CBSE Class 12 Mathematics Three Dimensional Geometry MCQs Set 09

Question. If a line makes an angle \( \theta_1, \theta_2, \theta_3 \) with the axes respectively, then the value of \( \cos 2\theta_1 + \cos 2\theta_2 + \cos 2\theta_3 \) is
(a) \( 1 \)
(b) \( -1 \)
(c) \( 4 \)
(d) \( 3 \)
Answer: (b) \( -1 \)

Question. If the equation of a line AB is \( \frac{x - 3}{1} = \frac{y + 2}{-2} = \frac{z - 5}{4} \), find the direction ratios of a line parallel to AB.
(a) \( 1, 2, 4 \)
(b) \( 1, 2, -4 \)
(c) \( 1, -2, -4 \)
(d) \( 1, -2, 4 \)
Answer: (d) \( 1, -2, 4 \)

Question. If \( \alpha, \beta, \gamma \) are the angles made by a line with the co-ordinate axes. Then \( \sin^2\alpha + \sin^2\beta + \sin^2\gamma \) is
(a) \( 0 \)
(b) \( -1 \)
(c) \( 2 \)
(d) \( 1 \)
Answer: (c) \( 2 \)

Question. If \( \alpha, \beta, \gamma \) are the direction angles of a vector and \( \cos \alpha = \frac{14}{15} \), \( \cos \beta = \frac{1}{3} \), then \( \cos \gamma = \)
(a) \( \pm \frac{2}{15} \)
(b) \( \pm \frac{1}{5} \)
(c) \( \pm \frac{1}{15} \)
(d) \( \pm \frac{4}{15} \)
Answer: (a) \( \pm \frac{2}{15} \)

Question. If a line makes angles \( 90^\circ, 60^\circ \) and \( 30^\circ \) with the positive directions of \( x, y \) and \( z \)-axis respectively, then its direction cosines are
(a) \( \left( \frac{1}{2}, 0, \frac{\sqrt{3}}{2} \right) \)
(b) \( \left( \frac{\sqrt{3}}{2}, \frac{1}{2}, 0 \right) \)
(c) \( \left( \frac{\sqrt{3}}{2}, 0, \frac{1}{2} \right) \)
(d) \( \left( 0, \frac{1}{2}, \frac{\sqrt{3}}{2} \right) \)
Answer: (d) \( \left( 0, \frac{1}{2}, \frac{\sqrt{3}}{2} \right) \)

Question. The direction cosines of the line passing through two points \( (2, 1, 0) \) and \( (1, -2, 3) \) are
(a) \( \left( \frac{1}{\sqrt{19}}, \frac{3}{\sqrt{19}}, \frac{3}{\sqrt{19}} \right) \)
(b) \( \left( \frac{-3}{\sqrt{19}}, \frac{3}{\sqrt{19}}, \frac{-1}{\sqrt{19}} \right) \)
(c) \( \left( \frac{-1}{\sqrt{19}}, \frac{-3}{\sqrt{19}}, \frac{3}{\sqrt{19}} \right) \)
(d) \( \left( \frac{1}{\sqrt{19}}, \frac{3}{\sqrt{19}}, \frac{-3}{\sqrt{19}} \right) \)
Answer: (c) \( \left( \frac{-1}{\sqrt{19}}, \frac{-3}{\sqrt{19}}, \frac{3}{\sqrt{19}} \right) \)

Question. Find the direction cosines of the line that makes equal angles with the three axes in space.
(a) \( \pm\frac{1}{\sqrt{2}} \)
(b) \( \pm 1 \)
(c) \( \pm\frac{1}{\sqrt{3}} \)
(d) \( \sqrt{3} \)
Answer: (c) \( \pm\frac{1}{\sqrt{3}} \)

