Class 11 Mathematics Introduction To Three-Dimensional Geometry MCQs Set 06

Straight Line

Question. The equation of straight line passing through the point (a, b, c) and parallel to z-axis, is
(a) \(\frac{x-a}{1} = \frac{y-b}{1} = \frac{z-c}{0}\)
(b) \(\frac{x-a}{0} = \frac{y-b}{1} = \frac{z-c}{1}\)
(c) \(\frac{x-a}{0} = \frac{y-b}{0} = \frac{z-c}{1}\)
(d) \(\frac{x-a}{0} = \frac{y-b}{0} = \frac{z-c}{0}\)
Answer: (c)

Question. Equation of x-axis is
(a) \(\frac{x}{1} = \frac{y}{1} = \frac{z}{1}\)
(b) \(\frac{x}{0} = \frac{y}{1} = \frac{z}{1}\)
(c) \(\frac{x}{1} = \frac{y}{0} = \frac{z}{0}\)
(d) \(\frac{x}{0} = \frac{y}{0} = \frac{z}{1}\)
Answer: (c)

Question. The equation of straight line passing through the points (a, b, c) and (a-b, b-c, c-a), is
(a) \(\frac{x-a}{a-b} = \frac{y-b}{b-c} = \frac{z-c}{c-a}\)
(b) \(\frac{x-a}{b} = \frac{y-b}{c} = \frac{z-c}{a}\)
(c) \(\frac{x-a}{a} = \frac{y-b}{b} = \frac{z-c}{c}\)
(d) \(\frac{x-a}{b-a} = \frac{y-b}{c-b} = \frac{z-c}{a-c}\)
Answer: (b)

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) \(\frac{x+3}{1} = \frac{y-2}{1} = \frac{z+4}{1}\)
(d) None of these
Answer: (c)

Question. The straight line through (a, b, c) and parallel to x-axis are
(a) \(\frac{x-a}{1} = \frac{y-b}{0} = \frac{z-c}{0}\)
(b) \(\frac{x-a}{0} = \frac{y-b}{1} = \frac{z-c}{0}\)
(c) \(\frac{x-a}{0} = \frac{y-b}{0} = \frac{z-c}{1}\)
(d) \(\frac{x-a}{1} = \frac{y-b}{1} = \frac{z-c}{1}\)
Answer: (a)

Question. Equation of the line passing through the point (1, 2, 3) and parallel to the line \(\frac{x-6}{12} = \frac{y-2}{4} = \frac{z+7}{5}\) is given by
(a) \(\frac{x+1}{12} = \frac{y+2}{4} = \frac{z+3}{5}\)
(b) \(\frac{x-1}{l} = \frac{y-2}{m} = \frac{z-3}{n}\), where \(12l+4m+5n=0\)
(c) \(\frac{x-1}{12} = \frac{y-2}{4} = \frac{z-3}{5}\)
(d) None of these
Answer: (c)

Question. Let G be the centroid of the triangle formed by the points (1, 2, 0), (2, 1, 1), (0, 0, 2). Then equation of the line OG is given by
(a) \(x = y = z\)
(b) \(x-1 = y-1 = z-1\)
(c) \(\frac{x-1}{1} = \frac{y-1}{1} = \frac{z-1}{1}\)
(d) None of these
Answer: (a)

Question. The direction cosines of the line \(\frac{x+1}{-3} = \frac{3y+2}{6} = \frac{z-3}{12}\) are
(a) \(\left(\frac{1}{3}, \frac{2}{3}, 0\right)\)
(b) \((1, \frac{2}{3}, -1)\)
(c) \(\left(\frac{1}{2}, 1, -\frac{1}{2}\right)\)
(d) \(\left(-\frac{1}{\sqrt{6}}, \frac{2}{\sqrt{6}}, -\frac{1}{\sqrt{6}}\right)\)
Answer: (d)

Question. The direction cosines of the line \(x=y=z\) are
(a) \(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\)
(b) \(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\)
(c) 1, 1, 1
(d) None of these
Answer: (a)

