Bihar Board 12th Maths Objective Questions and Answers

## Bihar Board 12th Maths Objective Answers Chapter 11 Three Dimensional Geometry

Question 1.

Direction cosines of the line that makes equal angles with the three axes in space are

Answer:

(c) \(\pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}}\)

Question 2.

If the direction ratios of a line are 1, -3, 2, then its direction cosines are

Answer:

(a) \(\frac{1}{\sqrt{14}}, \frac{-3}{\sqrt{14}}, \frac{2}{\sqrt{14}}\)

Question 3.

The cosines of the angle between any two diagonals of a cube is

(a) \(\frac{1}{3}\)

(b) \(\frac{1}{2}\)

(c) \(\frac{2}{3}\)

(d) \(\frac{1}{\sqrt{3}}\)

Answer:

(a) \(\frac{1}{3}\)

Question 4.

Which of the following is false?

(a) 30°, 45°, 60° can be the direction angles of a line is space.

(b) 90°, 135°, 45° can be the direction angles of a line is space.

(c) 120°, 60°, 45° can be the direction angles of a line in space.

(d) 60°, 45°, 60° can be the direction angles of a line in space.

Answer:

(a) 30°, 45°, 60° can be the direction angles of a line is space.

Question 5.

A line makes angles α, β and γ with the co-ordinate axes. If α + β = 90°, then γ is equal to

(a) 0°

(b) 90°

(c) 180°

(d) None of these

Answer:

(b) 90°

Question 6.

If a line makes an angle θ_{1}, θ_{2}, θ_{3} with the axis respectively, then cos 2θ_{1} + cos 2θ_{2} + cos 2θ_{3} =

(a) -4

(b) -2

(c) -3

(d) -1

Answer:

(d) -1

Question 7.

The coordinates of a point P are (3, 12, 4) w.r.t. origin O, then the direction cosines of OP are

Answer:

(d) \(\frac{3}{13}, \frac{12}{13}, \frac{4}{13}\)

Question 8.

Find the direction cosines of the line joining A(0, 7, 10) and B(-1, 6, 6).

Answer:

(b) \(\frac{1}{3 \sqrt{2}}, \frac{1}{3 \sqrt{2}}, \frac{4}{3 \sqrt{2}}\)

Question 9.

The direction cosines of a line passing through two points P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) are

Answer:

(c) \(\frac{x_{2}-x_{1}}{P Q}, \frac{y_{2}-y_{1}}{P Q}, \frac{z_{2}-z_{1}}{P Q}\)

Question 10.

The equation of a line which passes through the point (1, 2, 3) and is parallel to the vector \(3 \hat{i}+2 \hat{j}-2 \hat{k}\), is

Answer:

(b) \(r=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(3 \hat{i}+2 \hat{j}-2 \hat{k})\)

Question 11.

The equation of 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) \(\frac{x+3}{1}=\frac{y-2}{2}=\frac{z+4}{3}\)

(d) None of these

Answer:

(b) x + 3 = y – 2 = z + 4

Question 12.

If l, m and n are the direction cosines of line l, then the equation of the line (l) passing through (x_{1}, y_{1}, z_{1}) is

Answer:

(a) \(\frac{x-x_{1}}{l}=\frac{y-y_{1}}{m}=\frac{z-z_{1}}{n}\)

Question 13.

In the figure, a be the position vector of the point A with respect to the origin O. l is a line parallel to a

vector b. The vector equation of line l is

Answer:

(c) r = a + λb

Question 14.

The certesian equation of the line l when it passes through the point (x_{1}, y_{1}, z_{1}) and parallel to the vector

b = \(a \hat{i}+b \hat{j}+c \hat{k}\), is

(a) x – x_{1} = y – y_{1} = z – z_{1}

(b) x + x_{1} = y + y_{1} = z + z_{1}

(c) \(\frac{x+x_{1}}{a}=\frac{y+y_{1}}{b}=\frac{z+z_{1}}{c}\)

(d) \(\frac{x-x_{1}}{a}=\frac{y-y_{1}}{b}=\frac{z-z_{1}}{c}\)

Answer:

(d) \(\frac{x-x_{1}}{a}=\frac{y-y_{1}}{b}=\frac{z-z_{1}}{c}\)

Question 15.

