Class 11 Mathematics Introduction To Three-Dimensional Geometry MCQs Set 10

Question. Perpendiculars are drawn from points on the line \( \frac{x+2}{2} = \frac{y+1}{-1} = \frac{z}{3} \) to the plane \( x + y + z = 3 \). The feet of perpendiculars lie on the line
(a) \( \frac{x-1}{5} = \frac{y-1}{8} = \frac{z-2}{-13} \)
(b) \( \frac{x-1}{2} = \frac{y-1}{3} = \frac{z-2}{-5} \)
(c) \( \frac{x-1}{4} = \frac{y-1}{3} = \frac{z-2}{-7} \)
(d) \( \frac{x-1}{2} = \frac{y-1}{-7} = \frac{z-2}{5} \)
Answer: (d) \( \frac{x-1}{2} = \frac{y-1}{-7} = \frac{z-2}{5} \)

Question. The shortest distance from the point (1,2,3) to \( x^2 + y^2 + z^2 - xy - yz - zx = 0 \) is
(a) \( \frac{1}{2} \)
(b) 1
(c) \( \sqrt{2} \)
(d) \( \frac{1}{\sqrt{2}} \)
Answer: (d) \( \frac{1}{\sqrt{2}} \)

Question. A rigid body rotates about an axis through the origin with an angular velocity \( 10\sqrt{3} \) radians/s if \( \vec{\omega} \) points in the direction of \( \hat{i} + \hat{j} + \hat{k} \) then the equation to the locus of the points having tangential speed 20 m/sec. is
(a) \( x^2 + y^2 + z^2 - xy - yz - zx - 1 = 0 \)
(b) \( x^2 + y^2 + z^2 - 2xy - 2yz - 2zx - 1 = 0 \)
(c) \( x^2 + y^2 + z^2 - xy - yz - zx - 2 = 0 \)
(d) \( x^2 + y^2 + z^2 - 2xy - 2yz - 2zx - 2 = 0 \)
Answer: (c) \( x^2 + y^2 + z^2 - xy - yz - zx - 2 = 0 \)

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)

Question. Let PM be the perpendicular from the point P(1, 2, 3) to XY plane. If OP makes an angle \( \theta \) with the positive direction of the z-axis and OM makes an angle \( \phi \) with the positive direction of x-axis, where O is the origin then (\( \theta \) and \( \phi \) are acute angles)
(a) \( \tan \theta = \frac{\sqrt{5}}{3} \)
(b) \( \sin \theta \sin \phi = \frac{2}{\sqrt{14}} \)
(c) \( \tan \phi = 2 \)
(d) \( \cos \theta \cos \phi = \frac{1}{\sqrt{14}} \)
Answer: (c) \( \tan \phi = 2 \)

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

Question. The line \( x + 2y - z - 3 = 0, x + 3y - z - 4 = 0 \) is parallel to
(a) XY plane
(b) YZ plane
(c) ZX plane
(d) Z-axis
Answer: (d) Z-axis

Question. A variable plane makes with the coordinate planes, a tetrahedron of constant volume \( 64k^3 \). Then the locus of the centroid of tetrahedron is the surface
(a) \( xyz = 6k^3 \)
(b) \( xy + yz + zx = 6k^2 \)
(c) \( x^2 + y^2 + z^2 = 8k^2 \)
(d) \( x^{-2} + y^{-2} + z^{-2} = 8k^{-2} \)
Answer: (a) \( xyz = 6k^3 \)

Question. The angle between the line \( x + 2y + 3z = 0 = 3x + 2y + z \) and the y-axis is
(a) \( \frac{1}{2} \sec^{-1} 3 \)
(b) \( 2 \sec^{-1} 3 \)
(c) \( \cos^{-1} \left( \frac{2}{\sqrt{6}} \right) \)
(d) \( 2 \sec^{-1} 4 \)
Answer: (c) \( \cos^{-1} \left( \frac{2}{\sqrt{6}} \right) \)

