_{2} + cos² Q_{3}

(a) 2

(b) 1

(c) 4

(d) \(\frac{3}{2}\)

## Answer

Answer: (b) 1

Question 18.

The direction ratios of a line are 1,3,5 then its direction cosines are

(a) \(\frac{1}{\sqrt{35}}\), \(\frac{3}{\sqrt{35}}\), \(\frac{5}{\sqrt{35}}\)

(b) \(\frac{1}{9}\), \(\frac{1}{3}\), \(\frac{5}{9}\)

(c) \(\frac{5}{\sqrt{35}}\), \(\frac{3}{\sqrt{35}}\), \(\frac{1}{\sqrt{35}}\)

(d) None of these

## Answer

Answer: (a) \(\frac{1}{\sqrt{35}}\), \(\frac{3}{\sqrt{35}}\), \(\frac{5}{\sqrt{35}}\)

Question 19.

The direction ratios of the normal to the plane 7x + 4y – 2z + 5 = 0 are

(a) 7, 4,-2

(b)7, 4, 5

(c) 7, 4, 2

(d) 4, -2, 5

## Answer

Answer: (a) 7, 4,-2

Question 20.

The direction ratios of the line of intersection of the planes 3x + 2y – z = 5 and x – y + 2z = 3 are

(a) 3, 2, -1

(b) -3, 7, 5

(c) 1, -1, 2

(d) – 11, 4, -5

## Answer

Answer: (b) -3, 7, 5

Question 21.

The lines of intersection of the planes \(\vec{r}\)(3\(\hat{i}\) – \(\hat{j}\) + \(\hat{k}\)) = 1 and \(\vec{r}\)(\(\hat{i}\) +4\(\hat{j}\) – 2\(\hat{k}\)) = 2 is parallel to the vector

(a) 2\(\hat{i}\) + 7\(\hat{j}\) + 13\(\hat{k}\)

(b) -2\(\hat{i}\) + 7\(\hat{j}\) + 13\(\hat{k}\)

(c) 2\(\hat{i}\) – 7\(\hat{j}\) + 13\(\hat{i}\)

(b) -2\(\hat{i}\) – 7\(\hat{j}\) – 13\(\hat{k}\)

## Answer

Answer: (b) -2\(\hat{i}\) + 7\(\hat{j}\) + 13\(\hat{k}\)

Question 22.

The equation of the plane through the origin and parallel to the plane 3x – 4y + 5z + 6 = 0

(a) 3x – 4y – 5z – 6 = 0

(b) 3x – 4y + 5z + 6 = 0

(c) 3x – 4y + 5z = 0

(d) 3x + 4y – 5z + 6 = 0

## Answer

Answer: (c) 3x – 4y + 5z = 0

Question 23.

The locus of xy + yz = 0 is

(a) A pair of st. lines

(b) A pair of parallel lines

(c) A pair of parallel planes

(d) A pair of perpendicular planes

## Answer

Answer: (d) A pair of perpendicular planes

Question 24.

The plane x + y = 0

(a) is parallel to z-axis

(b) is perpendicular to z-axis

(c) passes through z-axis

(d) None of these

## Answer

Answer: (c) passes through z-axis

Question 25.

If α, β, γ are the angle which a half ray makes with the positive directions of the axis then sin²α + sin²β + sin²γ =

(a) 1

(b) 2

(c) 0

(d) -1

## Answer

Answer: (b) 2

Question 26.

If a line makes angles α, β, γ with the axis then cos 2α + cos 2β + cos 2γ =

(a) -2

(b) -1

(c) 1

(d) 2

## Answer

Answer: (b) -1

Question 27.

The line x = 1, y = 2 is

(a) parallel to x-axis

(b) parallel to y-axis

(c) parallel to z-axis

(d) None of these

## Answer

Answer: (c) parallel to z-axis

Question 28.

The points A (1, 1, 0), B(0, 1, 1), C(1, 0, 1) and D(\(\frac{2}{3}\), \(\frac{2}{3}\), \(\frac{2}{3}\))

(a) Coplanar

(b) Non-coplanar

(c) Vertices of a parallelogram

(d) None of these

## Answer

Answer: (a) Coplanar

Question 29.

