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) \( x^2 + y^2 + z^2 - 2cz = 0 \)

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

PARAGRAPH - II
Consider the lines represented parametrically as
\( L_1 : x = 1 - 2t, y = t, z = -1 + t \)
\( L_2 : x = 4 + S, y = 5 + 4S, z = -2 - S \)
Let \( \pi \) be the plane containing the line \( L_2 \) and parallel to line \( L_1 \)

Question. The acute angle between the lines \( L_1 \) and \( L_2 \) is
(a) \( \cos^{-1}\left(\frac{1}{18}\right) \)
(b) \( \cos^{-1}\left(\frac{1}{6\sqrt{3}}\right) \)
(c) \( \cos^{-1}\left(\frac{1}{3\sqrt{2}}\right) \)
(d) \( \cos^{-1}\left(\frac{1}{3\sqrt{6}}\right) \)
Answer: (d) \( \cos^{-1}\left(\frac{1}{3\sqrt{6}}\right) \)

Question. The equation of plane \( \pi \) is
(a) \( 5x + y + 9z - 7 = 0 \)
(b) \( 9x - 5y - z - 13 = 0 \)
(c) \( 2x - 3y - 4z - 15 = 0 \)
(d) \( 5x - y + 9z + 3 = 0 \)
Answer: (a) \( 5x + y + 9z - 7 = 0 \)

Question. The distance between the plane \( \pi \) and the line \( L_1 \) is
(a) \( \frac{17}{\sqrt{19}} \)
(b) \( \frac{3}{\sqrt{87}} \)
(c) \( \frac{1}{\sqrt{107}} \)
(d) \( \frac{11}{\sqrt{107}} \)
Answer: (d) \( \frac{11}{\sqrt{107}} \)

PARAGRAPH - III
Let \( A \) denote the plane consisting of all points that are equidistant from the points \( P(-4, 2, 1) \) and \( Q(2, -4, 3) \) and \( B \) be the plane, \( x - y + cz = 1 \) where \( c \in R \)

Question. If the angle between the planes A and B is \( 45^\circ \) then the product of all possible values of c is
(a) -17
(b) -2
(c) 17
(d) \( \frac{24}{17} \)
Answer: (b) -2

Question. If the line L with equation \( \frac{x-2}{1} = \frac{y-1}{3} = \frac{z-5}{-1} \) intersects the plane A at the point \( M(\lambda, \mu, \nu) \), then coordinate of M is
(a) \( \left( \frac{8}{7}, \frac{11}{7}, \frac{41}{7} \right) \)
(b) \( \left( -\frac{8}{7}, -\frac{11}{7}, -\frac{41}{7} \right) \)
(c) \( \left( \frac{8}{7}, -\frac{11}{7}, \frac{41}{7} \right) \)
(d) None of the options
Answer: (a) \( \left( \frac{8}{7}, \frac{11}{7}, \frac{41}{7} \right) \)

PARAGRAPH - IV
The line of greatest slope on an inclined plane \( P_1 \) is the line in the plane \( P_1 \) which is perpendicular to the line of intersection of the plane \( P_1 \) and a horizontal plane \( P_2 \).

Question. Assuming the plane \( 4x - 3y + 7z = 0 \) to be horizontal, the direction cosine of the line of greatest slope in the plane \( 2x + y - 5z = 0 \) are
(a) \( \frac{3}{\sqrt{11}}, \frac{-1}{\sqrt{11}}, \frac{1}{\sqrt{11}} \)
(b) \( \frac{3}{\sqrt{11}}, \frac{1}{\sqrt{11}}, \frac{-1}{\sqrt{11}} \)
(c) \( \frac{-3}{\sqrt{11}}, \frac{1}{\sqrt{11}}, \frac{1}{\sqrt{11}} \)
(d) None of the options
Answer: (b) \( \frac{3}{\sqrt{11}}, \frac{1}{\sqrt{11}}, \frac{-1}{\sqrt{11}} \)

