Class 11 Mathematics Introduction To Three-Dimensional Geometry MCQs Set 09

Question. If the three points with position vectors \( (1,a,b), (a,2,b) \) and \( (a,b,3) \) are collinear in space, then the value of \( a + b \) is
(a) 3
(b) 4
(c) 5
(d) None of the options
Answer: (b) 4

Question. Consider the three points P, Q, R whose coordinates are respectively \( (2, 5, -4), (1, 4, -3), (4, 7, -6) \) then the ratio in which the point Q divides PR.
(a) 1 : 3
(b) 1 : 2
(c) -1 : 3
(d) -1 : 2
Answer: (c) -1 : 3

Question. Let \( \vec{r} = \vec{a} + \lambda \vec{l} \) and \( \vec{r} = \vec{b} + \mu \vec{m} \) be two lines in space where \( \vec{a} = 5\hat{i} + \hat{j} + 2\hat{k}, \vec{b} = -\hat{i} + 7\hat{j} + 8\hat{k}, \vec{l} = -4\hat{i} + \hat{j} - \hat{k} \) and \( \vec{m} = 2\hat{i} - 5\hat{j} - 7\hat{k} \) then the p.v. of a point which lies on both of these lines is
(a) \( \hat{i} + 2\hat{j} + \hat{k} \)
(b) \( 2\hat{i} + \hat{j} + \hat{k} \)
(c) \( \hat{i} + \hat{j} + 2\hat{k} \)
(d) non existent s the lines are skew
Answer: (a) \( \hat{i} + 2\hat{j} + \hat{k} \)

Question. \( L_1 \) and \( L_2 \) are two lines whose vector equations are \( L_1 : \vec{r} = \lambda [(\cos \theta + \sqrt{3})\hat{i} + (\sqrt{2} \sin \theta)\hat{j} + (\cos \theta - \sqrt{3})\hat{k}] \), \( L_2 : \vec{r} = \mu ( a\hat{i} + b\hat{j} + c\hat{k} ) \), Where, \( \lambda \) and \( \mu \) are scalars and \( \alpha \) is the acute angle between \( L_1 \) and \( L_2 \). If the angle \( \alpha \) is independent of \( \theta \) then the value of \( \alpha \) is
(a) \( \frac{\pi}{6} \)
(b) \( \frac{\pi}{4} \)
(c) \( \frac{\pi}{3} \)
(d) \( \frac{\pi}{2} \)
Answer: (a) \( \frac{\pi}{6} \)

Question. If three lines \( L_1 : x = y = z \), \( L_2 : x = \frac{y}{2} = \frac{z}{3} \), \( L_3 : \frac{x-1}{a} = \frac{y-1}{b} = \frac{z-1}{c} \) form a triangle of area \( \sqrt{6} \) sq.units, then for the point of intersection \( (a, \beta, \gamma) \) of \( L_2 \) and \( L_3 \), \( \beta = \)
(a) 2
(b) 4
(c) 6
(d) 8
Answer: (b) 4

Question. Image of the point P with position vector \( 7\hat{i} - \hat{j} + 2\hat{k} \) in the line whose vector equation is \( \vec{r} = 9\hat{i} + 5\hat{j} + 5\hat{k} + \lambda (\hat{i} + 3\hat{j} + 5\hat{k}) \) has the position vector.
(a) \( (-9, 5, 2) \)
(b) \( (9, 5, -2) \)
(c) \( (9, -5, -2) \)
(d) None of the options
Answer: (b) \( (9, 5, -2) \)

Question. The intercept made by the plane \( \vec{r} \cdot \vec{n} = q \) on the x-axis is
(a) \( \frac{q}{\hat{i} \cdot \vec{n}} \)
(b) \( \frac{\hat{i} \cdot \vec{n}}{q} \)
(c) \( (\hat{i} \cdot \vec{n}) q \)
(d) \( \frac{q}{|\vec{n}|} \)
Answer: (a) \( \frac{q}{\hat{i} \cdot \vec{n}} \)

Question. ABC is any triangle and O is any point in the plane of the triangle. AO, BO, CO meet the sides BC, CA, AB in D, E, F respectively. Find \( \frac{OD}{AD} + \frac{OE}{BE} + \frac{OF}{CF} \).
(a) 1
(b) 2
(c) -1
(d) -2
Answer: (a) 1

