JEE Mathematics Straight Lines MCQs Set 05

Type – 1

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

Question. The direction cosines of a line whose equations are \( \frac{x - 1}{2} = \frac{y + 3}{4} = \frac{z - 2}{-3} \) are
(a) \( \frac{1}{\sqrt{14}}, \frac{-3}{\sqrt{14}}, \frac{2}{\sqrt{14}} \)
(b) \( \frac{2}{\sqrt{29}}, \frac{4}{\sqrt{29}}, \frac{-3}{\sqrt{29}} \)
(c) \( \frac{1}{\sqrt{29}}, \frac{-3}{\sqrt{29}}, \frac{2}{\sqrt{29}} \)
(d) \( 2, 4, -3 \)
Answer: (b) \( \frac{2}{\sqrt{29}}, \frac{4}{\sqrt{29}}, \frac{-3}{\sqrt{29}} \)

Question. The equations of the line passing through the point (1, 2, 3) having the direction ratios 3, 2, 1 are
(a) \( \frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z - 3}{3} \)
(b) \( \frac{x}{3} = \frac{y}{6} = \frac{z}{9} \)
(c) \( \frac{x + 2}{3} = \frac{y}{2} = \frac{z - 2}{1} \)
(d) None of the options
Answer: (c) \( \frac{x + 2}{3} = \frac{y}{2} = \frac{z - 2}{1} \)

Question. The equations of the line passing through the points (-2, 1, 0) and (3, 4, -1) are
(a) \( \frac{x + 7}{5} = \frac{y + 2}{3} = \frac{z - 1}{-1} \)
(b) \( \frac{x + 2}{1} = \frac{y - 1}{3} = \frac{z}{5} \)
(c) \( \frac{x + 3}{5} = \frac{y + 4}{3} = \frac{z - 1}{-1} \)
(d) None of the options
Answer: (a) \( \frac{x + 7}{5} = \frac{y + 2}{3} = \frac{z - 1}{-1} \)

Question. The coordinates of a point on the line \( \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z}{\sqrt{3}} \) at a distance 1 unit from the point \( (-1, -1, -\sqrt{3}) \) are
(a) \( (-\frac{3}{2}, -\frac{7}{4}, -\frac{5}{4}\sqrt{3}) \)
(b) \( (\frac{3}{2}, \frac{11}{4}, \frac{\sqrt{3}}{4}) \)
(c) \( (\frac{1}{2}, \frac{5}{4}, \frac{\sqrt{3}}{4}) \)
(d) None of the options
Answer: (a) \( (-\frac{3}{2}, -\frac{7}{4}, -\frac{5}{4}\sqrt{3}) \)

Question. The equation of the locus of the point \( (1 + \frac{r}{4}, -1 + \frac{r}{3}, 2) \), where \( r \in R \), is given by
(a) \( \frac{x - 1}{4} = \frac{y + 1}{3} = \frac{z - 2}{0} \)
(b) \( \frac{x - 1}{3} = \frac{y + 1}{4} = \frac{z - 2}{0} \)
(c) \( 4x - 3y = 7 \)
(d) \( z = 2 \)
Answer: (b) \( \frac{x - 1}{3} = \frac{y + 1}{4} = \frac{z - 2}{0} \)

Question. The lines \( \frac{x + 3}{-2} = \frac{y}{1} = \frac{z - 4}{3} \) and \( \frac{x}{\lambda} = \frac{y - 1}{\lambda + 1} = \frac{z}{\lambda + 2} \) are perpendicular to each other. Then \( \lambda \) is equal to
(a) \( -\frac{5}{3} \)
(b) 4
(c) \( -\frac{1}{4} \)
(d) -4
Answer: (d) -4

Question. If the lines \( \frac{x + 2}{4\lambda + 1} = \frac{y - 1}{4} = \frac{z}{-18} \) and \( \frac{x}{-3} = \frac{y + 1}{5\mu - 3} = \frac{z - 1}{6} \) are parallel to each other then the value of the pair \( (\lambda, \mu) \) is
(a) \( (-2, \frac{1}{3}) \)
(b) \( (2, -\frac{1}{3}) \)
(c) \( (2, \frac{1}{3}) \)
(d) None of the options
Answer: (c) \( (2, \frac{1}{3}) \)