Question. If \( p_1, p_2, p_3 \) denote the perpendicular distances of the plane \( 2x - 3y + 4z + 2 = 0 \) from the parallel planes, \( 2x - 3y + 4z + 6 = 0, 4x - 6y + 8z + 3 = 0 \) and \( 2x - 3y + 4z - 6 = 0 \) respectively, then
(a) \( p_1 + 8p_2 - p_3 = 0 \)
(b) \( p_3 = 16p_2 \)
(c) \( 8p_2 = p_1 \)
(d) \( p_1 + 2p_2 + 3p_3 = \sqrt{29} \)
Answer: (a) \( p_1 + 8p_2 - p_3 = 0 \)

Question. The line whose vector equations are \( \vec{r} = 2\hat{i} - 3\hat{j} + 7\hat{k} + \lambda (2\hat{i} + p\hat{j} + 5\hat{k}) \) and \( \vec{r} = \hat{i} + 2\hat{j} + 3\hat{k} + \mu (3\hat{i} - p\hat{j} + p\hat{k}) \) are perpendicular for all values of \( \lambda \) and \( \mu \) if p is equal to
(a) -1
(b) 2
(c) 5
(d) 6
Answer: (d) 6

Question. Consider 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) Angle between two lines 90°
(b) the second line passes through \( (1, -\frac{3}{2}, -5) \)
(c) Angle between two lines 45°
(d) Angle between two lines is 30°
Answer: (a) Angle between two lines 90°

Question. The equation of the bisector planes of an angle between the planes \( 2x-3y+6z+8=0 \) and \( x-2y+2z+5=0 \)
(a) \( x+5y+4z+11=0 \)
(b) \( x-5y-4z+11=0 \)
(c) \( 13x - 23y+32z+59=0 \)
(d) \( x + 5y + 4z + 11 = 0 \)
Answer: (a) \( x+5y+4z+11=0 \)

Question. Let \( \vec{A} \) be vector parallel to line of intersection of planes \( P_1 \) and \( P_2 \). Plane \( P_1 \) is parallel to the vectors \( 2\hat{j} + 3\hat{k} \) and \( 4\hat{j} - 3\hat{k} \) and that \( P_2 \) is parallel to \( \hat{j} - \hat{k} \) and \( 3\hat{i} + 3\hat{j} \), then the angle between vector \( \vec{A} \) and a given vector \( 2\hat{i} + \hat{j} - 2\hat{k} \) is
(a) \( \frac{\pi}{2} \)
(b) \( \frac{\pi}{4} \)
(c) \( \frac{\pi}{6} \)
(d) \( \frac{3\pi}{4} \)
Answer: (a) \( \frac{\pi}{2} \)

Question. Consider the lines \( x = y = z \) and the line \( 2x + y + z - 1 = 0 = 3x + y + 2z - 2 \) is
(a) The shortest distance between the two lines is \( \frac{1}{\sqrt{2}} \)
(b) The shortest distance between the two lines is \( \sqrt{2} \)
(c) plane containing 2nd line parallel to 1st line is \( y - z + 1 = 0 \)
(d) The shortest distance between the two lines is \( \frac{\sqrt{3}}{2} \)
Answer: (a) The shortest distance between the two lines is \( \frac{1}{\sqrt{2}} \)

Question. Two systems of rectangular axes have the same origin. If plane cut the intercepts a, b, c on co-ordinate axes for 1st system and intercepts a', b', c' on 2nd system then pick the correct alternatives
(a) \( \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} - \frac{1}{a'^2} - \frac{1}{b'^2} - \frac{1}{c'^2} = 0 \)
(b) \( \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} + \frac{1}{a'^2} + \frac{1}{b'^2} + \frac{1}{c'^2} = 0 \)
(c) \( \frac{1}{a^2} + \frac{1}{b^2} - \frac{1}{c^2} + \frac{1}{a'^2} + \frac{1}{b'^2} - \frac{1}{c'^2} = 0 \)
(d) \( \frac{1}{a^2} - \frac{1}{b^2} - \frac{1}{c^2} + \frac{1}{a'^2} - \frac{1}{b'^2} - \frac{1}{c'^2} = 0 \)
Answer: (a) \( \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} - \frac{1}{a'^2} - \frac{1}{b'^2} - \frac{1}{c'^2} = 0 \)