The equation of the straight line passing through the point (a, b, c) and parallel to Z-axis is

Answer:

(d) \(\frac{x-a}{0}=\frac{y-b}{0}=\frac{z-c}{1}\)

Question 16.

The coordinates of a point on the line \(\frac{x+2}{3}=\frac{y+1}{2}=\frac{z-3}{2}\) at a distance of \(\frac{6}{\sqrt{12}}\) from the point (1, 2, 3) is

(a) (56, 43, 111)

(b) \(\left(\frac{56}{17}, \frac{43}{17}, \frac{111}{17}\right)\)

(c) (2, 1, 3)

(d) (-2, -1, -3)

Answer:

(b) \(\left(\frac{56}{17}, \frac{43}{17}, \frac{111}{17}\right)\)

Question 17.

Find the coordinatets of the point where the line through the points (5, 1, 6) and (3, 4, 1) crosses the yz-plane.

(a) \(\left(0,-\frac{17}{2}, \frac{13}{2}\right)\)

(b) \(\left(0, \frac{17}{2},-\frac{13}{2}\right)\)

(c) \(\left(10, \frac{19}{2}, \frac{13}{2}\right)\)

(d) (0, 17, 13)

Answer:

(b) \(\left(0, \frac{17}{2},-\frac{13}{2}\right)\)

Question 18.

The point A(1, 2, 3), B(-1, -2, -1) and C(2, 3, 2) are three vertices of a parallelogram ABCD. Find the equation of CD.

Answer:

(d) \(\frac{x-2}{1}=\frac{y-3}{2}=\frac{z-2}{2}\)

Question 19.

The equation of the line joining the points (-3, 4, 11) and (1, -2, 7) is

Answer:

(b) \(\frac{x+3}{-2}=\frac{y-4}{3}=\frac{z-11}{2}\)

Question 20.

The vector equation of the line through the points A(3, 4, -7) and B(1, -1, 6) is

Answer:

(c) \(r=(3 \hat{i}+4 \hat{j}-7 \hat{k})+\lambda(-2 \hat{i}-5 \hat{j}+13 \hat{k})\)

Question 21.

The vactor equation of the symmetrical form of equation of straight line \(\frac{x+5}{3}=\frac{y+4}{7}=\frac{z-6}{2}\) is

Answer:

(d) \(r=(5 \hat{i}-4 \hat{j}+6 \hat{k})+\mu(3 \hat{i}+7 \hat{j}+2 \hat{k})\)

Question 22.

Vector equation of the line 6x – 3 = 3y + 4 = 2z – 2 is

Answer:

(c) \(r=\frac{1}{2} \hat{i}-\frac{4}{3} \hat{j}+\hat{k}+\lambda\left(\frac{1}{6} \hat{i}+\frac{1}{3} \hat{j}+\frac{1}{2} \hat{k}\right)\)

Question 23.

Answer:

(b) \(\frac{9}{2}\)

Question 24.

Answer:

(a) -5

Question 25.

The angle between the straight lines

Answer:

(a) 45°

(b) 30°

(c) 60°

(d) 90°

Answer:

(d) 90°

Question 26.

Answer:

(d) \(\frac{\pi}{6}\)

Question 27.

The angle between the line 2x = 3y = -z and 6x = -y = -4z is

(a) 30°

(b) 45°

(c) 90°

(d) 0°

Answer:

(c) 90°

Question 28.

The angle between the lines 3x = 6y = 2z and \(\frac{x-2}{-5}=\frac{y-1}{7}=\frac{z-3}{1}\) is

(a) \(\frac{\pi}{6}\)

(b) \(\frac{\pi}{4}\)

(c) \(\frac{\pi}{3}\)

(d) \(\frac{\pi}{2}\)

Answer:

(d) \(\frac{\pi}{2}\)

Question 29.

