JEE Mathematics Three Dimensional Geometry MCQs Set 04

Question. Equation of the plane passing through A(x₁, y₁, z₁) and containing the line \( \frac{x - x_2}{d_1} = \frac{y - y_2}{d_2} = \frac{z - z_2}{d_3} \) is
(a) \( \begin{vmatrix} x - x_1 & y - y_1 & z - z_1 \\ x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\ d_1 & d_2 & d_3 \end{vmatrix} = 0 \)
(b) \( \begin{vmatrix} x - x_2 & y - y_2 & z - z_2 \\ x_1 - x_2 & y_1 - y_2 & z_1 - z_2 \\ d_1 & d_2 & d_3 \end{vmatrix} = 0 \)
(c) \( \begin{vmatrix} x - d_1 & y - d_2 & z - d_3 \\ x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \end{vmatrix} = 0 \)
(d) \( \begin{vmatrix} x & y & z \\ x_1 - x_2 & y_1 - y_2 & z_1 - z_2 \\ d_1 & d_2 & d_3 \end{vmatrix} = 0 \)
Answer: (a) \( \begin{vmatrix} x - x_1 & y - y_1 & z - z_1 \\ x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\ d_1 & d_2 & d_3 \end{vmatrix} = 0 \) and (b) \( \begin{vmatrix} x - x_2 & y - y_2 & z - z_2 \\ x_1 - x_2 & y_1 - y_2 & z_1 - z_2 \\ d_1 & d_2 & d_3 \end{vmatrix} = 0 \)

Question. The equation of the line x + y + z - 1 = 0, 4x + y - 2z + 2 = 0 written in the symmetrical form is
(a) \( \frac{x + 1}{1} = \frac{y - 2}{-2} = \frac{z - 0}{1} \)
(b) \( \frac{x}{1} = \frac{y}{-2} = \frac{z}{1} \)
(c) \( \frac{x + 1/2}{1} = \frac{y - 1}{-2} = \frac{z - 1/2}{1} \)
(d) \( \frac{x - 1}{2} = \frac{y + 2}{-1} = \frac{z - 2}{2} \)
Answer: (a) \( \frac{x + 1}{1} = \frac{y - 2}{-2} = \frac{z - 0}{1} \) and (c) \( \frac{x + 1/2}{1} = \frac{y - 1}{-2} = \frac{z - 1/2}{1} \)

Question. The acute angle that the vector \( 2\hat{i} - 2\hat{j} + \hat{k} \) makes with the plane contained by the two vectors \( 2\hat{i} + 3\hat{j} - \hat{k} \) and \( \hat{i} - \hat{j} + 2\hat{k} \) is given by
(a) \( \cos^{-1} \left( \frac{1}{\sqrt{3}} \right) \)
(b) \( \sin^{-1} \left( \frac{1}{\sqrt{3}} \right) \)
(c) \( \tan^{-1} (\sqrt{2}) \)
(d) \( \cot^{-1} (\sqrt{2}) \)
Answer: (c) \( \tan^{-1} (\sqrt{2}) \) and (d) \( \cot^{-1} (\sqrt{2}) \)

Question. The ratio in which the sphere x² + y² + z² = 504 divides the line joining the points (12, -4, 8) and (27, -9, 18) is
(a) 2 : 3 internally
(b) 3 : 4 internally
(c) 2 : 3 externally
(d) 3 : 4 externally
Answer: (a) 2 : 3 internally and (c) 2 : 3 externally

Question. The equations of the planes through the origin which are parallel to the line \( \frac{x - 1}{2} = \frac{y + 3}{-1} = \frac{z + 1}{-2} \) and distance \( \frac{5}{3} \) from it are
(a) 2x + 2y + z = 0
(b) x + 2y + 2z = 0
(c) 2x - 2y + z = 0
(d) x - 2y + 2z = 0
Answer: (a) 2x + 2y + z = 0 and (d) x - 2y + 2z = 0

Question. If the edges of a rectangular parallelopiped are 3, 2, 1 then the angle between a pair of diagonals is given by
(a) \( \cos^{-1} \frac{6}{7} \)
(b) \( \cos^{-1} \frac{3}{7} \)
(c) \( \cos^{-1} \frac{2}{7} \)
(d) None of the options
Answer: (a) \( \cos^{-1} \frac{6}{7} \), (b) \( \cos^{-1} \frac{3}{7} \), and (c) \( \cos^{-1} \frac{2}{7} \)

Question. Consider the lines \( \frac{x}{2} = \frac{y}{3} = \frac{z}{5} \) and \( \frac{x}{1} = \frac{y}{2} = \frac{z}{3} \) the equation of the line which
(a) bisects the angle between the lines is \( \frac{x}{3} = \frac{y}{3} = \frac{z}{8} \)
(b) bisects the angle between the lines is \( \frac{x}{1} = \frac{y}{2} = \frac{z}{3} \)
(c) passes through origin and is perpendicular to the given lines is x = y = -z
(d) None of the options
Answer: (c) passes through origin and is perpendicular to the given lines is x = y = -z