Question. A line \( l \) passing through the origin is perpendicular to the line \( l_1 : (3+t)\hat{i} + (-1+2t)\hat{j} + (4+2t)\hat{k}, -\infty < t < \infty \) and \( l_2 : (3+2s)\hat{i} + (3+2s)\hat{j} + (2+s)\hat{k}, -\infty < s < \infty \). Then the coordinate (s) of the point (s) on \( l_2 \) at a distance of \( \sqrt{17} \) from the point of intersection of \( l \) and \( l_1 \) is (are)
(a) \( \left( \frac{7}{3}, \frac{7}{3}, \frac{5}{3} \right) \)
(b) \( (-1, -1, 0) \)
(c) \( (1, 1, 1) \)
(d) \( \left( \frac{7}{9}, \frac{7}{9}, \frac{8}{3} \right) \)
Answer: (a) \( \left( \frac{7}{3}, \frac{7}{3}, \frac{5}{3} \right) \)

Question. Consider the planes, \( 2x + 5y + 3z = 0 \), \( x - y + 4z = 2 \) and \( 7y - 5z + 4 = 0 \)
(a) Planes will meet at a point
(b) Planes will meet on a line
(c) The distance from (1, 1, 1) to one of the planes to \( \frac{\sqrt{2}}{3} \)
(d) Planes are equidistant from origin
Answer: (b) Planes will meet on a line

Question. A plane passes through a fixed point (a, b, c) and cuts the axes in A, B, C. The locus of a point equidistant from origin, A, B and C must be
(a) \( ayz + bzx + cxy = 2xyz \)
(b) \( \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 1 \)
(c) \( \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 2 \)
(d) \( \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 3 \)
Answer: (c) \( \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 2 \)

Question. Two lines \( L_1 : x = 5, \frac{y}{3 - \alpha} = \frac{z}{-2} \) and \( L_2 : x = \alpha, \frac{y}{-1} = \frac{z}{2 - \alpha} \) are coplanar. Then \( \alpha \) can take value (s)
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (d) 4

PASSAGE - I
Suppose direction cosines of two lines are given by \( ul + vm + wn = 0 \) and \( al^2 + bn^2 + cn^2 = 0 \), where u, v, w, a, b, c are arbitrary constants and l, m, n are direction cosines of the lines.
On the basis of above information, answer the following questions:

Question. For \( u = v = w = 1 \), both lines satisfies the relation
(a) \( (b+c)\left(\frac{n}{l}\right)^2 + 2b\left(\frac{n}{l}\right) + (a+b) = 0 \)
(b) \( (c+a)\left(\frac{l}{m}\right)^2 + 2c\left(\frac{l}{m}\right) + (b+c) = 0 \)
(c) \( (a+b)\left(\frac{m}{n}\right)^2 + 2a\left(\frac{m}{n}\right) + (c+a) = 0 \)
(d) All of the options
Answer: (d) All of the options

Question. For \( u = v = w = 1 \), if \( \frac{n_1 n_2}{l_1 l_2} = \frac{a+b}{b+c} \), then
(a) \( \frac{m_1 m_2}{l_1 l_2} = \frac{b+c}{c+a} \)
(b) \( \frac{m_1 m_2}{l_1 l_2} = \frac{c+a}{b+c} \)
(c) \( \frac{m_1 m_2}{l_1 l_2} = \frac{a+b}{c+a} \)
(d) \( \frac{m_1 m_2}{l_1 l_2} = \frac{c+a}{a+b} \)
Answer: (b) \( \frac{m_1 m_2}{l_1 l_2} = \frac{c+a}{b+c} \)