Find the angle between the pair of lines given by

Answer:

(a) \(\cos ^{-1}\left(\frac{19}{21}\right)\)

Question 30.

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) 90°

Question 31.

Answer:

(b) \(\frac{\pi}{2}\)

Question 32.

The angle between the lines passing through the points (4, 7, 8), (2, 3, 4) and (-1, -2, 1), (1, 2, 5) is

(a) 0

(b) \(\frac{\pi}{2}\)

(c) \(\frac{\pi}{4}\)

(d) \(\frac{\pi}{6}\)

Answer:

(a) 0

Question 33.

Answer:

(d) Both (a) and (b)

Question 34.

Answer:

(a) \(\frac{x-1}{1}=\frac{y-2}{3}=\frac{z+3}{4}\)

Question 35.

Answer:

(a) \(-\frac{10}{7}\)

Question 36.

Answer:

(b) \(\frac{70}{11}\)

Question 37.

The shortest distance between the lines

Answer:

(d) \(\frac{1}{\sqrt{6}}\)

Question 38.

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 equal

(a) 3√30

(b) √30

(c) 2√30

(d) None of these

Answer:

(a) 3√30

Question 39.

The shortest distance between the lines x = y = z and x + 1 – y = \(\frac{z}{0}\) is

(a) \(\frac{1}{2}\)

(b) \(\frac{1}{\sqrt{2}}\)

(c) \(\frac{1}{\sqrt{3}}\)

(d) \(\frac{1}{\sqrt{6}}\)

Answer:

(d) \(\frac{1}{\sqrt{6}}\)

Question 40.

The shortest distance between the lines x = y + 2 = 6z – 6 and x + 1 = 2y = -12z is

(a) \(\frac{1}{2}\)

(b) 2

(c) 1

(d) \(\frac{3}{2}\)

Answer:

(b) 2

Question 41.

Answer:

(a) 0

Question 42.

Answer:

(b) \(\frac{\left|\left(\tilde{a}_{2}-a_{1}\right) \times b\right|}{|b|}\)

Question 43.

Answer:

(b) \(\sqrt{\frac{59}{7}}\)

Question 44.

Answer:

(d) \(\sqrt{\frac{129}{5}}\)

Question 45.

The direction cosines of the unit vector perpendicular to the plane \(r \cdot(6 \hat{i}-3 \hat{j}-2 \hat{k})+1=0\) passing through the origin are

(a) \(\frac{6}{7}, \frac{3}{7}, \frac{2}{7}\)

(b) 6, 3, 2

(c) \(-\frac{6}{7}, \frac{3}{7}, \frac{2}{7}\)

(d) -6, 3, 2

Answer:

(c) \(-\frac{6}{7}, \frac{3}{7}, \frac{2}{7}\)

Question 46.

The coordinate of the foot of perpendicular drawn from origin to the plane 2x – 3y + 4z – 6 = 0 is

Answer:

(d) \(\left(\frac{12}{\sqrt{29}}, \frac{-18}{\sqrt{29}}, \frac{24}{\sqrt{29}}\right)\)

Question 47.

The vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector

\(3 \hat{i}+5 \hat{j}-6 \hat{k}\) is

Answer:

(d) \(r \cdot\left(\frac{3 \hat{i}}{70}+\frac{5 \hat{j}}{70}-\frac{6 \hat{k}}{70}\right)=7\)

Question 48.

Find the vector equation of the plane which is at a distance of 8 units from the origin and which is normal to the vector \(2 \hat{i}+\hat{j}+2 \hat{k}\).

Answer:

(c) \(r_{\cdot}(2 \hat{i}+\hat{j}+2 \hat{k})=24\)

Question 49.

Find the length of perpendicular from the origin to the plane \(r(3 \hat{i}-4 \hat{j}+12 \hat{k})\).

