CBSE Class 12 Mathematics Three Dimensional Geometry MCQs Set 08

Question. The angle which the line \( \frac{x}{1} = \frac{y}{-1} = \frac{z}{0} \) makes with the positive direction of Y-axis is
(a) \( \frac{5\pi}{6} \)
(b) \( \frac{3\pi}{4} \)
(c) \( \frac{5\pi}{4} \)
(d) \( \frac{7\pi}{4} \)
Answer: (b) \( \frac{3\pi}{4} \)

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

Question. The direction cosines of vector \( \vec{BA} \), where coordinates of A and B are \( (1, 2, -1) \) and \( (3, 4, 0) \) respectively, are
(a) \( -2, -2, -1 \)
(b) \( -\frac{2}{3}, -\frac{2}{3}, -\frac{1}{3} \)
(c) \( 2, 2, 1 \)
(d) \( \frac{2}{3}, \frac{2}{3}, \frac{1}{3} \)
Answer: (b) \( -\frac{2}{3}, -\frac{2}{3}, -\frac{1}{3} \)

Question. Direction cosines of the line \( \frac{x-1}{2} = \frac{1-y}{3} = \frac{2z-1}{12} \) are
(a) \( \frac{2}{7}, \frac{3}{7}, \frac{6}{7} \)
(b) \( \frac{2}{\sqrt{157}}, -\frac{3}{\sqrt{157}}, \frac{12}{\sqrt{157}} \)
(c) \( \frac{2}{7}, -\frac{3}{7}, -\frac{6}{7} \)
(d) \( \frac{2}{7}, -\frac{3}{7}, \frac{6}{7} \)
Answer: (d) \( \frac{2}{7}, -\frac{3}{7}, \frac{6}{7} \)

Question. A vector makes equal angles with positive directions of X, Y and Z-axes. The direction cosines of the vector are
(a) \( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \)
(b) \( \frac{-1}{\sqrt{3}}, \frac{-1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \)
(c) \( \frac{-1}{2}, \frac{1}{2}, \frac{-1}{2} \)
(d) \( \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{-2}{\sqrt{6}} \)
Answer: (a) \( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \)

Question. If O is the origin and \( OP = 3 \) with direction ratios \( -1, 2 \) and \( -2 \), then coordinates of P are
(a) \( (-1, 2, -2) \)
(b) \( \left(-\frac{1}{3}, \frac{2}{3}, -\frac{2}{3}\right) \)
(c) \( (-3, 6, 9) \)
(d) \( (1, 2, 2) \)
Answer: (a) \( (-1, 2, -2) \)

Question. The cartesian equation of the line \( \vec{r} = (\hat{i} + 2\hat{j} - 4\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k}) \) is
(a) \( \frac{x-2}{1} = \frac{y-3}{2} = \frac{z-6}{-4} \)
(b) \( \frac{x+1}{2} = \frac{y+2}{3} = \frac{z-4}{6} \)
(c) \( \frac{x-1}{2} = \frac{y-2}{3} = \frac{z+4}{5} \)
(d) None of the options
Answer: (d) None of the options

Question. The lines \( \vec{r} = \hat{i} + \hat{j} - \hat{k} + \lambda(2\hat{i} + 3\hat{j} - 6\hat{k}) \) and \( \vec{r} = 2\hat{i} - \hat{j} - \hat{k} + \mu(6\hat{i} + 9\hat{j} - 18\hat{k}) \), (where \( \lambda \) and \( \mu \) are scalars) are
(a) coincident
(b) skew
(c) intersecting
(d) parallel
Answer: (d) parallel

Question. The cartesian equation of the line passing through the point \( (1, -3, 2) \) and parallel to the line \( \vec{r} = (2 + \lambda)\hat{i} + \lambda\hat{j} + (2\lambda - 1)\hat{k} \) is
(a) \( \frac{x-1}{2} = \frac{y+3}{0} = \frac{z-2}{-1} \)
(b) \( \frac{x+1}{1} = \frac{y-3}{1} = \frac{z+2}{2} \)
(c) \( \frac{x+1}{2} = \frac{y-3}{0} = \frac{z+2}{-1} \)
(d) \( \frac{x-1}{1} = \frac{y+3}{1} = \frac{z-2}{2} \)
Answer: (d) \( \frac{x-1}{1} = \frac{y+3}{1} = \frac{z-2}{2} \)