Question. Find the direction cosines of the line joining \( A(0, 7, 10) \) and \( B(-1, 6, 6) \).
(a) \( \left( -\frac{1}{3\sqrt{2}}, -\frac{1}{3\sqrt{2}}, -\frac{4}{3\sqrt{2}} \right) \)
(b) \( \left( \frac{1}{3\sqrt{2}}, \frac{4}{3\sqrt{2}}, \frac{1}{3\sqrt{2}} \right) \)
(c) \( \left( \frac{1}{3\sqrt{2}}, \frac{1}{3\sqrt{2}}, \frac{1}{3\sqrt{2}} \right) \)
(d) \( \left( \frac{4}{3\sqrt{2}}, \frac{1}{\sqrt{2}}, \frac{1}{3\sqrt{2}} \right) \)
Answer: (a) \( \left( -\frac{1}{3\sqrt{2}}, -\frac{1}{3\sqrt{2}}, -\frac{4}{3\sqrt{2}} \right) \)

Question. Find the equation of a line passing through a point \( (2, -1, 3) \) and parallel to the line \( \vec{r} = (\hat{i} + \hat{j}) + \lambda(2\hat{i} + \hat{j} - 2\hat{k}) \).
(a) \( \vec{r} = (\hat{i} + \hat{j}) + \mu(2\hat{i} - \hat{j} + 3\hat{k}) \)
(b) \( \vec{r} = (2\hat{i} - \hat{j} + 3\hat{k}) + \mu(2\hat{i} + \hat{j} - 2\hat{k}) \)
(c) \( \vec{r} = (\hat{i} - \hat{j}) + \mu(2\hat{i} - \hat{j} + 3\hat{k}) \)
(d) \( \vec{r} = (2\hat{i} + \hat{j} + 3\hat{k}) + \mu(2\hat{i} + \hat{j} - 2\hat{k}) \)
Answer: (b) \( \vec{r} = (2\hat{i} - \hat{j} + 3\hat{k}) + \mu(2\hat{i} + \hat{j} - 2\hat{k}) \)

Question. If \( \left( \frac{1}{2}, \frac{1}{3}, n \right) \) are the direction cosines of a line, then the value of \( n \) is
(a) \( \frac{\sqrt{23}}{6} \)
(b) \( \frac{23}{6} \)
(c) \( \frac{2}{3} \)
(d) \( \frac{3}{2} \)
Answer: (a) \( \frac{\sqrt{23}}{6} \)

Question. The distance of the plane \( 3x - 6y + 2z + 11 = 0 \) from the origin is
(a) \( \frac{11}{7} \) units
(b) \( \frac{1}{7} \) unit
(c) \( \frac{7}{11} \) unit
(d) \( \frac{13}{7} \) units
Answer: (a) \( \frac{11}{7} \) units

Question. Find the distance of the point \( (2, 3, 4) \) from the plane \( \vec{r} \cdot (3\hat{i} - 6\hat{j} + 2\hat{k}) + 11 = 0 \).
(a) \( 1 \) unit
(b) \( 4 \) units
(c) \( 2 \) units
(d) \( 3 \) units
Answer: (a) \( 1 \) unit

Question. Write the direction cosines of a line parallel to the line \( \frac{3-x}{3} = \frac{y+2}{-2} = \frac{z+2}{6} \).
(a) \( \left( \frac{1}{7}, \frac{2}{7}, \frac{3}{7} \right) \)
(b) \( \left( -\frac{3}{7}, -\frac{2}{7}, \frac{6}{7} \right) \)
(c) \( \left( \frac{3}{7}, \frac{2}{7}, \frac{6}{7} \right) \)
(d) \( \left( \frac{3}{7}, -\frac{2}{7}, \frac{6}{7} \right) \)
Answer: (b) \( \left( -\frac{3}{7}, -\frac{2}{7}, \frac{6}{7} \right) \)

Question. If lines \( \frac{x-1}{-3} = \frac{y-2}{2k} = \frac{z-3}{2} \) and \( \frac{x-1}{3k} = \frac{y-5}{1} = \frac{z-6}{-5} \) are mutually perpendicular, then \( k \) is equal to
(a) \( -\frac{10}{7} \)
(b) \( -\frac{7}{10} \)
(c) \( -10 \)
(d) \( -7 \)
Answer: (a) \( -\frac{10}{7} \)