Question. The direction ratio's of the line \(x-y+z-5 = 0 = x-3y-6\) are
(a) 3, 1, –2
(b) 2, –4, 1
(c) \(\frac{3}{\sqrt{14}}, \frac{1}{\sqrt{14}}, -\frac{2}{\sqrt{14}}\)
(d) \(\frac{2}{\sqrt{41}}, -\frac{4}{\sqrt{41}}, \frac{1}{\sqrt{41}}\)
Answer: (a)

Question. The angle between two lines \(\frac{x+1}{2} = \frac{y+3}{2} = \frac{z-4}{-1}\) and \(\frac{x-4}{1} = \frac{y+4}{2} = \frac{z+1}{2}\) is
(a) \(\cos^{-1}\left(\frac{1}{9}\right)\)
(b) \(\cos^{-1}\left(\frac{2}{9}\right)\)
(c) \(\cos^{-1}\left(\frac{3}{9}\right)\)
(d) \(\cos^{-1}\left(\frac{4}{9}\right)\)
Answer: (d)

Question. The angle between the lines \(\frac{x+4}{1} = \frac{y-3}{2} = \frac{z+2}{3}\) and \(\frac{x}{3} = \frac{y-1}{-2} = \frac{z}{1}\) is
(a) \(\sin^{-1}\left(\frac{1}{7}\right)\)
(b) \(\cos^{-1}\left(\frac{2}{7}\right)\)
(c) \(\cos^{-1}\left(\frac{1}{7}\right)\)
(d) None of these
Answer: (d)

Question. The angle between the lines \(\frac{x}{1} = \frac{y}{0} = \frac{z}{-1}\) and \(\frac{x}{3} = \frac{y}{4} = \frac{z}{5}\) is
(a) \(\cos^{-1} \frac{1}{5}\)
(b) \(\cos^{-1} \frac{1}{3}\)
(c) \(\cos^{-1} \frac{1}{2}\)
(d) \(\cos^{-1} \frac{1}{4}\)
Answer: (a)

Question. The value of \(\lambda\) for which the lines \(\frac{x-1}{1} = \frac{y-2}{\lambda} = \frac{z+1}{-1}\) and \(\frac{x+1}{-\lambda} = \frac{y+1}{2} = \frac{z-2}{1}\) are perpendicular to each other is
(a) 0
(b) 1
(c) –1
(d) None of these
Answer: (b)

Question. The angle between the straight lines \(\frac{x+1}{2} = \frac{y-2}{5} = \frac{z+3}{4}\) and \(\frac{x-1}{1} = \frac{y+2}{2} = \frac{z-3}{-3}\) is
(a) 45°
(b) 30°
(c) 60°
(d) 90°
Answer: (d)

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

Question. The angle between the lines \(x=1, y=2\) and \(y=-1, z=0\) is
(a) 90°
(b) 30°
(c) 60°
(d) 0°
Answer: (a)

Question. The straight line \(\frac{x-3}{3} = \frac{y-2}{1} = \frac{z-1}{0}\) is
(a) Parallel to x-axis
(b) Parallel to y-axis
(c) Parallel to z-axis
(d) Perpendicular to z-axis
Answer: (d)

Question. The lines \(\frac{x-1}{2} = \frac{y-1}{3} = \frac{z-3}{0}\) and \(\frac{x-2}{0} = \frac{y-3}{0} = \frac{z-4}{1}\) are
(a) Parallel
(b) Skew
(c) Coincident
(d) Perpendicular
Answer: (b)

Question. The straight lines \(\frac{x-1}{1} = \frac{y-2}{2} = \frac{z-3}{3}\) and \(\frac{x-1}{2} = \frac{y-2}{2} = \frac{z-3}{-2}\) are
(a) Parallel lines
(b) Intersecting at 60°
(c) Skew lines
(d) Intersecting at right angle
Answer: (d)

Question. The angle between the lines \(\frac{x-2}{3} = \frac{y+1}{-2}, z=2\) and \(\frac{x-1}{1} = \frac{2y+3}{3} = \frac{z+5}{2}\) is
(a) \(\pi/2\)
(b) \(\pi/3\)
(c) \(\pi/6\)
(d) None of these
Answer: (a)

Question. The lines \(\frac{x}{1} = \frac{y}{2} = \frac{z}{3}\) and \(\frac{x-1}{-2} = \frac{y-2}{-4} = \frac{z-3}{-6}\) are
(a) Parallel
(b) Intersecting
(c) Skew
(d) Coincident
Answer: (a)