Question. The equation of the line in vector form passing through the point \( (-1, 3, 5) \) and parallel to line \( \frac{x-3}{2} = \frac{y-4}{3}, z = 2 \) is
(a) \( \vec{r} = (-\hat{i} + 3\hat{j} + 5\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + \hat{k}) \)
(b) \( \vec{r} = (-\hat{i} + 3\hat{j} + 5\hat{k}) + \lambda(2\hat{i} + 3\hat{j}) \)
(c) \( \vec{r} = (2\hat{i} + 3\hat{j} - 2\hat{k}) + \lambda(-\hat{i} + 3\hat{j} + 5\hat{k}) \)
(d) \( \vec{r} = (2\hat{i} + 3\hat{j}) + \lambda(-\hat{i} + 3\hat{j} + 5\hat{k}) \)
Answer: (b) \( \vec{r} = (-\hat{i} + 3\hat{j} + 5\hat{k}) + \lambda(2\hat{i} + 3\hat{j}) \)

Question. The angle between the lines \( 2x = 3y = -z \) and \( 6x = -y = -4z \) is
(a) \( 0^\circ \)
(b) \( 30^\circ \)
(c) \( 45^\circ \)
(d) \( 90^\circ \)
Answer: (d) \( 90^\circ \)

Question. Direction ratios of the line which is perpendicular to the lines with direction ratios \( -1, 2, 2 \) and \( 0, 2, 1 \) are
(a) \( 1, 1, 2 \)
(b) \( 2, -1, 2 \)
(c) \( -2, 1, -2 \)
(d) \( 2, 1, -2 \)
Answer: (c) \( -2, 1, -2 \)

Question. The value of \( \lambda \) for which the angle between the lines \( \vec{r} = \hat{i} + \hat{j} + \hat{k} + p(2\hat{i} + \hat{j} + 2\hat{k}) \) and \( \vec{r} = (1+q)\hat{i} + (1+q\lambda)\hat{j} + (1+q)\hat{k} \) is \( \frac{\pi}{2} \), is
(a) -4
(b) 4
(c) 2
(d) -2
Answer: (a) -4

Question. The lines \( \frac{x-2}{1} = \frac{y-3}{1} = \frac{4-z}{k} \) and \( \frac{x-1}{k} = \frac{y-4}{2} = \frac{z-5}{-2} \) are mutually perpendicular, if the value of \( k \) is
(a) \( -\frac{2}{3} \)
(b) \( \frac{2}{3} \)
(c) -2
(d) 2
Answer: (a) \( -\frac{2}{3} \)

Question. Find the angle between the lines \( \frac{x-3}{3} = \frac{y-1}{5} = \frac{z+3}{4} \) and \( \frac{x+1}{1} = \frac{y-4}{1} = \frac{z-5}{2} \)
(a) \( \cos^{-1}\left(\frac{8\sqrt{3}}{15}\right) \)
(b) \( \cos^{-1}\left(\frac{16}{10\sqrt{2}}\right) \)
(c) \( \cos^{-1}\left(\frac{16}{15\sqrt{3}}\right) \)
(d) None of the options
Answer: (a) \( \cos^{-1}\left(\frac{8\sqrt{3}}{15}\right) \)

Question. The angle between any two diagonals of a cube is
(a) \( 60^\circ \)
(b) \( \cos^{-1}\left(\frac{1}{3}\right) \)
(c) \( \sin^{-1}\left(\frac{1}{3}\right) \)
(d) \( \frac{\pi}{2} \)
Answer: (b) \( \cos^{-1}\left(\frac{1}{3}\right) \)

Question. Find the distance between the lines \( l_1 \) and \( l_2 \) given by \( \vec{r} = (\hat{i} + 2\hat{j} - 4\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k}) \) and \( \vec{r} = (3\hat{i} + 3\hat{j} - 5\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k}) \)
(a) \( \frac{15}{7} \)
(b) \( \frac{\sqrt{293}}{7} \)
(c) \( \frac{17}{7} \)
(d) \( \frac{\sqrt{296}}{7} \)
Answer: (b) \( \frac{\sqrt{293}}{7} \)

Assertion-Reason Based Questions

Question. Assertion (A): If a line has direction ratios \( 2, -1, -2 \), then its direction cosines are \( \frac{2}{3}, \frac{-1}{3}, \frac{-2}{3} \).
Reason (R): If \( a, b \) and \( c \) are the direction ratios of a line, then the direction cosines are \( \frac{a}{\sqrt{a^2+b^2+c^2}}, \frac{b}{\sqrt{a^2+b^2+c^2}}, \frac{c}{\sqrt{a^2+b^2+c^2}} \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Assertion (A): The direction cosines of a line passing through the points \( (1, 3, 5) \) and \( (2, 4, 6) \) are \( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \).
Reason (R): The direction ratios of a line passing through two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) are \( [(x_2 - x_1), (y_2 - y_1), (z_2 - z_1)] \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (b) Both A and R are correct; R is not the correct explanation of A.