Question. The direction cosines of the lines bisecting the angle between the lines whose direction cosines are \( l_1, m_1, n_1 \) and \( l_2, m_2, n_2 \) and the angle between these lines is \( \theta \), are
(a) \( \frac{l_1 + l_2}{\cos \frac{\theta}{2}}, \frac{m_1 + m_2}{\cos \frac{\theta}{2}}, \frac{n_1 + n_2}{\cos \frac{\theta}{2}} \)
(b) \( \frac{l_1 + l_2}{2\cos \frac{\theta}{2}}, \frac{m_1 + m_2}{2\cos \frac{\theta}{2}}, \frac{n_1 + n_2}{2\cos \frac{\theta}{2}} \)
(c) \( \frac{l_1 + l_2}{\sin \frac{\theta}{2}}, \frac{m_1 + m_2}{\sin \frac{\theta}{2}}, \frac{n_1 + n_2}{\sin \frac{\theta}{2}} \)
(d) \( \frac{l_1 + l_2}{2\sin \frac{\theta}{2}}, \frac{m_1 + m_2}{2\sin \frac{\theta}{2}}, \frac{n_1 + n_2}{2\sin \frac{\theta}{2}} \)
Answer: (b) \( \frac{l_1 + l_2}{2\cos \frac{\theta}{2}}, \dots \) and (d) \( \frac{l_1 + l_2}{2\sin \frac{\theta}{2}}, \dots \)

Question. The equation of line AB is \( \frac{x}{2} = \frac{y}{-3} = \frac{z}{6} \). Through a point P(1, 2, 5), line PN is drawn perpendicular to AB and line PQ is drawn parallel to the plane 3x + 4y + 5z = 0 to meet AB is Q. Then
(a) co-ordinate of N is \( \left( \frac{52}{49}, \frac{78}{49}, \frac{156}{49} \right) \)
(b) the equation of PN is \( \frac{x - 1}{3} = \frac{y - 2}{-176} = \frac{z - 5}{-89} \)
(c) the co-ordinates of Q is \( \left( 3, \frac{-9}{2}, 9 \right) \)
(d) the equation of PQ is \( \frac{x - 1}{4} = \frac{y - 2}{-13} = \frac{z - 5}{8} \)
Answer: (a), (b), (c) and (d)

Question. The planes 2x - 3y - 7z = 0, 3x - 14y - 13z = 0 and 8x - 31y - 33z = 0
(a) pass through origin
(b) intersect in a common line
(c) form a triangular prism
(d) None of the options
Answer: (a) pass through origin and (b) intersect in a common line

Question. If the length of perpendicular drawn from origin on a plane is 7 units and its direction ratios are -3, 2, 6, then that plane is
(a) -3x + 2y + 6z - 7 = 0
(b) -3x + 2y + 6z - 49 = 0
(c) 3x - 2y - 6z - 49 = 0
(d) -3x + 2y - 6z - 49 = 0
Answer: (b) -3x + 2y + 6z - 49 = 0 and (c) 3x - 2y - 6z - 49 = 0

Question. Let a perpendicular PQ be drawn from P(5, 7, 3) to the line \( \frac{x - 15}{3} = \frac{y - 2}{8} = \frac{z - 6}{-5} \) when Q is the foot. Then
(a) Q is (9, 13, -15)
(b) PQ = 14
(c) the equation of plane containing PQ and the given line is 9x - 4y - z - 14 = 0
(d) None of the options
Answer: (b) PQ = 14 and (c) the equation of plane containing PQ and the given line is 9x - 4y - z - 14 = 0

Question. If the lines \( \frac{x-1}{2} = \frac{y+1}{3} = \frac{z-1}{4} \) and \( \frac{x-3}{1} = \frac{y-k}{2} = \frac{z}{1} \) intersect, then k equals 
(a) 2/9
(b) 9/2
(c) 0
(d) -1
Answer: (b) 9/2

Question. Let P be the plane passing through (1, 1, 1) and parallel to the lines \( L_1 \) and \( L_2 \) having direction ratios (1, 0, -1) and (-1, 1, 0) respectively. If A, B and C are the points at which P intersects the coordinate axes, find the volume of the tetrahedron whose vertices are A, B, C and the origin. 
Answer: Direction of plane = \( \vec{L_1} \times \vec{L_2} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 0 & -1 \\ -1 & 1 & 0 \end{vmatrix} = \hat{i} + \hat{j} + \hat{k} \)
\( \vec{n} = (1, 1, 1) \)
Equation of plane \( x + y + z = d \) passes through (1, 1, 1)
\( \implies d = 3 \)
\( x + y + z = 3 \)
Points of intersection with axes are A(3, 0, 0), B(0, 3, 0), C(0, 0, 3)
Volume of OABC = \( \frac{1}{6} \begin{vmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{vmatrix} = \frac{27}{6} = \frac{9}{2} \) cubic units.