Question. The equation of a plane with intercepts \( 2, 3 \) and \( 4 \) on the \( X, Y \) and \( Z \)-axes respectively is \( A \). Here, \( A \) refers to
(a) \( 2x + 3y + 4z = 12 \)
(b) \( 6x + 4y + 3z = 12 \)
(c) \( 2x + 3y + 4z = 1 \)
(d) \( 6x + 4y + 3z = 1 \)
Answer: (b) \( 6x + 4y + 3z = 12 \)

Question. What is the distance between the planes \( 2x + 2y - z + 2 = 0 \) and \( 4x + 4y - 2z + 5 = 0 \)?
(a) \( 2/6 \) unit
(b) \( 3/2 \) units
(c) \( 1/6 \) unit
(d) \( 1/4 \) unit
Answer: (c) \( 1/6 \) unit

Question. The equation of a line is given by \( \frac{4-x}{2} = \frac{y+3}{3} = \frac{z+2}{6} \), the direction cosines of line parallel to the given line is
(a) \( \left( -\frac{2}{7}, -\frac{3}{7}, -\frac{6}{7} \right) \)
(b) \( \left( \frac{2}{7}, -\frac{3}{7}, -\frac{6}{7} \right) \)
(c) \( \left( \frac{2}{7}, \frac{3}{7}, \frac{6}{7} \right) \)
(d) \( \left( -\frac{2}{7}, \frac{3}{7}, \frac{6}{7} \right) \)
Answer: (d) \( \left( -\frac{2}{7}, \frac{3}{7}, \frac{6}{7} \right) \)

Question. A line makes angles \( \alpha, \beta \) and \( \gamma \) with the co-ordinate axes. If \( \alpha + \beta = 90^\circ \), then the value of angle \( \gamma \) is
(a) \( 60^\circ \)
(b) \( 90^\circ \)
(c) \( 45^\circ \)
(d) \( 30^\circ \)
Answer: (b) \( 90^\circ \)

Question. Find the equation of a line passing through \( (1, 2, -3) \) and parallel to the line \( \frac{x-2}{1} = \frac{y+1}{3} = \frac{z-1}{4} \).
(a) \( \frac{x-2}{-1} = \frac{y+1}{1} = \frac{z-1}{1} \)
(b) \( \frac{x-1}{1} = \frac{y-2}{3} = \frac{z+3}{4} \)
(c) \( \frac{x+1}{1} = \frac{y-2}{3} = \frac{z+3}{4} \)
(d) \( \frac{x-2}{2} = \frac{y+1}{-1} = \frac{z-1}{1} \)
Answer: (b) \( \frac{x-1}{1} = \frac{y-2}{3} = \frac{z+3}{4} \)

Question. The vector equation of the plane passing through a point having position vector \( 2\hat{i} + 3\hat{j} + 4\hat{k} \) and perpendicular to the vector \( 2\hat{i} + \hat{j} - 2\hat{k} \) is
(a) \( \vec{r} \cdot (2\hat{i} + \hat{j} - 2\hat{k}) = -1 \)
(b) \( \vec{r} \cdot (2\hat{i} + \hat{j} - 2\hat{k}) = 8 \)
(c) \( \vec{r} \cdot (2\hat{i} + \hat{j} - 2\hat{k}) = 9 \)
(d) \( \vec{r} \cdot (2\hat{i} + \hat{j} - 2\hat{k}) = 15 \)
Answer: (a) \( \vec{r} \cdot (2\hat{i} + \hat{j} - 2\hat{k}) = -1 \)

Case-I

Case-I: Read the following passage and answer the questions.
In a diamond exhibition, a diamond is covered in cubical glass box having coordinates \( O(0, 0, 0) \), \( A(1, 0, 0) \), \( B(1, 2, 0) \), \( C(0, 2, 0) \), \( O'(0, 0, 3) \), \( A'(1, 0, 3) \), \( B'(1, 2, 3) \) and \( C'(0, 2, 3) \).