Question. Assertion (A): If a line makes \( \alpha, \beta \) and \( \gamma \) with X, Y and Z-axes respectively, then \( \cos 2\alpha + \cos 2\beta + \cos 2\gamma = -1 \).
Reason (R): Sum of the square of the direction cosines of a line is equal to 1.
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Assertion (A): The points \( (1, 2, 3) \), \( (-2, 3, 4) \) and \( (7, 0, 1) \) are collinear.
Reason (R): 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) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (b) Both A and R are correct; R is not the correct explanation of A.

Question. Assertion (A): Equation of the line passing through the point \( (1, -2, 3) \) and parallel to the vector \( (3\hat{i} + 2\hat{j} + 6\hat{k}) \) is \( \vec{r} = (\hat{i} - 2\hat{j} + 3\hat{k}) + \lambda(3\hat{i} + 2\hat{j} + 6\hat{k}) \).
Reason (R): The vector equation of a straight line passing through a given point with a position vector \( \vec{r}_1 \) and parallel to a given vector \( \vec{m} \) is \( \vec{r} = \vec{r}_1 + \lambda \vec{m} \) (where \( \lambda \) is a scalar).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Assertion (A): The equation of a line which passes through the point \( (1, 2, 3) \) and is parallel to the vector \( 3\hat{i} + 2\hat{j} - 2\hat{k} \) is \( \vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(2\hat{i} + 3\hat{j} - 2\hat{k}) \).
Reason (R): The vector equation of a line passing through a point with position vector \( \vec{a} \) and parallel to a given vector \( \vec{b} \) is \( \vec{r} = \vec{a} + \lambda \vec{b} \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (d) R is correct; A is incorrect.

Question. Assertion (A): Line \( \frac{-x+2}{-2} = \frac{y-1}{7} = \frac{z+3}{-3} \) and \( \frac{x+2}{-1} = \frac{2y-8}{4} = \frac{z-5}{4} \) are perpendicular to each other.
Reason (R): If dot product of the direction ratios of two lines is zero, then lines are perpendicular.
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Assertion (A): The lines \( \vec{r} = \vec{a}_1 + \lambda \vec{b}_1 \) and \( \vec{r} = \vec{a}_2 + \mu \vec{b}_2 \) are perpendicular, when \( \vec{b}_1 \cdot \vec{b}_2 = 0 \).
Reason (R): The angle \( \theta \) between the lines \( \vec{r} = \vec{a}_1 + \lambda \vec{b}_1 \) and \( \vec{r} = \vec{a}_2 + \mu \vec{b}_2 \) is given by \( \cos \theta = \frac{|\vec{b}_1 \cdot \vec{b}_2|}{|\vec{b}_1||\vec{b}_2|} \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Assertion (A): Angle between \( \frac{x+2}{2} = \frac{y-1}{3} = \frac{z-2}{1} \) and \( \frac{x-3}{-3} = \frac{y}{-2} = \frac{z+1}{2} \) is \( \frac{\pi}{6} \).
Reason (R): If direction ratios of two lines are in same ratio, then lines are parallel.
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (d) R is correct; A is incorrect.

Case Study Based Questions - I

A cricket match is organized between two clubs A and B for which a team from each club is chosen. Remaining players of club A and club B, respectively sitting on the lines, represented by the equation \( \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} \) and \( \frac{x-4}{3} = \frac{y+2}{2} = \frac{z}{6} \) and to cheer the team of their club.

