CBSE Class 12 Mathematics Three Dimensional Geometry MCQs Set 10

Question. The cartesian equation of a line is \( \frac{x+3}{2} = \frac{y-5}{4} = \frac{z+6}{2} \). The vector equation for the line is
(a) \( \vec{r} = (2\hat{i} + 3\hat{j} - 6\hat{k}) + \lambda(2\hat{i} - 3\hat{j} + 2\hat{k}) \)
(b) \( \vec{r} = (-3\hat{i} + 5\hat{j} - 6\hat{k}) + \lambda(2\hat{i} + 4\hat{j} + 2\hat{k}) \)
(c) \( \vec{r} = (-3\hat{i} - 5\hat{j} + 6\hat{k}) + \lambda(2\hat{i} - 3\hat{j} - 2\hat{k}) \)
(d) \( \vec{r} = (3\hat{i} + 5\hat{j} + 6\hat{k}) + \lambda(2\hat{i} - 4\hat{j} - 2\hat{k}) \)
Answer: (b) \( \vec{r} = (-3\hat{i} + 5\hat{j} - 6\hat{k}) + \lambda(2\hat{i} + 4\hat{j} + 2\hat{k}) \)

Question. Find the equation of plane passing through the point \( (1, 2, 3) \) and the direction cosines of the normal as \( l, m, n \).
(a) \( lx + my + nz = l + 2m + 3n \)
(b) \( lx + my + nz + (l + 2m + 3n) = 0 \)
(c) \( lx + my + nz = \frac{1}{2} (l + 2m + 3n) \)
(d) None of the options
Answer: (a) \( lx + my + nz = l + 2m + 3n \)

Question. The lines \( \frac{x-1}{2} = \frac{y+1}{-3} = \frac{z+10}{8} \) and \( \frac{x-4}{1} = \frac{y+3}{k} = \frac{z+1}{7} \) are coplanar if \( k = \)
(a) \( 4 \)
(b) \( -4 \)
(c) \( 2 \)
(d) \( -2 \)
Answer: (b) \( -4 \)

Question. Find the equation of the plane passing through \( (2, 3, -1) \) and is perpendicular to the vector \( 3\hat{i} - 4\hat{j} + 7\hat{k} \).
(a) \( 3x - 4y + 7z + 13 = 0 \)
(b) \( 3x + 4y - 7z - 13 = 0 \)
(c) \( 3x + 4y + 7z - 13 = 0 \)
(d) \( 3x - 4y - 7z + 13 = 0 \)
Answer: (a) \( 3x - 4y + 7z + 13 = 0 \)

Question. The equation of a line passing through the point \( (-3, 2, -4) \) and equally inclined to the axes are
(a) \( x - 3 = y + 2 = z - 4 \)
(b) \( x + 3 = y - 2 = z + 4 \)
(c) \( \frac{x+3}{1} = \frac{y-2}{2} = \frac{z+4}{3} \)
(d) None of the options
Answer: (b) \( x + 3 = y - 2 = z + 4 \)

Question. Find the direction cosines of the line \( \frac{x-2}{2} = \frac{2y-5}{-3} = \frac{z+1}{0} \).
(a) \( -2, -5, 1 \)
(b) \( 2, -3, 0 \)
(c) \( 2, -\frac{3}{2}, 0 \)
(d) \( \frac{4}{5}, -\frac{3}{5}, 0 \)
Answer: (d) \( \frac{4}{5}, -\frac{3}{5}, 0 \)

Question. The distance of the plane \( \vec{r} \cdot \left( \frac{2}{7}\hat{i} + \frac{3}{7}\hat{j} - \frac{6}{7}\hat{k} \right) = 1 \) from the origin is
(a) \( 1 \) unit
(b) \( 7 \) units
(c) \( \frac{1}{7} \) unit
(d) \( 2 \) units
Answer: (a) \( 1 \) unit

Question. An equation of the plane passing through the points \( (3, 2, -1) \), \( (3, 4, 2) \) and \( (7, 0, 6) \) is \( 5x + 3y - 2z = \lambda \), where \( \lambda \) is
(a) \( 23 \)
(b) \( 21 \)
(c) \( 19 \)
(d) \( 27 \)
Answer: (a) \( 23 \)

Question. The distance of the plane \( 2x - 3y + 4z - 6 = 0 \) from the origin is \( A \). Here, \( A \) refers to
(a) \( 6 \)
(b) \( -6 \)
(c) \( -\frac{6}{\sqrt{29}} \)
(d) \( \frac{6}{\sqrt{29}} \)
Answer: (d) \( \frac{6}{\sqrt{29}} \)

Question. What is the distance (in units) between the two planes \( 3x + 5y + 7z = 3 \) and \( 9x + 15y + 21z = 9 \)?
(a) \( 0 \)
(b) \( 3 \)
(c) \( \frac{6}{\sqrt{83}} \)
(d) \( 6 \)
Answer: (a) \( 0 \)