Question. The coordinates of a point on the plane \( 2x + y - 5z = 0 \), \( 2\sqrt{11} \) units away from the line of intersection of \( 2x + y - 5z = 0 \) and \( 4x - 3y + 7z = 0 \) are
(a) (6, 2, -2)
(b) (3, 1, -1)
(c) (6, -2, 2)
(d) (1, 3, -1)
Answer: (a) (6, 2, -2)

PARAGRAPH - V
In three dimensions there may be more than one point, which are equidistant from three given non-colliner points A,B,C. One of these points will be circumcentre of the triangle ABC

Question. The circumcentre of the triangle ABC where A,B,C are (a,0,0), (0,b,0) and (0,0,c) will lie in the plane
(a) \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \)
(b) \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 2 \)
(c) \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 3 \)
(d) None of the options
Answer: (a) \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \)

Question. y coordinate of the circumcentre of triangle ABC must be
(a) \( \frac{ac}{a + b + c} \)
(b) \( \frac{a^2c^2 - b^4}{a^3 + b^3 + c^3} \)
(c) \( \frac{b(c^2 + a^2 - b^2)}{a^2b^2 + b^2c^2 + c^2a^2} \)
(d) \( \frac{b^3(a^2 + c^2)}{2(b^2c^2 + a^2c^2 + a^2b^2)} \)
Answer: (d) \( \frac{b^3(a^2 + c^2)}{2(b^2c^2 + a^2c^2 + a^2b^2)} \)

Question. The y coordinate of orthocentre of the triangle ABC
(a) \( \frac{3a^2c^2 - a^2b^2 - b^2c^2}{a^2b^2 + b^2c^2 + c^2a^2} \)
(b) \( \frac{ab + b^2 - ac}{a + b + c} \)
(c) \( b - \frac{2(a^2c^2 - b)}{a^3 + b^3 + c^3} \)
(d) \( \frac{a^2bc^2}{b^2c^2 + c^2a^2 + a^2b^2} \)
Answer: (d) \( \frac{a^2bc^2}{b^2c^2 + c^2a^2 + a^2b^2} \)

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-coplaner 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) Statement -1 is true, statement -2 is true, statement -2 is a correct explanation for statement-1

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: (a) Statement -1 is true, statement -2 is true, statement -2 is a correct explanation for statement-1

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.
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: (a) Statement -1 is true, statement -2 is true, statement -2 is a correct explanation for statement-1

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) Statement -1 is true, statement -2 is true, statement -2 is a correct explanation for statement-1

Question. Statement 1 : The shortest distance between the skew lines \( \frac{x+3}{-4} = \frac{y-6}{3} = \frac{z}{2} \) and \( \frac{x+2}{-4} = \frac{y}{1} = \frac{z-7}{1} \) is 9.
Statement 2 : Two lines are skew lines if there exists no plane passing through them.
(a) A
(b) B
(c) C
(d) D
Answer: (b) Statement-1 is true, statement-2 is true, statement -2 is not correct explanation for statement-1

Question. Statement 1 : The equation of the plane through the intersection of the planes \( x + y + z = 6 \) and \( 2x + 3y + 4z + 5 = 0 \) and the point (4, 4, 4) is \( 29x + 23y + 17z = 276 \).
Statement 2 : Equation of the plane through the line of intersection of the planes \( P_1 = 0 \) and \( P_2 = 0 \) is \( P_1 + \lambda P_2 = 0 \)
(a) A
(b) B
(c) C
(d) D
Answer: (a) Statement -1 is true, statement -2 is true, statement -2 is a correct explanation for statement-1

Question. Consider the planes \( P_1 : x - y + z = 1 \); \( P_2 : x + y - z = -1 \) and \( P_3 : x - 3y + 3z = 2 \). Let \( L_1, L_2, L_3 \) be the lines of intersection of the planes \( P_2 \) and \( P_3 \), \( P_3 \) and \( P_1 \), \( P_1 \) and \( P_2 \) respectively.
Statement 1 : At least two of the lines \( L_1, L_2 \) and \( L_3 \) are non-parallel.
Statement II : The three planes do not have a common point.
(a) A
(b) B
(c) C
(d) D
Answer: (d) Statement -1 is false, statement-2 is true