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

Question. If \(\left(\frac{1}{2}, \frac{1}{3}, n\right)\) are the direction cosines of a line, then the value of n is
(a) \(\frac{\sqrt{23}}{6}\)
(b) \(\frac{23}{6}\)
(c) \(\frac{2}{3}\)
(d) \(\frac{3}{2}\)
Answer: (a)

Question. If a line makes the angle \(\alpha, \beta, \gamma\) with three dimensional coordinate axes respectively, then \(\cos 2\alpha + \cos 2\beta + \cos 2\gamma =\)
(a) –2
(b) –1
(c) 1
(d) 2
Answer: (b)

Question. A line makes angles of 45° and 60° with the positive axes of X and Y respectively. The angle made by the same line with the positive axis of Z, is
(a) 30° or 60°
(b) 60° or 90°
(c) 90° or 120°
(d) 60° or 120°
Answer: (d)

Question. If \(\alpha, \beta, \gamma\) be the angles which a line makes with the positive direction of coordinate axes, then \(\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma =\)
(a) 2
(b) 1
(c) 3
(d) 0
Answer: (a)

Question. A line makes angles \(\alpha, \beta, \gamma\) with the coordinate axes. If \(\alpha + \beta = 90^\circ\), then \(\gamma =\)
(a) 0°
(b) 90°
(c) 180°
(d) None of these
Answer: (b)

Question. The coordinates of the points P and Q are \((x_1,y_1,z_1)\) and \((x_2,y_2,z_2)\) respectively, then the projection of the line PQ on the line whose direction cosines are l, m, n, will be
(a) \(l(x_2-x_1) + m(y_2-y_1) + n(z_2-z_1)\)
(b) \(\left(\frac{x_2-x_1}{l}\right) + \left(\frac{y_2-y_1}{m}\right) + \left(\frac{z_2-z_1}{n}\right)\)
(c) \(\frac{x_1}{l} + \frac{y_1}{m} + \frac{z_1}{n}\)
(d) \(\frac{x_2}{l} + \frac{y_2}{m} + \frac{z_2}{n}\)
Answer: (a)

Question. The projection of the line segment joining the points (–1, 0, 3) and (2, 5, 1) on the line whose direction ratios are 6, 2, 3, is
(a) 10/7
(b) 22/7
(c) 18/7
(d) None of these
Answer: (b)

Question. The projection of any line on coordinate axes be respectively 3, 4, 5, then its length is
(a) 12
(b) 50
(c) \(5\sqrt{2}\)
(d) None of these
Answer: (c)

Question. If \(\theta\) is the angle between the lines AB and CD, then projection of line segment AB on line CD is
(a) \(AB \sin \theta\)
(b) \(AB \cos \theta\)
(c) \(AB \tan \theta\)
(d) \(CD \cos \theta\)
Answer: (b)

Question. The projections of a line on the co-ordinate axes are 4, 6, 12. The direction cosines of the line are
(a) \(\frac{2}{7}, \frac{3}{7}, \frac{6}{7}\)
(b) 2, 3, 6
(c) \(\frac{2}{11}, \frac{3}{11}, \frac{6}{11}\)
(d) None of these
Answer: (a)

Question. The projections of segment PQ on the coordinate planes are –9, 12, –8 respectively. The direction cosines of PQ are
(a) \(\left<-\frac{9}{\sqrt{17}}, \frac{12}{\sqrt{17}}, -\frac{8}{\sqrt{17}}\right>\)
(b) \(\left<-9, 12, -8\right>\)
(c) \(\left<\frac{9}{289}, \frac{12}{289}, -\frac{8}{289}\right>\)
(d) \(\left<-\frac{9}{17}, \frac{12}{17}, -\frac{8}{17}\right>\)
Answer: (d)

Question. The projections of a line segment on \(x,y,z\) axes are 12, 4, 3. The length and the direction cosines of the line segments are
(a) \(13, \left<12/13, 4/13, 3/13\right>\)
(b) \(19, \left<12/19, 4/19, 3/19\right>\)
(c) \(11, \left<12/11, 4/11, 3/11\right>\)
(d) None of these
Answer: (a)

Question. The coordinates of A and B be (1, 2, 3) and (7, 8, 7), then the projections of the line segment AB on the coordinate axes are
(a) 6, 6, 4
(b) 4, 6, 4
(c) 3, 3, 2
(d) 2, 3, 2
Answer: (a)

Question. A line segment (vector) has length 21 and direction ratios (2, –3, 6). If the line makes an obtuse angle with x-axis, the components of the line (vector) are
(a) 6, –9, 18
(b) 2, –3, 6
(c) –18, 27, –54
(d) –6, 9, –18
Answer: (d)