Question. Direction ratios of OA are
(a) \( < 0, 1, 0 > \)
(b) \( < 1, 0, 0 > \)
(c) \( < 0, 0, 1 > \)
(d) None of the options
Answer: (b) \( < 1, 0, 0 > \)

Question. Equation of diagonal OB' is
(a) \( \frac{x}{1} = \frac{y}{2} = \frac{z}{3} \)
(b) \( \frac{x}{0} = \frac{y}{1} = \frac{z}{2} \)
(c) \( \frac{x}{1} = \frac{y}{0} = \frac{z}{2} \)
(d) None of the options
Answer: (a) \( \frac{x}{1} = \frac{y}{2} = \frac{z}{3} \)

Question. Equation of plane OABC is
(a) \( x = 0 \)
(b) \( y = 0 \)
(c) \( z = 0 \)
(d) None of the options
Answer: (c) \( z = 0 \)

Question. Equation of plane O'A'B'C' is
(a) \( x = 3 \)
(b) \( y = 3 \)
(c) \( z = 3 \)
(d) \( z = 2 \)
Answer: (c) \( z = 3 \)

Question. Equation of plane ABB'A' is
(a) \( x = 1 \)
(b) \( y = 1 \)
(c) \( z = 2 \)
(d) \( x = 3 \)
Answer: (a) \( x = 1 \)

Assertion & Reasoning Based MCQs

Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given. Choose the correct answer out of the following choices:
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.

Question. Assertion: The points \( (1, 2, 3), (-2, 3, 4) \) and \( (7, 0, 1) \) are collinear.
Reason: If a line makes angles \( \frac{\pi}{2}, \frac{3\pi}{4} \) and \( \frac{\pi}{4} \) with \( X, Y, \) and \( Z \)-axes respectively, then its direction cosines are \( 0, -\frac{1}{\sqrt{2}} \) and \( \frac{1}{\sqrt{2}} \).
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.

Question. Assertion: If the cartesian equation of a line is \( \frac{x-5}{3} = \frac{y+4}{7} = \frac{z-6}{2} \), then its vector form is \( \vec{r} = 5\hat{i} - 4\hat{j} + 6\hat{k} + \lambda(3\hat{i} + 7\hat{j} + 2\hat{k}) \).
Reason: The cartesian equation of the line which passes through the point \( (-2, 4, -5) \) and parallel to the line given by \( \frac{x+3}{3} = \frac{y-4}{5} = \frac{z+8}{6} \) is \( \frac{x+3}{-2} = \frac{y-4}{4} = \frac{z+8}{-5} \).
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (c) Assertion is correct statement but Reason is wrong statement.

Question. Assertion: The three lines with direction cosines \( \frac{12}{13}, -\frac{3}{13}, -\frac{4}{13} \); \( \frac{4}{13}, \frac{12}{13}, -\frac{3}{13} \); \( \frac{3}{13}, -\frac{4}{13}, \frac{12}{13} \) are mutually perpendicular.
Reason: The line through the points \( (1, -1, 2) \) and \( (3, 4, -2) \) is perpendicular to the line through the points \( (0, 3, 2) \) and \( (3, 5, 6) \).
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.

Question. Assertion: The pair of lines given by \( \vec{r} = \hat{i} - \hat{j} + \lambda(2\hat{i} + \hat{k}) \) and \( \vec{r} = 2\hat{i} - \hat{k} + \mu(\hat{i} + \hat{j} - \hat{k}) \) intersect.
Reason: Two lines intersect each other, if they are not parallel and shortest distance = 0.
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.

Question. Assertion: There exists only one plane that is perpendicular to the given vector.
Reason: Through a given point perpendicular to the given vector only one plane exists.
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (d) Assertion is wrong statement but Reason is correct statement.