Question. The lines \(\frac{x-1}{2} = \frac{y-2}{4} = \frac{z-3}{7}\) and \(\frac{x-1}{4} = \frac{y-2}{5} = \frac{z-3}{7}\) are
(a) Parallel
(b) Intersecting
(c) Skew
(d) Perpendicular
Answer: (b)

Question. Lines \(r = a_1 + t b_1\) and \(r = a_2 + s b_2\) are parallel iff
(a) \(b_1\) is parallel to \(a_2 - a_1\)
(b) \(b_2\) is parallel to \(a_2 - a_1\)
(c) \(b_1 = \lambda b_2\) for some real \(\lambda\)
(d) None of these
Answer: (c)

Question. The equation of the line passing through the points \(a_1i+a_2j+a_3k\) and \(b_1i+b_2j+b_3k\)
(a) \((a_1i+a_2j+a_3k) + t(b_1i+b_2j+b_3k)\)
(b) \((a_1i+a_2j+a_3k) - t(b_1i+b_2j+b_3k)\)
(c) \((1-t)(a_1i+a_2j+a_3k) + t(b_1i+b_2j+b_3k)\)
(d) None of these
Answer: (c)

Question. The vector equation of the line joining the points \(i-2j+k\) and \(-2j+3k\) is
(a) \(r = t(i+j+k)\)
(b) \(r = (1-t)(i-2j+k) + t(3k-2j)\)
(c) \(r = (i-2j+k) + t(2k-i)\)
(d) \(r = t(2k-i)\)
Answer: (b)

Question. The acute angle between the line joining the points (2, 1, –3), (–3, 1, 7) and a line parallel to \(\frac{x-1}{3} = \frac{y}{4} = \frac{z+3}{5}\) through the point (–1, 0, 4) is
(a) \(\cos^{-1} \left(\frac{7}{5\sqrt{10}}\right)\)
(b) \(\cos^{-1} \left(\frac{1}{\sqrt{10}}\right)\)
(c) \(\cos^{-1} \left(\frac{3}{5\sqrt{10}}\right)\)
(d) \(\cos^{-1} \left(\frac{1}{5\sqrt{10}}\right)\)
Answer: (d)

Question. The shortest distance between the lines \(\frac{x-3}{3} = \frac{y-8}{-1} = \frac{z-3}{1}\) and \(\frac{x+3}{-3} = \frac{y+7}{2} = \frac{z-6}{4}\) is
(a) \(\sqrt{30}\)
(b) \(2\sqrt{30}\)
(c) \(5\sqrt{30}\)
(d) \(3\sqrt{30}\)
Answer: (d)

Question. Shortest distance between lines \(\frac{x-6}{1} = \frac{y-2}{-2} = \frac{z-2}{2}\) and \(\frac{x+4}{3} = \frac{y}{-2} = \frac{z+1}{-2}\) is
(a) 108
(b) 9
(c) 27
(d) None of these
Answer: (b)

Question. The lines \(l_1\) and \(l_2\) intersect. The shortest distance between them is
(a) Positive
(b) Zero
(c) Negative
(d) Infinity
Answer: (b)

Question. The shortest distance between two straight lines given by \(\frac{x-4}{1} = \frac{y+1}{2} = \frac{z-0}{-3}\) and \(\frac{x-1}{1} = \frac{y+1}{4} = \frac{z-2}{-5}\) is
(a) \(\frac{2}{\sqrt{5}}\)
(b) \(\frac{3}{\sqrt{5}}\)
(c) \(\frac{6}{\sqrt{5}}\)
(d) None of these
Answer: (c)

Question. The shortest distance between the lines \(r = (3i-2j-2k)+it\) and \(r = i-j+2k+js\) (t and s being parameters) is
(a) \(\sqrt{21}\)
(b) \(\sqrt{102}\)
(c) 4
(d) 3
Answer: (d)