Angle between Two Lines

Question. The angle between the pair of lines with direction ratios (1, 1, 2) and \((\sqrt{3}-1, -\sqrt{3}-1, 4)\) is
(a) 30°
(b) 45°
(c) 60°
(d) 90°
Answer: (c)

Question. The angle between a line with direction ratios \(2:2:1\) and a line joining (3, 1, 4) to (7, 2, 12) is
(a) \(\cos^{-1}(2/3)\)
(b) \(\cos^{-1}(-2/3)\)
(c) \(\tan^{-1}(2/3)\)
(d) None of these
Answer: (a)

Question. The angle between the lines whose direction cosines are proportional to (1, 2, 1) and (2, –3, 6) is
(a) \(\cos^{-1}\left(\frac{2}{\sqrt{6} \cdot 7}\right)\)
(b) \(\cos^{-1}\left(\frac{1}{\sqrt{6} \cdot 7}\right)\)
(c) \(\cos^{-1}\left(\frac{3}{\sqrt{6} \cdot 7}\right)\)
(d) \(\cos^{-1}\left(\frac{5}{\sqrt{6} \cdot 7}\right)\)
Answer: (d)

Question. If the vertices of a triangle are A (1, 4, 2), B(–2, 1, 2), C(2, –3, 4), then the angle B is equal to
(a) \(\cos^{-1}(1/\sqrt{3})\)
(b) \(\pi/2\)
(c) \(\cos^{-1}(\sqrt{6}/3)\)
(d) \(\cos^{-1} \sqrt{3}\)
Answer: (b)

Question. If the coordinates of the points P, Q, R, S be (1, 2, 3), (4, 5, 7), (–4, 3, –6) and (2, 0, 2) respectively, then
(a) \(PQ \parallel RS\)
(b) \(PQ \perp RS\)
(c) \(PQ = RS\)
(d) None of these
Answer: (a)

Question. If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (–4, 3, –6) and (2, 9, 2) respectively, then the angle between the lines AB and CD is
(a) \(\frac{\pi}{6}\)
(b) \(\frac{\pi}{4}\)
(c) \(\frac{\pi}{3}\)
(d) None of these
Answer: (d)

Question. If the angle between the lines whose direction ratios are \(2, -1, 2\) and \(a, 3, 5\) be 45°, then \(a =\)
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b)

Question. If O be the origin and \(P(2, 3, 4)\) and \(Q(1, b, 1)\) be two points such that \(OP \perp OQ\), then \(b =\)
(a) 2
(b) –2
(c) No such real b exists
(d) None of these
Answer: (b)

Question. If d.r.'s of two straight lines are 5, –12, 13 and –3, 4, 5 then, angle between them is
(a) \(\cos^{-1}\left(\frac{2}{65}\right)\)
(b) \(\cos^{-1}\left(\frac{1}{65}\right)\)
(c) \(\cos^{-1}\left(\frac{3}{65}\right)\)
(d) \(\frac{\pi}{3}\)
Answer: (b)

Question. If direction ratio of two lines are \(a_1, b_1, c_1\) and \(a_2, b_2, c_2\) then these lines are parallel if and only if
(a) \(a_1/a_2 = b_1/b_2 = c_1/c_2\)
(b) \(a_1 a_2 + b_1 b_2 + c_1 c_2 = 0\)
(c) \(a_1 = a_2, b_1 = b_2, c_1 = c_2\)
(d) None of these
Answer: (a)

Question. If \(A(k, 1, -1), B(2, 0, 2)\) and \(C(2k+2, 2k, 1)\) be such that the line \(AB \perp BC\), then the value of k will be
(a) 1
(b) 2
(c) 3
(d) 0
Answer: (a)

Question. \(A(a, 7, 10), B(-1, 6, 6)\) and \(C(-4, 9, 6)\) are the vertices of a right angled isosceles triangle. If \(\angle ABC = 90^\circ\), then \(a =\)
(a) 0
(b) 2
(c) –1
(d) –3
Answer: (b)

Advance Level

Question. The angle between two diagonals of a cube will be
(a) \(\sin^{-1} \frac{1}{3}\)
(b) \(\cos^{-1} \frac{1}{3}\)
(c) Constant
(d) Variable
Answer: (b)

Question. If a line makes angles \(\alpha, \beta, \gamma, \delta\) with the four diagonals of a cube, then the value of \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma + \cos^2 \delta =\)
(a) 1
(b) \(4/3\)
(c) Constant
(d) Variable
Answer: (b)

