JEE Mathematics Points Direction Cosines and Direction Rations MCQs

Type – 1

Choose the most appropriate option (a, b, c or d).

Question. A line makes angles \( \alpha, \beta, \gamma \) with the positive directions of the axes of reference. The value of \( \cos 2\alpha + \cos 2\beta + \cos 2\gamma \) is
(a) 1
(b) 2
(c) -1
(d) 0
Answer: (c) -1

Question. If a line makes angles of 60° and 45° with the positive directions of the x-axis and y-axis respectively then the acute angle between the line and the z-axis is
(a) 60°
(b) 45°
(c) 75°
(d) 15°
Answer: (a) 60°

Question. If \( \theta \) is an angle given by \( \cos \theta = \frac{\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma}{\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma} \) where \( \alpha, \beta, \gamma \) are the angles made by a line with the positive directions of the axes of reference then the measure of \( \theta \) is
(a) \( \frac{\pi}{4} \)
(b) \( \frac{\pi}{6} \)
(c) \( \frac{\pi}{2} \)
(d) \( \frac{\pi}{3} \)
Answer: (d) \( \frac{\pi}{3} \)

Question. If the direction cosines of the line joining the origin and a point at unit distance from the origin are \( \frac{1}{\sqrt{3}}, -\frac{1}{2}, \lambda \), then the point has coordinates
(a) \( \left( \frac{1}{\sqrt{3}}, -\frac{1}{2}, \frac{\sqrt{5}}{2\sqrt{3}} \right) \)
(b) \( \left( \frac{2}{\sqrt{3}}, -1, \frac{\sqrt{5}}{4} \right) \)
(c) \( \left( \frac{-2}{\sqrt{3}}, 1, \frac{\sqrt{5}}{4} \right) \)
(d) None of the options
Answer: (a) \( \left( \frac{1}{\sqrt{3}}, -\frac{1}{2}, \frac{\sqrt{5}}{2\sqrt{3}} \right) \)

Question. The direction ratios of two parallel lines are 4, -3, -1 and \( \lambda + \mu, 1 + \mu, 2 \). The value of the pair \( (\lambda, \mu) \) is
(a) (1, 7)
(b) (-1, -7)
(c) (7, 1)
(d) no fixed value
Answer: (b) (-1, -7)

Question. The direction ratios of two perpendicular lines are 1, -3, 5 and \( \lambda, 1 + \lambda, 2 + \lambda \). Then \( \lambda \) is
(a) \( -\frac{7}{3} \)
(b) \( \frac{7}{2} \)
(c) \( -\frac{1}{4} \)
(d) \( \frac{1}{2} \)
Answer: (a) \( -\frac{7}{3} \)

Question. The direction cosines of the line joining the points (2, 3, -1) and (3, -2, 1)
(a) –1, 5, –2
(b) \( \frac{1}{\sqrt{30}}, -\frac{5}{\sqrt{30}}, \frac{2}{\sqrt{30}} \)
(c) \( -\frac{1}{\sqrt{30}}, \frac{5}{\sqrt{30}}, -\frac{2}{\sqrt{30}} \)
(d) None of the options
Answer: (b) \( \frac{1}{\sqrt{30}}, -\frac{5}{\sqrt{30}}, \frac{2}{\sqrt{30}} \)

Question. If A = (1, 2, 3), B = (-2, 4, \( \lambda \)) and \( \angle AOB = \pi/2 \) where O is the origin then \( \lambda \) is
(a) 6
(b) -6
(c) 0
(d) -2
Answer: (d) -2

Question. Let A = (2, 3, -1), B = (1, -1, 2). A point C on the y-axis such that AB \( \perp \) BC has the coordinates
(a) \( \left( 0, \frac{9}{4}, 0 \right) \)
(b) \( \left( 0, -\frac{4}{9}, 0 \right) \)
(c) \( \left( 0, -\frac{9}{4}, 0 \right) \)
(d) \( \left( 0, \frac{9}{4}, 0 \right) \)
Answer: (c) \( \left( 0, -\frac{9}{4}, 0 \right) \)

Question. If the points A(1, 2, -1), B(2, 6, 2) and C(\( \lambda \), -2, -A) are collinear then \( \lambda \) is
(a) 0
(b) 2
(c) -2
(d) 1
Answer: (a) 0

Question. If A = (1, 1, 1), B = (2, -1, 3), C = (0, 4, -2), D = (1, 2, X) and AB, AC and AD are coplanar then \( \lambda \) is
(a) 1
(b) 0
(c) -1
(d) 3
Answer: (b) 0