Question. The angle between the lines \( \frac{x + 2}{1} = \frac{y + 3}{-2} = \frac{z - 4}{1} \) and \( \frac{x + 1}{-1} = \frac{y}{1} = \frac{z}{0} \) is
(a) \( \frac{\pi}{6} \)
(b) \( \frac{\pi}{3} \)
(c) \( \frac{\pi}{2} \)
(d) 0
Answer: (a) \( \frac{\pi}{6} \)

Question. If \( \frac{\alpha}{\alpha'}, \frac{\beta}{\beta'}, \frac{\gamma}{\gamma'} \) are not equal, then point of intersection of the lines \( \frac{x - \alpha'}{\alpha} = \frac{y - \beta'}{\beta} = \frac{z - \gamma'}{\gamma} \) and \( \frac{x - \alpha}{\alpha'} = \frac{y - \beta}{\beta'} = \frac{z - \gamma}{\gamma'} \) is
(a) \( (\alpha - \alpha', \beta', \gamma - \gamma') \)
(b) \( (\alpha + \alpha', \beta + \beta', \gamma + \gamma') \)
(c) \( (\alpha\alpha', \beta\beta', \gamma\gamma') \)
(d) None of the options because they are nonintersecting
Answer: (b) \( (\alpha + \alpha', \beta + \beta', \gamma + \gamma') \)

Question. The equation of the straight line passing through the origin and perpendicular to the lines \( \frac{x + 1}{-3} = \frac{y - 2}{2} = \frac{z}{1} \) and \( \frac{x - 1}{1} = \frac{y}{-3} = \frac{z + 1}{2} \) has the equation
(a) \( x = y = z \)
(b) \( \frac{x}{4} = \frac{y}{3} = \frac{z}{6} \)
(c) \( \frac{x}{3} = \frac{y}{1} = \frac{z}{0} \)
(d) None of the options
Answer: (a) \( x = y = z \)

Question. The shortest distance between the line \( \frac{x - 3}{3} = \frac{y}{0} = \frac{z}{-4} \) and the y-axis is
(a) \( \frac{1}{5} \)
(b) 1
(c) 0
(d) \( \frac{12}{5} \)
Answer: (d) \( \frac{12}{5} \)

Question. The equations of the line of shortest distance between the lines \( \frac{x + 4}{4} = \frac{y - 3}{0} \) and \( \frac{x - 5}{4} = \frac{y - 3}{3} = \frac{z}{0} \) are
(a) \( \frac{x + 4}{0} = \frac{y - 2}{0} = \frac{z - 3}{1} \)
(b) \( \frac{x - 5}{0} = \frac{y - 3}{0} = \frac{z}{1} \)
(c) \( \frac{x}{0} = \frac{y}{0} = \frac{z - 3}{1} \)
(d) None of the options
Answer: (c) \( \frac{x}{0} = \frac{y}{0} = \frac{z - 3}{1} \)

Question. The projection of the line segment joining the point (6,-2,1) and the origin on the line \( \frac{x - 2}{4} = \frac{y + 1}{-3} = \frac{z - 1}{0} \) is
(a) 30
(b) 6
(c) 5
(d) None of the options
Answer: (b) 6

Question. If \( A = (p, q, r) \) and \( B = (p', q', r') \) are two points on the line \( \frac{x}{l} = \frac{y}{m} = \frac{z}{n} \) such that OA = a, OB = b then \( pp' + qq' + rr' \) is equal to
(a) \( a + b \)
(b) \( ab \)
(c) \( \sqrt{a^2 + b^2} \)
(d) None of the options
Answer: (b) \( ab \)

Question. The number of real values of k for which the lines \( \frac{x - k}{4} = \frac{y - 1}{2} = \frac{z + 1}{1} \) and \( \frac{x(k + 1)}{1} = \frac{y}{1} = \frac{z - 1}{2} \) are intersecting, is
(a) 0
(b) 2
(c) 1
(d) infinite
Answer: (d) infinite

Question. The distance between the lines \( \frac{x - 4}{2} = \frac{y + 1}{-3} = \frac{z}{6} \) and \( \frac{x}{-1} = \frac{y - 1}{3/2} = \frac{z + 1}{-3} \) is
(a) \( \sqrt{\frac{629}{7}} \)
(b) \( \frac{39}{7} \)
(c) \( \frac{\sqrt{629}}{7} \)
(d) None of the options
Answer: (c) \( \frac{\sqrt{629}}{7} \)