Question. The angle between the lines whose direction cosines satisfy the equations \(l+m+n=0, l^2+m^2-n^2=0\) is given by
(a) \(\frac{2\pi}{3}\)
(b) \(\frac{\pi}{6}\)
(c) \(\frac{5\pi}{6}\)
(d) \(\frac{\pi}{3}\)
Answer: (a)

Question. If three mutually perpendicular lines have direction cosines \((l_1,m_1,n_1), (l_2,m_2,n_2)\) and \((l_3,m_3,n_3)\), then the line having direction cosines \(l_1+l_2+l_3, m_1+m_2+m_3\) and \(n_1+n_2+n_3\) make an angle of ……….with each other
(a) 0°
(b) 30°
(c) 60°
(d) 90°
Answer: (a)

Question. The straight lines whose direction cosines are given by \(al+bm+cn=0, fmn+gnl+hlm=0\) are perpendicular, if
(a) \(\frac{f}{a} + \frac{g}{b} + \frac{h}{c} = 0\)
(b) \(\frac{a}{f} + \frac{b}{g} + \frac{c}{h} = 0\)
(c) \(\sqrt{af} = \sqrt{bg} = \sqrt{ch}\)
(d) \(\frac{a}{f} = \frac{b}{g} = \frac{c}{h}\)
Answer: (a)

Question. The angle between the lines whose direction cosines are connected by the relations \(l+m+n=0\) and \(2lm + 2nl - mn = 0\), is
(a) \(\frac{\pi}{3}\)
(b) \(\frac{2\pi}{3}\)
(c) \(\pi\)
(d) None of these
Answer: (d)

Question. \(A(3, 2, 0), B(5, 3, 2), C(-9, 6, -3)\) are three points forming a triangle and AD is the bisector of the \(\angle BAC\), then coordinates of D are
(a) \(\left(\frac{38}{16}, \frac{17}{16}, \frac{57}{16}\right)\)
(b) \(\left(\frac{17}{16}, \frac{38}{16}, \frac{57}{16}\right)\)
(c) \(\left(\frac{38}{16}, \frac{57}{16}, \frac{17}{16}\right)\)
(d) \(\left(\frac{57}{16}, \frac{38}{16}, \frac{17}{16}\right)\)
Answer: (d)

Question. The direction cosines of two lines at right angles are \(\left\) and \(\left\). Then the d.c. of a line \(\perp\) to both the given lines are
(a) \(\left\)
(b) \(\left\)
(c) \(\left\)
(d) None of these
Answer: (a)

Question. Three lines drawn from origin with direction cosines \(l_1,m_1,n_1; l_2,m_2,n_2; l_3,m_3,n_3\) are coplanar iff \(\begin{vmatrix} l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3 \end{vmatrix} = 0\), since
(a) All lines pass through origin
(b) It is possible to find a line perpendicular to all these lines
(c) Intersecting lines are coplanar
(d) None of these
Answer: (b)

Question. The direction cosines of a variable line in two adjacent positions are \(l, m, n\) and \(l+\delta l, m+\delta m, n+\delta n\). If angle between these two positions is \(\delta \theta\), where \(\delta \theta\) is a small angle, then \(\delta \theta^2\) is equal to
(a) \(\delta l^2 + \delta m^2 + \delta n^2\)
(b) \(\delta l + \delta m + \delta n\)
(c) \(\delta l \cdot \delta m + \delta m \cdot \delta n + \delta n \cdot \delta l\)
(d) None of these
Answer: (a)

Question. If direction cosines of two lines OA and OB are respectively proportional to 1, –2, –1 and 3, –2, 3 then direction cosine of line perpendicular to given both lines are
(a) \(\pm 4/\sqrt{29}, \pm 3/\sqrt{29}, \pm 2/\sqrt{29}\)
(b) \(\pm 4/\sqrt{29}, \pm 3/\sqrt{29}, \mp 2/\sqrt{29}\)
(c) \(\pm 4/\sqrt{29}, \pm 2/\sqrt{29}, \pm 3/\sqrt{29}\)
(d) None of these
Answer: (b)

Question. A mirror and a source of light are situated at the origin O and at a point on OX respectively. A ray of light from the source strikes the mirror and is reflected. If the d.r'.s of the normal to the plane are 1, –1, 1, then d.c'.s of the reflected ray are
(a) \( \frac{1}{3}, \frac{2}{3}, \frac{2}{3} \)
(b) \( -\frac{1}{3}, \frac{2}{3}, \frac{2}{3} \)
(c) \( -\frac{1}{3}, -\frac{2}{3}, -\frac{2}{3} \)
(d) \( -\frac{1}{3}, -\frac{2}{3}, \frac{2}{3} \)
Answer: (b)