Question. Distance of the point \( (\alpha, \beta, \gamma) \) from \( y \)-axis is
(a) \( \beta \)
(b) \( |\beta| \)
(c) \( |\beta| + |\gamma| \)
(d) \( \sqrt{\alpha^2 + \gamma^2} \)
Answer: (d) \( \sqrt{\alpha^2 + \gamma^2} \)

Question. The reflection of the point \( (\alpha, \beta, \gamma) \) in the \( xy \)-plane is
(a) \( (\alpha, \beta, 0) \)
(b) \( (0, 0, \gamma) \)
(c) \( (-\alpha, -\beta, \gamma) \)
(d) \( (\alpha, \beta, -\gamma) \)
Answer: (d) \( (\alpha, \beta, -\gamma) \)

Question. \( P \) is a point on the line segment joining the points \( (3, 2, -1) \) and \( (6, 2, -2) \). If \( x \) co-ordinate of \( P \) is \( 5 \), then its \( y \) co-ordinate is
(a) \( 2 \)
(b) \( 1 \)
(c) \( -1 \)
(d) \( -2 \)
Answer: (a) \( 2 \)

Question. The equation of the line joining the points \( (-3, 4, 11) \) and \( (1, -2, 7) \) is
(a) \( \frac{x+3}{2} = \frac{y-4}{3} = \frac{z-11}{4} \)
(b) \( \frac{x+3}{-2} = \frac{y-4}{3} = \frac{z-11}{2} \)
(c) \( \frac{x+3}{-2} = \frac{y+4}{3} = \frac{z+11}{4} \)
(d) \( \frac{x+3}{2} = \frac{y+4}{-3} = \frac{z+11}{2} \)
Answer: (b) \( \frac{x+3}{-2} = \frac{y-4}{3} = \frac{z-11}{2} \)

Question. The vector equation of the line through the points \( A(3, 4, -7) \) and \( B(1, -1, 6) \) is
(a) \( \vec{r} = (3\hat{i} - 4\hat{j} - 7\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}) \)
(b) \( \vec{r} = (\hat{i} - \hat{j} + 6\hat{k}) + \lambda(3\hat{i} - 4\hat{j} - 7\hat{k}) \)
(c) \( \vec{r} = (3\hat{i} + 4\hat{j} - 7\hat{k}) + \lambda(-2\hat{i} - 5\hat{j} + 13\hat{k}) \)
(d) \( \vec{r} = (\hat{i} - \hat{j} + 6\hat{k}) + \lambda(4\hat{i} + 3\hat{j} - \hat{k}) \)
Answer: (c) \( \vec{r} = (3\hat{i} + 4\hat{j} - 7\hat{k}) + \lambda(-2\hat{i} - 5\hat{j} + 13\hat{k}) \)

Question. If the line joining \( (2, 3, -1) \) and \( (3, 5, -3) \) is perpendicular to the line joining \( (1, 2, 3) \) and \( (3, 5, \lambda) \), then \( \lambda = \)
(a) \( -3 \)
(b) \( 2 \)
(c) \( 5 \)
(d) \( 7 \)
Answer: (d) \( 7 \)

Question. The value of \( p \), so that the lines \( \frac{1-x}{3} = \frac{7y-14}{2p} = \frac{z-3}{2} \) and \( \frac{7-7x}{3p} = \frac{y-5}{1} = \frac{6-z}{5} \) intersect at right angle, is
(a) \( \frac{10}{11} \)
(b) \( \frac{70}{11} \)
(c) \( \frac{10}{7} \)
(d) \( \frac{70}{9} \)
Answer: (b) \( \frac{70}{11} \)

Question. Two lines \( \vec{r} = \vec{a}_1 + \lambda \vec{b}_1 \) and \( \vec{r} = \vec{a}_2 + \mu \vec{b}_2 \) are said to be coplanar, if
(a) \( (\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) = 0 \)
(b) \( \begin{vmatrix} x_2-x_1 & y_2-y_1 & z_2-z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} = 0 \), where \( (x_1, y_1, z_1) \) are the coordinates of a point on any of the line, and \( a_1, b_1, c_1 \) and \( a_2, b_2, c_2 \) are the direction ratios of \( \vec{b}_1 \) and \( \vec{b}_2 \)
(c) both (a) and (b)
(d) None of the options
Answer: (c) both (a) and (b)

Question. The lines \( \frac{x+3}{-3} = \frac{y-1}{1} = \frac{z-5}{5} \) and \( \frac{x+1}{-1} = \frac{y-2}{2} = \frac{z-5}{5} \) are
(a) coplanar
(b) non-coplanar
(c) perpendicular
(d) None of the options
Answer: (a) coplanar

Question. If the lines \( \frac{x-2}{1} = \frac{y-9}{2} = \frac{z-13}{3} \) and \( \frac{x-a}{1} = \frac{y-1}{-2} = \frac{z+2}{3} \) are coplanar, then \( a = \)
(a) \( 2 \)
(b) \( -2 \)
(c) \( 3 \)
(d) \( -3 \)
Answer: (d) \( -3 \)

Case-I

Case-I: Read the following passage and answer the questions.
A football match is organised between students of class XII of two schools, say school A and school B. For which a team from each school is chosen. Remaining students of class XII of school A and B are respectively sitting on the plane represented by the equation \( \vec{r} \cdot (\hat{i} + \hat{j} + 2\hat{k}) = 5 \) and \( \vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 6 \), to cheer up the team of their respective schools.