Question. The angle between the lines joining the points (1, 0, -3), (2, -1, 2), and (1, 1, 11), (3, 2, 0) is
(a) \( \cos^{-1} \frac{2}{9} \)
(b) 90°
(c) 0°
(d) \( \cos^{-1} \frac{2\sqrt{2}}{9} \)
Answer: (d) \( \cos^{-1} \frac{2\sqrt{2}}{9} \)

Question. The angle between the lines joining the points (1, 1, 0), (-3, -\( \sqrt{3} \) + 1, 3), and (0, -1, 0), (-1, \( \sqrt{3} \) - 1, \( \lambda \)) is \( \cos^{-1} \left( \frac{1}{8} \right) \). If X is an integer then its value is
(a) 1
(b) 0
(c) -1
(d) 2
Answer: (b) 0

Question. \( l_r, m_r, n_r \); r = 1, 2, 3; are the direction cosines of three mutually perpendicular lines. The direction cosines of the line equally inclined to them are
(a) \( l_1 + l_2 + l_3, m_1 + m_2 + m_3, n_1 + n_2 + n_3 \)
(b) \( \frac{l_1 + l_2 + l_3}{3}, \frac{m_1 + m_2 + m_3}{3}, \frac{n_1 + n_2 + n_3}{3} \)
(c) \( \frac{1}{\sqrt{3}}(l_1 + l_2 + l_3), \frac{1}{\sqrt{3}}(m_1 + m_2 + m_3), \frac{1}{\sqrt{3}}(n_1 + n_2 + n_3) \)
(d) \( \frac{l_1 l_2 l_3}{\sqrt{3}}, \frac{m_1 m_2 m_3}{\sqrt{3}}, \frac{n_1 n_2 n_3}{\sqrt{3}} \)
Answer: (c) \( \frac{1}{\sqrt{3}}(l_1 + l_2 + l_3), \frac{1}{\sqrt{3}}(m_1 + m_2 + m_3), \frac{1}{\sqrt{3}}(n_1 + n_2 + n_3) \)

Question. Let A = (1, 2, 2), B = (2, 3, 6) and C = (3, 4, 12). The direction cosines of a line equally inclined with OA, OB and OC where O is the origin, are
(a) \( \frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}, 0 \)
(b) \( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0 \)
(c) \( \frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \)
(d) \( -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} \)
Answer: (d) \( -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} \)

Question. The angle between the lines whose direction cosines satisfy the equations \( l + m + n = 0 \) and \( l^2 = m^2 + n^2 \) is
(a) \( \frac{\pi}{6} \)
(b) \( \frac{\pi}{2} \)
(c) \( \frac{\pi}{3} \)
(d) \( \frac{\pi}{4} \)
Answer: (c) \( \frac{\pi}{3} \)

Question. The angle between two lines whose direction cosines satisfy the equations \( n = l + m \) and \( m = 2l + 3n \) is
(a) 0°
(b) 90°
(c) 60°
(d) None of the options
Answer: (a) 0°

Question. The value of \( \lambda \) for which the triangle ABC whose vertices are A(6, 10, 10), B(1, 0, -5) and C(6, -10, \( \lambda \)) is right-angled at B, is
(a) 0
(b) 30
(c) \( \frac{70}{3} \)
(d) \( \frac{3}{70} \)
Answer: (a) 0

Question. Let A = (1, 2, 3), B = (-1, -2, -1), C = (2, 3, 2) and D = (4, 7, 6). Then ABCD is a
(a) rectangle
(b) square
(c) parallelogram
(d) None of the options
Answer: (c) parallelogram

Question. The points (0, 0, 0), (0, 2, 0), (1, 0, 0), (0, 0, 4) are
(a) coplanar
(b) vertices of a parallelogram
(c) vertices of a rectangle
(d) on a sphere
Answer: (d) on a sphere

Question. If A = (5, -1, 1), B = (7, -4, 7), C = (1, -6, 10), D = (-1, -3, 4) then ABCD is a
(a) square
(b) rectangle
(c) rhombus
(d) None of the options
Answer: (c) rhombus

Question. If A = (0, 0, 2), B = \( (\sqrt{2}, \sqrt{2}, 2) \), C = \( (\sqrt{2}, \sqrt{2}, 0) \) and \( D = \left( \frac{8\sqrt{2}-20}{17}, \frac{12\sqrt{2}+4}{17}, \frac{20-8\sqrt{2}}{17} \right) \) then BCD is a
(a) rhombus
(b) square
(c) parallelogram
(d) None of the options
Answer: (d) None of the options

Question. If the vertices of a triangle are (-1, 6, -A), (2, 1, 1) and (5, -1, 0) then the centroid of the triangle is
(a) (6, 6, -3)
(b) (2, 2, -1)
(c) \( \left( 3, 3, -\frac{3}{2} \right) \)
(d) None of the options
Answer: (b) (2, 2, -1)