Question. If from the point \( P(f, g, h) \) perpendicular PL, PM be drawn to YZ and ZX planes then the equation of the plane OLM is
(a) \( \frac{x}{f} + \frac{y}{g} - \frac{z}{h} = 0 \)
(b) \( \frac{x}{f} + \frac{y}{g} + \frac{z}{h} = 0 \)
(c) \( \frac{x}{f} - \frac{y}{g} + \frac{z}{h} = 0 \)
(d) \( -\frac{x}{f} + \frac{y}{g} + \frac{z}{h} = 0 \)
Answer: (a) \( \frac{x}{f} + \frac{y}{g} - \frac{z}{h} = 0 \)

Question. If the distance from point \( P(1,1,1) \) to the line passing through the points \( Q(0, 6, 8) \) and \( R(-1, 4, 7) \) is expressed in the form \( \sqrt{\frac{p}{q}} \) where p and q are coprime, then the value \( \frac{(p+q)(p+q-1)}{2} \) equals
(a) 4950
(b) 5050
(c) 5150
(d) None of the options
Answer: (a) 4950

Question. Consider the following 3 lines in space
\( L_1 : \vec{r} = 3\hat{i} - \hat{j} + 2\hat{k} + \lambda (2\hat{i} + 4\hat{j} - \hat{k}) \)
\( L_2 : \vec{r} = \hat{i} + \hat{j} - 3\hat{k} + \mu (4\hat{i} + 2\hat{j} + 4\hat{k}) \)
\( L_3 : \vec{r} = 3\hat{i} + 2\hat{j} - 2\hat{k} + t (2\hat{i} + \hat{j} + 2\hat{k}) \)
Then which one of the following pair(s) are in the same plane.
(a) only \( L_1 L_2 \)
(b) only \( L_2 L_3 \)
(c) only \( L_3 L_1 \)
(d) \( L_1 L_2 \) and \( L_2 L_3 \)
Answer: (d) \( L_1 L_2 \) and \( L_2 L_3 \)

Question. Position vectors of the four angular points of a tetrahedron ABCD are \( A(3, -2, 1); B(3, 1, 5); C(4, 0, 3) \) and \( D(1, 0, 0) \). Acute angle between the plane faces ADC and ABC is
(a) \( \tan^{-1}\left(\frac{5}{2}\right) \)
(b) \( \cos^{-1}\left(\frac{2}{5}\right) \)
(c) \( \csc^{-1}\left(\frac{5}{2}\right) \)
(d) \( \cot^{-1}\left(\frac{3}{2}\right) \)
Answer: (a) \( \tan^{-1}\left(\frac{5}{2}\right) \)

Question. If a plane passing through the point \( (1,2,3) \) cuts positive directions of co-ordinate axes in A, B & C, then the minimum volume of the tetrahedron formed by origin and A, B, C is cubic units
(a) \( \frac{9}{2} \)
(b) 9
(c) 18
(d) 27
Answer: (d) 27

Question. A, B, C, D are 4 coplanar points and A', B', C', D' are their projections on any plane. If \( \alpha \) is the angle between plane of ABCD and plane of projections then \( \frac{\text{Volume of tetrahedron } AB'C'D'}{\text{Volume of tetrahedron } A'BCD} = \)
(a) 1
(b) 2
(c) 2 cos \( \alpha \)
(d) cos \( \alpha \)
Answer: (d) cos \( \alpha \)

Question. Let a point R lies on the plane \( x - y + z - 3 = 0 \) and P be the point (1, 1, 1). A point Q lies on PR such that \( PQ^2 + PR^2 = k \) (\( \neq 0 \)) then the equation of locus of Q is
(a) \( \left[ (x-1)^2 + (y-1)^2 + (z-1)^2 \right] \left[ 1 + \frac{4}{(x-y+z-1)^2} \right] = k \)
(b) \( \left[ (x-1)^2 + (y-1)^2 + (z-1)^2 \right] \left[ 1 - \frac{4}{(x-y+z-1)^2} \right] = k \)
(c) \( \left[ (x-1)^2 + (y-1)^2 + (z-1)^2 \right] \left[ 1 - \frac{4}{(x-y+z-1)^2} \right] = k \)
(d) \( \frac{1}{(x-1)^2} + \frac{1}{(y-1)^2} + \frac{1}{(z-1)^2} + \frac{(x-y+z-1)^2}{4} = k \)
Answer: (a) \( \left[ (x-1)^2 + (y-1)^2 + (z-1)^2 \right] \left[ 1 + \frac{4}{(x-y+z-1)^2} \right] = k \)

Question. Let OABC be tetrahedron, Let the mid points of edges OA & OB and OC be \( A_1, B_1, C_1 \) respectively while those of edges AB, BC and AC be R, P and Q respectively. If OA is
(a) \( QB_1^2 = RC_1^2 \)
(b) \( QA_1^2 = RC_1^2 \)
(c) \( QC_1^2 = RC_1^2 \)
(d) None of the options
Answer: (a) \( QB_1^2 = RC_1^2 \)