Question. The point A(3, -2, 4) is shifted parallel to the line \( \frac{x}{\sqrt{3}} = \frac{y - 1}{2} = \frac{z + 1}{3} \) by a distance 1. The coordinates of P in the new position are
(a) \( (\frac{12 - \sqrt{3}}{4}, -\frac{5}{2}, \frac{13}{4}) \)
(b) \( (3 + \sqrt{3}, 3, 2) \)
(c) \( (3 - \sqrt{3}, -1, -4) \)
(d) None of the options
Answer: (a) \( (\frac{12 - \sqrt{3}}{4}, -\frac{5}{2}, \frac{13}{4}) \)

Question. The image of the origin in the line \( \frac{x + 1}{2} = \frac{y - 2}{3} = \frac{z}{\sqrt{3}} \) is
(a) \( (-1, \frac{11}{2}, \frac{\sqrt{3}}{2}) \)
(b) \( (3, -\frac{5}{2}, \frac{\sqrt{3}}{2}) \)
(c) \( (-3, \frac{5}{2}, -\frac{\sqrt{3}}{2}) \)
(d) \( (1, \frac{11}{2}, -\frac{\sqrt{3}}{2}) \)
Answer: (c) \( (-3, \frac{5}{2}, -\frac{\sqrt{3}}{2}) \)

Question. The distance of the point (1, 2, \( \lambda \)) from the line \( \frac{x}{3} = \frac{y}{0} = \frac{z}{4} \) is 2. Then \( \lambda \) is
(a) \( \frac{4}{3} \)
(b) \( \frac{3}{4} \)
(c) 1
(d) nonexistent
Answer: (a) \( \frac{4}{3} \)

Question. If the lines \( \frac{x}{1} = \frac{y}{2} = \frac{z}{3} \), \( \frac{x - 1}{3} = \frac{y - 2}{-1} = \frac{z - 3}{4} \) and \( \frac{x + k}{3} = \frac{y - 1}{2} = \frac{z - 2}{h} \) are concurrent then
(a) \( h = -2, k = -6 \)
(b) \( h = \frac{1}{2}, k = 2 \)
(c) \( h = 6, k = 2 \)
(d) \( h = 2, k = \frac{1}{2} \)
Answer: (d) \( h = 2, k = \frac{1}{2} \)

Question. The number of real values of k for which the lines \( \frac{x - 1}{4} = \frac{y + 1}{3} = \frac{z}{k} \) and \( \frac{x}{1} = \frac{y - k}{3} = \frac{z - 1}{-2} \) are coplanar, is
(a) 2
(b) 1
(c) 3
(d) 0
Answer: (a) 2

Type 2

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

Question. A point on the line \( \frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z + 1}{3} \) at a distance \( \sqrt{6} \) from the origin is
(a) \( (-\frac{5}{7}, -\frac{10}{7}, \frac{13}{7}) \)
(b) \( (1, 2, -1) \)
(c) \( (\frac{5}{7}, \frac{10}{7}, -\frac{13}{7}) \)
(d) \( (-1, -2, 1) \)
Answer: (b) (1, 2, -1), (c) \( (\frac{5}{7}, \frac{10}{7}, -\frac{13}{7}) \)

Question. The direction cosines of a line passing through the origin and cutting the line \( \frac{x + 2}{1} = \frac{y - 1}{2} = \frac{z}{-1} \) at \( \cos^{-1}\sqrt{\frac{6}{11}} \) are
(a) \( \frac{-1}{\sqrt{11}}, \frac{3}{\sqrt{11}}, \frac{-1}{\sqrt{11}} \)
(b) \( \frac{1}{\sqrt{11}}, \frac{3}{\sqrt{11}}, \frac{1}{\sqrt{11}} \)
(c) \( \frac{-1}{\sqrt{6}}, \frac{2}{\sqrt{6}}, \frac{1}{\sqrt{6}} \)
(d) \( \frac{-3}{\sqrt{11}}, \frac{-1}{\sqrt{11}}, \frac{1}{\sqrt{11}} \)
Answer: (a) \( \frac{-1}{\sqrt{11}}, \frac{3}{\sqrt{11}}, \frac{-1}{\sqrt{11}} \), (d) \( \frac{-3}{\sqrt{11}}, \frac{-1}{\sqrt{11}}, \frac{1}{\sqrt{11}} \)