The angle between the planes 2x – y + z = 6 and x + y + 2z = 7 is

(a) \(\frac{π}{4}\)

(b) \(\frac{π}{6}\)

(c) \(\frac{π}{3}\)

(d) \(\frac{π}{2}\)

## Answer

Answer: (c) \(\frac{π}{3}\)

Question 30.

The distance of the points (2, 1, -1) from the plane x- 2y + 4z – 9 is

(a) \(\frac{\sqrt{31}}{21}\)

(b) \(\frac{13}{21}\)

(c) \(\frac{13}{\sqrt{21}}\)

(d) \(\sqrt{\frac{π}{2}}\)

## Answer

Answer: (c) \(\frac{13}{\sqrt{21}}\)

Question 31.

The planes \(\vec{r}\)(2\(\hat{i}\) + 3\(\hat{j}\) – 6\(\hat{k}\)) = 7 and

\(\vec{r}\)(\(\frac{-2}{7}\)\(\vec{i}\) – \(\frac{3}{j}\)\(\vec{j}\) + \(\frac{6}{7}\)\(\vec{k}\)) = 0 are

(a) parallel

(b) at right angles

(c) equidistant front origin

(d) None of these

## Answer

Answer: (a) parallel

Question 32.

The equation of the plane through point (1, 2, -3) which is parallel to the plane 3x- 5y + 2z = 11 is given by

(a) 3x – 5y + 2z – 13 = 0

(b) 5x – 3y + 2z + 13 = 0

(c) 3x – 2y + 5z + 13 = 0

(d) 3x – 5y + 2z + 13 = 0

## Answer

Answer: (d) 3x – 5y + 2z + 13 = 0

Question 33.

Distance of the point (a, β, γ) from y-axis is

(a) β

(b) |β|

(c) |β + γ|

(d) \(\sqrt{α^2+γ^2}\)

## Answer

Answer: (d) \(\sqrt{α^2+γ^2}\)

Question 34.

If the directions cosines of a line are A, k, k, then

(a) k > 0

(b) 0 < k < 1

(c) k = 1

(d) k = \(\frac{1}{√3}\) or –\(\frac{1}{√3}\)

## Answer

Answer: (d) k = \(\frac{1}{√3}\) or –\(\frac{1}{√3}\)

Question 35.

The distance of the plane \(\vec{r}\)(\(\frac{-2}{7}\)\(\hat{i}\) – \(\frac{3}{7}\)\(\hat{j}\) + \(\frac{6}{7}\)\(\hat{k}\)) = 0 from the orgin is

(a) 1

(b) 7

(c) \(\frac{1}{7}\)

(d) None of these

## Answer

Answer: (a) 1

Question 36.

The sine of the angle between the straight line \(\frac{x-2}{3}\) = \(\frac{y-3}{4}\) = \(\frac{z-4}{5}\) and the plane 2x – 2y + z = 5 is

(a) \(\frac{10}{6√5}\)

(b) \(\frac{4}{5√2}\)

(c) \(\frac{2√3}{5}\)

(d) \(\sqrt{\frac{√2}{10}}\)

## Answer

Answer: (c) \(\frac{2√3}{5}\)

Question 37.

The reflection of the point (a, β, γ) in the xy-plane is

(a) (α, β, 0)

(b) (0, 0, γ)

(c) (- α, – β, γ)

(d) (α, β, γ)

## Answer

Answer: (d) (α, β, γ)

Question 38.

The area of the quadrilateral ABCD, where A(0, 4, 1), B(2, 3, -1), C(4, 5, 0) and D(2, 6, 2) is equal to

(a) 9 sq. units

(b) 18 sq. units

(c) 27 sq. units

(d) 81 sq. units

## Answer

Answer: (a) 9 sq. units

Question 39.

The plane 2x – 3y + 6z – 11 = 0 makes an angle sin^{-1} (α) with .e-axis. The value of a is equal to

(a) \(\frac{√3}{2}\)

(b) \(\frac{√2}{3}\)

(c) \(\frac{2}{7}\)

(d) \(\frac{3}{7}\)

## Answer

Answer: (c) \(\frac{2}{7}\)

Question 40.

The cosines of the angle between any two diagonals of a cube is

(a) \(\frac{1}{3}\)

(b) \(\frac{1}{2}\)

(c) \(\frac{2}{3}\)

(d) \(\frac{1}{√3}\)

## Answer

Answer: (a) \(\frac{1}{3}\)

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