Question. Let \( \Delta_1, \Delta_2, \Delta_3 \) and \( \Delta_4 \) be the areas of the triangular faces of tetrahedron and \( h_1, h_2, h_3, \& h_4 \) be the corresponding altitudes of the tetrahedron, then the minimum value of \( \sum_{1 \leq i < j \leq 4} \sum (\Delta_i h_j) = \) Given volume of the tetrahedron is 5 cubic units.
(a) 240
(b) 225
(c) 160
(d) 180
Answer: (d) 180

Question. A line is drawn from the point P(1,1,1) and Perpendicular to a line with direction ratios (1,1,1) to intersect the plane \( x + 2y + 3z = 4 \) at Q. The locus of point Q is
(a) \( \frac{x}{1} = \frac{y-5}{-2} = \frac{z+2}{1} \)
(b) \( \frac{x}{-2} = \frac{y-5}{1} = \frac{z+2}{1} \)
(c) \( x = y = z \)
(d) \( \frac{x}{2} = \frac{y}{3} = \frac{z}{5} \)
Answer: (a) \( \frac{x}{1} = \frac{y-5}{-2} = \frac{z+2}{1} \)

Question. Three positive real numbers x,y,z satisfy the equations \( x^2 + \sqrt{3}xy + y^2 = 25 \), \( y^2 + z^2 = 9 \) and \( x^2 + xz + z^2 = 16 \). Then the value of \( xy + 2yz + \sqrt{3}xz \) is
(a) 18
(b) 24
(c) 30
(d) 36
Answer: (b) 24

Question. Three straight lines mutually perpendicular to each other meet in a point P and one of them intersects the x-axis and another intersects the y-axis, while the third line passes through a fixed point (0,0,c) on the Z-axis. Then the locus of P is
(a) \( x^2 + y^2 + z^2 - 2cx = 0 \)
(b) \( x^2 + y^2 + z^2 - 2cy = 0 \)
(c) \( x^2 + y^2 + z^2 - 2cz = 0 \)
(d) \( x^2 + y^2 + z^2 - 2c(x+y+z) = 0 \)
Answer: (c) \( x^2 + y^2 + z^2 - 2cz = 0 \)

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 - 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) \( \cot^2 \alpha + \cot^2 \beta + \cot^2 \gamma \)

ASSERTION & REASON QUESTIONS
(a) Statement -1 is true, statement -2 is true, statement -2 is a correct explanation for statement-1
(b) Statement-1 is true, statement-2 is true, statement -2 is not correct explanation for statement-1
(c) Statement-1 is true, statement-2 is false
(d) Statement -1 is false, statement-2 is true

Question. Statement - 1: If \( \vec{a}, \vec{b} \) and \( \vec{c} \) are three non coplanar vectors, then the length of projection of vector \( \vec{a} \) in the plane of vectors \( \vec{b} \) and \( \vec{c} \) may be \( \frac{|\vec{a} \times (\vec{b} \times \vec{c})|}{|\vec{b} \times \vec{c}|} \)
Statement - 2: \( \hat{n} \) = unit vector normal to plane \( \vec{b} \) and \( \vec{c} \) is \( \frac{\vec{b} \times \vec{c}}{|\vec{b} \times \vec{c}|} \) & projection of \( \vec{a} \) in the plane of \( \vec{b} \) and \( \vec{c} \) is \( \frac{|\vec{a} \times (\vec{b} \times \vec{c})|}{|\vec{b} \times \vec{c}|} \)
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Answer: (a) (a)

Question. Statement-1: If \( \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} \), then equation \( \vec{r} \times (2\hat{i} - \hat{j} + 3\hat{k}) = 3\hat{i} + \hat{k} \) represents a straight line
Statement-2: If \( \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} \) then equation \( \vec{r} \times (\hat{i} + 2\hat{j} - 3\hat{k}) = 2\hat{i} - \hat{j} \) represents a straight line
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Answer: (d) (d)

Question. Statement 1 : Planes parallel to x-axis and passing through the point (2, 1, 3) will not be at a fixed distance from the x-axis.
because
Statement 2 : Such planes will be tangential to a cylinder with its axis as x-axis.
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Answer: (c) (c)

Question. Statement 1 : The equation \( 2x^2 - 6y^2 + 4z^2 + 18yz + 2zx + xy = 0 \) represents a pair of perpendicular planes.
Statement 2 : A pair of planes given by \( ax^2 + by^2 + cz^2 + 2fyz + 2gzx + 2hxy = 0 \) are perpendicular, if \( a + b + c = 0 \)
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Answer: (a) (a)