Advance Level

Question. The equation of the line passing through the point (1, 2, -4) and perpendicular to the two lines \(\frac{x-8}{2} = \frac{y+19}{-16} = \frac{z-10}{7}\) and \(\frac{x-15}{3} = \frac{y-29}{8} = \frac{z-5}{-5}\), will be
(a) \(\frac{x-1}{2} = \frac{y-2}{3} = \frac{z+4}{6}\)
(b) \(\frac{x-1}{2} = \frac{y-2}{3} = \frac{z+4}{8}\)
(c) \(\frac{x-1}{3} = \frac{y-2}{2} = \frac{z+4}{8}\)
(d) None of these
Answer: (a)

Question. The equation of straight line \(3x+2y-z-4=0 ; 4x+2y-2z+3=0\) in the symmetrical form is
(a) \(\frac{x-2}{3} = \frac{y-5}{2} = \frac{z}{5}\)
(b) \(\frac{x+2}{5} = \frac{y-5}{3} = \frac{z}{2}\)
(c) \(\frac{x+2}{3} = \frac{y-5}{2} = \frac{z}{5}\)
(d) None of these
Answer: (d)

Question. The point of intersection of lines \(\frac{x-4}{5} = \frac{y-1}{2} = \frac{z}{1}\) and \(\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}\) is
(a) (–1, –1, –1)
(b) (–1, –1, 1)
(c) (1, –1, –1)
(d) (–1, 1, –1)
Answer: (d)

Question. The length and foot of the perpendicular from the point (2, –1, 5) to the line \(\frac{x-11}{10} = \frac{y+2}{-4} = \frac{z+8}{-11}\) are
(a) \(\sqrt{14}, (1, 2, -3)\)
(b) \(\sqrt{14}, (1, -2, 3)\)
(c) \(\sqrt{14}, (1, 2, 3)\)
(d) None of these
Answer: (a)

Question. The perpendicular distance of the point (2, 4, –1) from the line \( \frac{x+5}{1} = \frac{y+3}{4} = \frac{z-6}{-9} \) is
(a) 3
(b) 5
(c) 7
(d) 9
Answer: (c)

Question. Distance of the point \( (x_1, y_1, z_1) \) from the line \( \frac{x - x_2}{l} = \frac{y - y_2}{m} = \frac{z - z_2}{n} \), where l, m and n are the direction cosines of line is
(a) \( \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2 - [l(x_1 - x_2) + m(y_1 - y_2) + n(z_1 - z_2)]^2} \)
(b) \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \)
(c) \( \sqrt{(x_2 - x_1)l + (y_2 - y_1)m + (z_2 - z_1)n} \)
(d) None of these
Answer: (a)

Question. The length of the perpendicular from point (1, 2, 3) to the line \( \frac{x-6}{3} = \frac{y-7}{2} = \frac{z-7}{-2} \) is
(a) 5
(b) 6
(c) 7
(d) 8
Answer: (c)

Question. The foot of the perpendicular from (0, 2, 3) to the line \( \frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3} \) is
(a) (–2, 3, 4)
(b) (2, –1, 3)
(c) (2, 3, –1)
(d) (3, 2, –1)
Answer: (b)

Question. The foot of the perpendicular from (1, 2, 3) to the line joining the points (6, 7, 7) and (9, 9, 5) is
(a) (5, 3, 9)
(b) (3, 5, 9)
(c) (3, 9, 5)
(d) (3, 9, 9)
Answer: (d)

Question. If the equation of a line through a point a and parallel to vector b is \( \mathbf{r} = \mathbf{a} + t\mathbf{b} \), where t is a parameter, then its perpendicular distance from the point c is
(a) \( |(\mathbf{c}-\mathbf{b}) \times \mathbf{a}| \div |\mathbf{a}| \)
(b) \( |(\mathbf{c}-\mathbf{a}) \times \mathbf{b}| \div |\mathbf{b}| \)
(c) \( |(\mathbf{a}-\mathbf{b}) \times \mathbf{c}| \div |\mathbf{c}| \)
(d) \( |(\mathbf{a}-\mathbf{b}) \times \mathbf{c}| \div |\mathbf{a}+\mathbf{c}| \)
Answer: (b)

Question. The distance of the point \( B(\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}) \) from the line which is passing through \( A(4\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}) \) and which is parallel to the vector \( \mathbf{C} = 2\mathbf{i} + 3\mathbf{j} + 6\mathbf{k} \) is
(a) 10
(b) \( \sqrt{10} \)
(c) 100
(d) None of these
Answer: (b)