Question. For \( u = v = w = 1 \) and if lines are perpendicular, then
(a) \( a + b + c = 0 \)
(b) \( ab + bc + ca = 0 \)
(c) \( ab + bc + ca = 3abc \)
(d) \( ab + bc + ca = abc \)
Answer: (a) \( a + b + c = 0 \)

Question. The given lines will be parallel if
(a) \( \sum u^2(b+c) = 0 \)
(b) \( \sum \frac{a^2}{u} = 0 \)
(c) \( \sum \frac{u^2}{a} = 0 \)
(d) \( \sum \frac{b+c}{u^2} = 0 \)
Answer: (b) \( \sum \frac{a^2}{u} = 0 \)

PASSAGE - II
The vector equation of a plane is a relation satisfied by position vectors of all the points on the plane. If P is a plane and \( \hat{n} \) is a unit vector through origin which is perpendicular to the plane P then vector equation of the plane must be \( \vec{r} \cdot \hat{n} = d \) where d represents perpendicular distance of plane P from origin.

Question. If A is a point vector \( \vec{a} \) then perpendicular distance of A from the plane \( \vec{r} \cdot \hat{n} = d \) must be
(a) \( |d + \vec{a} \cdot \hat{n}| \)
(b) \( |d - \vec{a} \cdot \hat{n}| \)
(c) \( |\vec{a} - d| \)
(d) \( |d - \vec{a}| \)
Answer: (b) \( |d - \vec{a} \cdot \hat{n}| \)

Question. If \( \vec{b} \) be the foot of perpendicular from A to the plane \( \vec{r} \cdot \hat{n} = d \) then \( \vec{b} \) must be
(a) \( \vec{a} + (d - \vec{a} \cdot \hat{n})\hat{n} \)
(b) \( \vec{a} - (d - \vec{a} \cdot \hat{n})\hat{n} \)
(c) \( \vec{a} + \vec{a} \cdot \hat{n} \)
(d) \( \vec{a} - \vec{a} \cdot \hat{n} \)
Answer: (a) \( \vec{a} + (d - \vec{a} \cdot \hat{n})\hat{n} \)

Question. The position vector of the image of the point \( \vec{a} \) in the plane \( \vec{r} \cdot \hat{n} = d \) must be \( (d \neq 0) \)
(a) \( -\vec{a} \cdot \hat{n} \)
(b) \( \vec{a} - 2(d - \vec{a} \cdot \hat{n})\hat{n} \)
(c) \( \vec{a} + 2(d - \vec{a} \cdot \hat{n})\hat{n} \)
(d) \( \vec{a} + d(-\vec{a} \cdot \hat{n}) \)
Answer: (c) \( \vec{a} + 2(d - \vec{a} \cdot \hat{n})\hat{n} \)

PASSAGE - III
Let the planes \( P_1 : 2x - y + z = 2 \) and \( P_2 : x + 2y - z = 3 \) are given. On the basis of the above information, answer the following questions

Question. The equation of the plane through the intersection of \( P_1 \) and \( P_2 \) and the point (3, 2, 1) is
(a) \( 3x - y + 2z - 9 = 0 \)
(b) \( x - 3y + 2z + 1 = 0 \)
(c) \( 2x - 3y + z - 1 = 0 \)
(d) \( 4x - 3y + 2z - 8 = 0 \)
Answer: (d) \( 4x - 3y + 2z - 8 = 0 \)

Question. Equation of the plane which passes through the point (-1, 3, 2) and is perpendicular to each of the planes \( P_1 \) and \( P_2 \) is
(a) \( x + 3y - 5z + 2 = 0 \)
(b) \( x + 3y + 5z - 18 = 0 \)
(c) \( x - 3y - 5z + 20 = 0 \)
(d) \( x - 3y + 5z = 0 \)
Answer: (c) \( x - 3y - 5z + 20 = 0 \)