Question. Find the vector equation of the line on which players of club A are seated.
Answer: \( \vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 4\hat{k}) \)

Question. Find the direction cosines of the line on which players of club B are seated.
Answer: \( \frac{3}{7}, \frac{2}{7}, \frac{6}{7} \)

Question. If the line on which players of club A are seated is perpendicular to the line \( \frac{x-3}{7} = \frac{7-y}{4} = \frac{kz-7}{1} \), then find the value of \( k \).
Answer: \( k = -2 \)

Question. Find the angle between the lines on which players of club A and club B are seated.
Answer: \( \theta = \cos^{-1}\left(\frac{36}{7\sqrt{29}}\right) \)

Question. Find the shortest distance between the lines on which players of club A and club B are seated.
Answer: \( \frac{9\sqrt{5}}{5} \text{ units} \)

Case Study Based Questions - II

A new railway line is to be created in the direction whose cartesian equation is \( 6x - 2 = 3y + 1 = 2z - 2 \). Based on this information, answer the following questions.

Question. Find the direction cosines of the line.
Answer: \( \frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}} \)

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

Question. If a city with coordinate \( \left(\frac{2}{3}, \frac{1}{3}, 2\right) \), then find whether it lie in the path or not.
Answer: Yes, the city lies on the railway line.

Question. If the path with equation \( \frac{x-1}{2} = \frac{y-2}{3} = \frac{z}{\lambda} \) is perpendicular to the original path find the value of \( \lambda \).
Answer: \( \lambda = -\frac{8}{3} \)

Question. Find the angle between the original path and \( \frac{x}{-1} = \frac{y+2}{2} = \frac{3-6z}{5} \).
Answer: \( \theta = \cos^{-1}\left(\frac{3}{\sqrt{2870}}\right) \)

Case Study Based Questions - III

A supply ship left a port to replenish food and equipment for the Indian navy. Enemies used a submarine to track the ship and scout the port. Assuming the port to be at the origin, the position of the ship is \( (5\text{ km}, 0\text{ km}, 9\text{ km}) \) and the position of the submarine is \( (11\text{ km}, -0.9\text{ km}, -10\text{ km}) \). (Note the figure is not to scale.)

Question. What angle does the line joining the ship and the port make with the Z-axis?
(a) \( \cos^{-1} 0 \)
(b) \( \cos^{-1}\pm 1 \)
(c) \( \cos^{-1}\pm \left(\frac{9}{\sqrt{106}}\right) \)
(d) \( \cos^{-1}\pm \left(\frac{10}{\sqrt{302}}\right) \)
Answer: (c) \( \cos^{-1}\pm \left(\frac{9}{\sqrt{106}}\right) \)

Question. The submarine squad wants to find the direction cosines of the line joining the submarine to the port. The captain of the submarine says that the direction cosines are \( \frac{-11}{\sqrt{221.81}}, \frac{0.9}{\sqrt{221.81}}, \frac{10}{\sqrt{221.81}} \) and his sub-ordinate says that the direction cosines are \( \frac{11}{\sqrt{221.81}}, \frac{-0.9}{\sqrt{221.81}}, \frac{-10}{\sqrt{221.81}} \). Who is correct?
(a) The captain
(b) The sub-ordinate
(c) Both of them
(d) Neither of them
Answer: (c) Both of them

Question. The distance between the port and the ship is \( \sqrt{106}\text{ km} \), the distance between submarine and the port is \( \sqrt{221.81}\text{ km} \) and the distance between the ship and the submarine is \( \sqrt{397.81}\text{ km} \). Which of the following represents the angle between the lines joining the submarine to the ship and the line joining the submarine to the port?
(a) \( \cos^{-1}\left(\frac{-35}{\sqrt{106} \times \sqrt{221.81}}\right) \)
(b) \( \cos^{-1}\left(\frac{201}{\sqrt{397.81} \times \sqrt{106}}\right) \)
(c) \( \cos^{-1}\left(\frac{256.81}{\sqrt{397.81} \times \sqrt{221.81}}\right) \)
(d) \( \cos^{-1}\left(\frac{2040}{\sqrt{397.81} \times \sqrt{106} \times \sqrt{221.81}}\right) \)
Answer: (c) \( \cos^{-1}\left(\frac{256.81}{\sqrt{397.81} \times \sqrt{221.81}}\right) \)

Question. Which of the following is the equation of the line joining the submarine and the port?
Equation I: \( 11\hat{i} - 0.9\hat{j} - 10\hat{k} + \lambda(-11\hat{i} + 0.9\hat{j} + 10\hat{k}) \)
Equation II: \( \lambda(-11\hat{i} - 0.9\hat{j} - 10\hat{k}) \)
(a) Equation I
(b) Equation II
(c) Both equation I and equation II
(d) Neither equation I nor equation II
Answer: (a) Equation I