Question. Find the equation of the plane containing the line \( 2x - y + z - 3 = 0 \), \( 3x + y + z = 5 \) and at a distance of \( 1/\sqrt{6} \) from the point (2, 1, -1). 
Answer: Required plane \( \pi_1 + \lambda\pi_2 = 0 \)
\( 2x - y + z - 3 + \lambda(3x + y + z - 5) = 0 \)
\( (3\lambda + 2)x + (\lambda - 1)y + (\lambda + 1)z - 5\lambda - 3 = 0 \) ....(1)
Distance of plane (1) from point (2, 1, -1) is \( \frac{1}{\sqrt{6}} \)
\( \implies \frac{|6\lambda + 4 + \lambda - 1 - \lambda - 1 - 5\lambda - 3|}{\sqrt{(3\lambda + 2)^2 + (\lambda - 1)^2 + (\lambda + 1)^2}} = \frac{1}{\sqrt{6}} \)
\( \implies 6(\lambda - 1)^2 = 11\lambda^2 + 12\lambda + 6 \)
\( \implies \lambda = 0, -\frac{24}{5} \)
The planes are \( 2x - y + z - 3 = 0 \) and \( 62x + 29y + 19z - 109 = 0 \).

Question. A plane passes through (1, -2, 1) and is perpendicular to two planes \( 2x - 2y + z = 0 \) and \( x - y + 2z = 4 \). The distance of the plane from the point (1, 2, 2) is 
(a) 0
(b) 1
(c) \( \sqrt{2} \)
(d) \( 2\sqrt{2} \)
Answer: (d) \( 2\sqrt{2} \)

Question. Match the following
Column-I
(A) Two rays in the first quadrant \( x + y = |a| \) and \( ax - y = 1 \) intersects each other in the interval \( a \in (a_0, \infty) \), the value of \( a_0 \) is
(B) Point \( (\alpha, \beta, \gamma) \) lies on the plane \( x + y + z = 2 \). Let \( \vec{a} = \alpha\hat{i} + \beta\hat{j} + \gamma\hat{k} \). \( \hat{k} \times (\hat{k} \times \vec{a}) = 0 \), then \( \gamma \) equal
(C) \( \int_0^1 (1-y^2) dy + \left| \int_0^1 (y^2-1) dy \right| \)
(D) In a \( \Delta ABC \), if \( \sin A \sin B \sin C + \cos A \cos B = 1 \), then the value of \( \sin C \) equal
Column-II
(P) 2
(Q) 4/3
(R) \( \int_0^1 \sqrt{1-x} dx + \int_{-1}^0 \sqrt{1+x} dx \)
(S) 1
Answer:
(A) matches (S)
(B) matches (P)
(C) matches (Q)
(D) matches (S)

Question. Consider the planes \( 3x - 6y - 2z = 15 \) and \( 2x + y - 2z = 5 \).
Statement-I : The parametric equations of the line of intersection of the given planes are \( x = 3 + 14t \), \( y = 1 + 2t \), \( z = 15t \).
Statement-II : The vector \( 14\hat{i} + 2\hat{j} + 15\hat{k} \) is parallel to the line of intersection of given planes.
(a) Statement-I is true, Statement-II is true; Statement-II is correct explanation for Statement-I
(b) Statement-I is true, Statement-II is true; Statement-II is NOT correct explanation for Statement-I
(c) Statement-I is true, Statement-II is False
(d) Statement-I is False, Statement-II is True
Answer: (d) Statement-I is False, Statement-II is True

Question. Consider the following linear equations
\( ax + by + cz = 0 \)
\( bx + cy + az = 0 \)
\( cx + ay + bz = 0 \)
Match the conditions/expressions in Column-I with statements in Column-II.
Column-I
(A) \( a + b + c \neq 0 \) and \( a^2 + b^2 + c^2 = ab + bc + ca \)
(B) \( a + b + c = 0 \) and \( a^2 + b^2 + c^2 \neq ab + bc + ca \)
(C) \( a + b + c \neq 0 \) and \( a^2 + b^2 + c^2 \neq ab + bc + ca \)
(D) \( a + b + c = 0 \) and \( a^2 + b^2 + c^2 = ab + bc + ca \)
Column-II
(P) the equation represent planes meeting only at a single point.
(Q) the equation represent the line \( x = y = z \)
(R) the equation represent identical planes
(S) the equation represent the whole of the three dimensional space.
Answer:
(A) matches (R)
(B) matches (Q)
(C) matches (P)
(D) matches (S)

Question. Consider three planes.
\( P_1 : x - y + z = 1 \)
\( P_2 : x + y - z = -1 \)
\( P_3 : x - 3y + 3z = 2 \)
Let \( L_1 \), \( L_2 \), \( L_3 \) be the lines of intersection of the planes \( P_2 \) and \( P_3 \), \( P_3 \) and \( P_1 \), and \( P_1 \) and \( P_2 \) respectively.
Statement-I : At least two of the lines \( L_1 \), \( L_2 \) and \( L_3 \) are non-parallel.
Statement-II : The three planes do not have a common point.
(a) Statement-I is true, Statement-II is true; Statement-II is correct explanation for Statement-I
(b) Statement-I is true, Statement-II is true; Statement-II is NOT correct explanation for Statement-I
(c) Statement-I is true, Statement-II is False
(d) Statement-I is False, Statement-II is True
Answer: (d) Statement-I is False, Statement-II is True