Question. The equation of the acute angle bisector of planes \( P_1 \) and \( P_2 \) is
(a) \( x - 3y + 2z + 1 = 0 \)
(b) \( 3x + y - 5 = 0 \)
(c) \( x + 3y - 2z + 1 = 0 \)
(d) \( 3x + z + 7 = 0 \)
Answer: (a) \( x - 3y + 2z + 1 = 0 \)

Question. The equation of the bisector of angle of the planes \( P_1 \) and \( P_2 \) which is not containing origin, is
(a) \( x - 3y + 2z + 1 = 0 \)
(b) \( x + 3y = 5 \)
(c) \( x + 3y + 2z + 2 = 0 \)
(d) \( 3x + y = 5 \)
Answer: (d) \( 3x + y = 5 \)

Question. The image of plane \( P_1 \) in the plane mirror \( P_2 \) is
(a) \( x + 7y - 4z + 5 = 0 \)
(b) \( 3x + 4y - 5z + 9 = 0 \)
(c) \( 7x - y + 4z - 9 = 0 \)
(d) None of the options
Answer: (c) \( 7x - y + 4z - 9 = 0 \)

PASSAGE - IV
If \( \alpha, \beta, \gamma \) are angles made by a line from x, y, z axis respectively, then \( \cos \alpha, \cos \beta, \cos \gamma \) are known as direction cosines of a line and represented by \( l, m, n \) respectively. Direction ratios are quantities which are directly proportional to direction cosines. If \( \frac{x-\alpha_1}{l_1} = \frac{y-\beta_1}{m_1} = \frac{z-\gamma_1}{n_1} \) and \( \frac{x-\alpha_2}{l_2} = \frac{y-\beta_2}{m_2} = \frac{z-\gamma_2}{n_2} \) are two lines, then angle between them is given by \( \cos \theta = |l_1 l_2 + m_1 m_2 + n_1 n_2| \) and shortest distance between two lines exists along a line which is normal to both of them. \( (l_1, m_1, n_1) \) and \( (l_2, m_2, n_2) \) are direction cosines of the two straight lines.

Question. Minimum distance between lines \( \frac{x}{1} = \frac{y}{2} = \frac{z}{3} \) and \( \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-4}{4} \) is
(a) \( \frac{1}{\sqrt{6}} \)
(b) \( \frac{5}{\sqrt{6}} \)
(c) \( \frac{7}{\sqrt{6}} \)
(d) \( \frac{9}{\sqrt{6}} \)
Answer: (a) \( \frac{1}{\sqrt{6}} \)

Question. If P and Q are points of intersection of line \( 1-x = \frac{y}{2} = z \) with x-axis and plane \( x + 2z = 0 \), then area of triangle OPQ (O is origin) is
(a) \( \frac{\sqrt{3}}{2} \)
(b) 2
(c) \( \frac{\sqrt{5}}{2} \)
(d) \( 2\sqrt{2} \)
Answer: (c) \( \frac{\sqrt{5}}{2} \)

Question. Values of ‘a’ for which lines whose dc’s are connected by the relations \( l + am + n = 0 \) \( (a \in R) \) and \( 2l^2 + m^2 - n^2 = 0 \) are parallel, is / are
(a) \( \pm \frac{1}{\sqrt{2}} \)
(b) \( \pm 2 \)
(c) 2, 3
(d) \( \pm 1 \)
Answer: (d) \( \pm 1 \)

PASSAGE-V
If the three plane \( x = y \sin \psi + z \sin \phi \), \( y = z \sin \theta + x \sin \psi \), \( z = x \sin \phi + y \sin \theta \), intersect in a line, then where \( \theta, \phi, \psi \in (0, \pi/2) \)

Question. \( \theta, \phi, \psi \) & satisfy
(a) \( \sin^2 \theta + \sin^2 \phi + \sin^2 \psi = 1 \)
(b) \( \sin^2 \theta + \sin^2 \phi + \sin^2 \psi + 2 \sin \theta \sin \phi \sin \psi = 1 \)
(c) \( \cos^2 \theta + \cos^2 \phi + \cos^2 \psi = 1 \)
(d) \( \sin^2 \theta + \sin^2 \phi + \sin^2 \psi = 1 \)
Answer: (b) \( \sin^2 \theta + \sin^2 \phi + \sin^2 \psi + 2 \sin \theta \sin \phi \sin \psi = 1 \)