Question. The cartesian equation of the plane on which students of school A are seated is
(a) \( 2x - y + z = 8 \)
(b) \( 2x + y + z = 8 \)
(c) \( x + y + 2z = 5 \)
(d) \( x + y + z = 5 \)
Answer: (c) \( x + y + 2z = 5 \)

Question. The magnitude of the normal to the plane on which students of school B are seated, is
(a) \( \sqrt{5} \)
(b) \( \sqrt{6} \)
(c) \( \sqrt{3} \)
(d) \( \sqrt{2} \)
Answer: (b) \( \sqrt{6} \)

Question. The intercept form of the equation of the plane on which students of school B are seated, is
(a) \( \frac{x}{6} + \frac{y}{6} + \frac{z}{6} = 1 \)
(b) \( \frac{x}{3} + \frac{y}{(-6)} + \frac{z}{6} = 1 \)
(c) \( \frac{x}{3} + \frac{y}{6} + \frac{z}{6} = 1 \)
(d) \( \frac{x}{3} + \frac{y}{6} + \frac{z}{3} = 1 \)
Answer: (b) \( \frac{x}{3} + \frac{y}{(-6)} + \frac{z}{6} = 1 \)

Question. Which of the following is a student of school B?
(a) Mohit sitting at \( (1, 2, 1) \)
(b) Ravi sitting at \( (0, 1, 2) \)
(c) Khushi sitting at \( (3, 1, 1) \)
(d) Shewta sitting at \( (2, -1, 2) \)
Answer: (c) Khushi sitting at \( (3, 1, 1) \)

Question. The distance of the plane, on which students of school B are seated, from the origin is
(a) \( 6 \) units
(b) \( \frac{1}{\sqrt{6}} \) units
(c) \( \frac{5}{\sqrt{6}} \) units
(d) \( \sqrt{6} \) units
Answer: (d) \( \sqrt{6} \) units

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: If a variable line in two adjacent positions has direction cosines \( l, m, n \) and \( l + \delta l, m + \delta m, n + \delta n \), then the small angle \( \delta\theta \) between the two positions is given by \( \delta\theta^2 = \delta l^2 + \delta m^2 + \delta n^2 \).
Reason: If \( O \) is the origin and \( A \) is \( (a, b, c) \), then the equation of plane through \( A \) at right angle to \( OA \) is given by \( ax + by + cz = a^2 + b^2 + c^2 \).
(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. Consider the lines \( L_1: \frac{x+1}{3} = \frac{y+2}{1} = \frac{z+1}{2} \), \( L_2: \frac{x-2}{1} = \frac{y+2}{2} = \frac{z-3}{3} \).
Assertion: The distance of point \( (1, 1, 1) \) from the plane passing through the point \( (-1, -2, -1) \) and whose normal is perpendicular to both the lines \( L_1 \) and \( L_2 \) is \( \frac{13}{5\sqrt{3}} \).
Reason: The unit vector perpendicular to both the lines \( L_1 \) and \( L_2 \) is \( \frac{-\hat{i} - 7\hat{j} + 5\hat{k}}{5\sqrt{3}} \).
(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: The equation of a plane which passes through \( (2, -3, 1) \) and normal to the line joining the points \( (3, 4, -1) \) and \( (2, -1, 5) \) is given by \( x + 5y - 6z + 19 = 0 \).
Reason: The length of perpendicular from the point \( (7, 14, 5) \) to the plane \( 2x + 4y - z = 2 \) is \( 2\sqrt{21} \).
(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: Two systems of rectangular axis have the same origin. If a plane cuts them at distances \( a, b, c \) and \( a', b', c' \) respectively from the origin, then \( \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} = \frac{1}{a'^2} + \frac{1}{b'^2} + \frac{1}{c'^2} \).
Reason: The points \( (\hat{i} - \hat{j} + 3\hat{k}) \) and \( 3(\hat{i} + \hat{j} + \hat{k}) \) are equidistant from the plane \( \vec{r} \cdot (5\hat{i} + 2\hat{j} - 7\hat{k}) + 9 = 0 \).
(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: (b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.

Question. Assertion: The straight line \( \frac{x-3}{-4} = \frac{y-4}{-7} = \frac{z+3}{13} \) lies in the plane \( 5x - y + z = 8 \).
Reason: The straight line \( \frac{x-x_1}{l} = \frac{y-y_1}{m} = \frac{z-z_1}{c} \) lies in the plane \( ax + by + cz + d = 0 \) iff normal to the plane is perpendicular to the line & every point of the line satisfies the equation of the plane.
(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.