Question. If two vertices of a triangle ABC are A(-1, 2, 4) and B(2, -3, 0), and the centroid is (2, 0, 2) then the vertex C has the coordinates
(a) (5, 1, 2)
(b) \( \left( 1, -\frac{1}{3}, \frac{7}{3} \right) \)
(c) \( \left( 3, -\frac{2}{3}, \frac{5}{3} \right) \)
(d) None of the options
Answer: (a) (5, 1, 2)

Question. Four vertices of a tetrahedron are (0, 0, 0), (4, 0, 0), (0, -8, 0) and (0, 0, 12). Its centroid has the coordinates
(a) \( \left( \frac{4}{3}, -\frac{8}{3}, 4 \right) \)
(b) (2, -4, 6)
(c) (1, -2, 3)
(d) None of the options
Answer: (c) (1, -2, 3)

Question. Three vertices of a tetrahedron are (0, 0, 0), (6, -5, -1) and (-4, 1, 3). If the centroid of the tetrahedron be (1, -2, 5) then the fourth vertex is
(a) (2, -4, 18)
(b) (2, -4, -18)
(c) \( \left( \frac{3}{4}, -\frac{3}{2}, \frac{7}{4} \right) \)
(d) None of the options
Answer: (a) (2, -4, 18)

Question. The points A(1, 2, -1), B(2, 5, -2), C(4, 4, -3) and D(3, 1, -2) are
(a) collinear
(b) vertices of a rectangle
(c) vertices of a square
(d) vertices of a rhombus
Answer: (b) vertices of a rectangle

Question. The projection of a line segment on the axes of reference are 3, 4 and 12 respectively. The length of the line segment is
(a) 19
(b) \( \frac{19}{3} \)
(c) 5
(d) 13
Answer: (d) 13

Question. ABC is a triangle where A = (2, 3, 5), B = (-1, 3, 2) and C = (\( \lambda, 5, \mu \)). If the median through A is equally inclined with the axes then
(a) \( \lambda = 14, \mu = 20 \)
(b) \( \lambda = 7, \mu = 10 \)
(c) \( \lambda = \frac{7}{2}, \mu = 5 \)
(d) \( \lambda = 10, \mu = 7 \)
Answer: (b) \( \lambda = 7, \mu = 10 \)

Question. The volume of the tetrahedron whose vertices are (0, 1, 2), (4, 3, 6), (2, 3, 2) and (3, 0, 1) is (in unit\(^3\))
(a) 0
(b) 1
(c) 6
(d) 3
Answer: (c) 6

Question. If (1, -1, 0), (-2, 1, 8) and (-1, 2, 7) are three consecutive vertices of a parallelogram then the fourth vertex is
(a) (2, 0, -1)
(b) (1, 0, -1)
(c) (1, -2, 0)
(d) (0, -2, 1)
Answer: (a) (2, 0, -1)

Question. Let \( P_r(x_r, y_r, z_r) \); r = 1, 2, 3; be three points where \( x_1, x_2, x_3, y_1, y_2, y_3 \) and \( z_1, z_2, z_3 \) are each in GP with the same common ratio. Then \( P_1, P_2, P_3 \) are
(a) coplanar points
(b) collinear points
(c) vertices of an equilateral triangle
(d) None of the options
Answer: (b) collinear points

Type 2

Choose the correct options. One or more options may be correct.

Question. A point Q at a distance 3 from the point P(1, 1, 1) lying on the line joining the points A(0, -1, 3) and P, has the coordinates
(a) (2, 3, -1)
(b) (4, 7, -5)
(c) (0, -1, 3)
(d) (-2, -5, 7)
Answer: (a) (2, 3, -1) and (c) (0, -1, 3)

Question. If A = (2, -3, 7), B = (-1, 4, -5) and P is a point on the line AB such that AP:BP = 3:2 then P has the coordinates
(a) \( \left( \frac{4}{5}, -\frac{1}{5}, \frac{11}{5} \right) \)
(b) \( \left( \frac{1}{5}, \frac{6}{5}, \frac{1}{5} \right) \)
(c) \( \left( -\frac{7}{5}, \frac{18}{5}, -\frac{29}{5} \right) \)
(d) (– 7, 18, – 29)
Answer: (b) \( \left( \frac{1}{5}, \frac{6}{5}, \frac{1}{5} \right) \) and (d) (– 7, 18, – 29)

Question. If the direction ratios of a line are 1 + X, 1 - X, 2, and it makes an angle of 60° with the y-axis then X is
(a) \( 1 + \sqrt{3} \)
(b) \( 2 + \sqrt{3} \)
(c) \( 1 - \sqrt{3} \)
(d) \( 2 - \sqrt{5} \)
Answer: (b) \( 2 + \sqrt{3} \) and (d) \( 2 - \sqrt{5} \)