Question. \( \theta + \phi + \psi = \)
(a) \( 90^\circ \)
(b) \( 120^\circ \)
(c) \( 150^\circ \)
(d) \( 180^\circ \)
Answer: (d) \( 180^\circ \)

Question. Equation of their common line is
(a) \( \frac{x}{\sin \theta} = \frac{y}{\sin \phi} = \frac{z}{\sin \psi} \)
(b) \( \frac{x}{\cos \theta} = \frac{y}{\cos \phi} = \frac{z}{\cos \psi} \)
(c) \( \frac{x}{\tan \theta} = \frac{y}{\tan \phi} = \frac{z}{\tan \psi} \)
(d) None of the options
Answer: (b) \( \frac{x}{\cos \theta} = \frac{y}{\cos \phi} = \frac{z}{\cos \psi} \)

PASSAGE - VI
A variable point \( P(\alpha, \beta, \gamma) \) moves on a fixed plane \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \). Then plane through P and perpendicular to OP meets the coordinate axes in A, B, C. If the planes through A, B, C respectively parallel to co-ordinate planes YOZ, XOZ, meet in point Q then

Question. Q =
(a) \( (\alpha, 0, 0) \)
(b) \( \left( \frac{d}{\alpha + \beta + \gamma}, 0, 0 \right) \)
(c) \( \left( \frac{\alpha^2 + \beta^2 + \gamma^2}{\alpha}, 0, 0 \right) \)
(d) None of the options
Answer: (c) \( \left( \frac{\alpha^2 + \beta^2 + \gamma^2}{\alpha}, \frac{\alpha^2 + \beta^2 + \gamma^2}{\beta}, \frac{\alpha^2 + \beta^2 + \gamma^2}{\gamma} \right) \) (Correcting based on context of point Q coordinates).

Question. If the surface generated by Q passes through (1, 1, 1), then \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \)
(a) 1
(b) 2
(c) 3
(d) None of the options
Answer: (a) 1

PASSAGE-VII
From any point \( P(a, b, c) \) perpendiculars PM & PN drawn to zx and xy-plane respectively. Let \( \alpha, \beta, \gamma \) be the angles which OP makes with coordinate planes and \( \theta \) be the angle which OP makes with the plane OMN must be

Question. Equation of plane OMN must be
(a) \( \frac{x}{a} - \frac{y}{b} + \frac{z}{c} = 0 \)
(b) \( \frac{x}{a} + \frac{y}{b} - \frac{z}{c} = 0 \)
(c) \( \frac{x}{a} - \frac{y}{b} + \frac{z}{c} = 0 \)
(d) None of the options
Answer: (b) \( \frac{x}{a} + \frac{y}{b} - \frac{z}{c} = 0 \)

Question. \( \sin \theta \) must be equal to
(a) \( \frac{abc}{\sqrt{a^2 + b^2 + c^2} \sqrt{a^2b^2 + b^2c^2 + c^2a^2}} \)
(b) \( \frac{ab + bc + ca}{a^2 + b^2 + c^2} \)
(c) \( \frac{a+b+c}{a^2 + b^2 + c^2} \)
(d) None of the options
Answer: (a) \( \frac{abc}{\sqrt{a^2 + b^2 + c^2} \sqrt{a^2b^2 + b^2c^2 + c^2a^2}} \)

Question. \( \csc^2 \theta = \)
(a) \( \cot^2 \alpha + \cot^2 \beta + \cot^2 \gamma \)
(b) \( \csc^2 \alpha + \csc^2 \beta + \csc^2 \gamma \)
(c) \( \csc \alpha + \csc \beta + \csc \gamma \)
(d) None of the options
Answer: (a) \( \cot^2 \alpha + \cot^2 \beta + \cot^2 \gamma \)