(a) \(\frac{5}{13}\)

(b) \(\frac{5}{\sqrt{13}}\)

(c) \(\frac{5}{23}\)

(d) \(\frac{\sqrt{5}}{13}\)

Answer:

(a) \(\frac{5}{13}\)

Question 50.

The equation of the plane passing through three non- collinear points with position vectors a, b, c is

(a) r.(b × c + c × a + a × b) = 0

(b) r.(b × c + c × a + a × b) = [abc]

(c) r.(a × (b + c)) = [abc]

(d) r.(a + b + c) = 0

Answer:

(b) r.(b × c + c × a + a × b) = [abc]

Question 51.

Equation of the plane passing through three points A, B, C with position vectors

Answer:

(a) \(\pi(\hat{i}-\hat{j}-2 \hat{k})+23=0\)

Question 52.

Four points (0, -1, -1) (-4, 4, 4) (4, 5, 1) and (3, 9, 4) are coplanar. Find the equation of the plane containing them.

(a) 5x + 7y + 11z – 4 =0

(b) 5x – 7y + 11z + 4 = 0

(c) 5x – 7y – 11z – 4 = 0

(d) 5x + 7y – 11z + 4 = 0

Answer:

(b) 5x – 7y + 11z + 4 = 0

Question 53.

Find the equation of plane passing through the points P(1, 1, 1), Q(3, -1, 2), R(-3, 5, -4).

(a) x + 2y = 0

(b) x – y = 2

(c) -x + 2y = 2

(d) x + y = 2

Answer:

(d) x + y = 2

Question 54.

The vector equation of the plane passing through the origin and the line of intersection of the plane r.a = λ and r.b = µ is

(a) r.(λa – µb) = 0

(b) r.(λb – µa) = 0

(c) r.(λa + µb)= 0

(d) r.(λb + µa) = 0

Answer:

(b) r.(λb – µa) = 0

Question 55.

The vector equation of a plane passing through the intersection of the planes \(r_{\cdot}(\hat{i}+\hat{j}+\hat{k})=6\) and \(r_{\cdot}(2 \hat{i}+3 \hat{j}+4 \hat{k})=-5\) and the point (1, 1, 1) is

Answer:

(c) \(r_{\cdot}(20 \hat{i}+23 \hat{j}+26 \hat{k})=69\)

Question 56.

Answer:

(b) -4

Question 57.

(a) coplanar

(b) non-coplanar

(c) perpendicular

(d) None of the above

Answer:

(a) coplanar

Question 58.

The angle between the planes 3x + 2y + z – 5 = 0 and x + y – 2z – 3 = 0 is

Answer:

(c) \(\cos ^{-1}\left(\frac{3}{2 \sqrt{21}}\right)\)

Question 59.

The equation of the plane through the point (0, -4, -6) and (-2, 9, 3) and perpendicular to the plane x – 4y – 2z = 8 is

(a) 3x + 3y – 2z = 0

(b) x – 2y + z = 2

(c) 2x + y – z = 2

(d) 5x – 3y + 2z = 0

Answer:

(c) 2x + y – z = 2

Question 60.

The angle between the planes \(r \cdot(\hat{i}+2 \hat{j}+\hat{k})=4\) and \(r(-\hat{i}+\hat{j}+2 \hat{k})=9\) is

(a) 30°

(b) 60°

(c) 45°

(d) None of these

Answer:

(b) 60°

Question 61.

The angle between the panes x + y = 0 and y – z = 1 is

(a) \(\frac{\pi}{6}\)

(b) \(\frac{\pi}{4}\)

(c) \(\frac{\pi}{3}\)

(d) \(\frac{\pi}{2}\)

Answer:

(c) \(\frac{\pi}{3}\)

Question 62.

If the angle between the planes 2x – y + 2z = 3 and 3x + 6y + cz = 4 is \(\cos ^{-1}\left(\frac{4}{21}\right)\), then c2 =

(a) 1

(b) 4

(c) 9

(d) 5

Answer:

(b) 4

Question 63.

The distance of the plane 2x – 3y + 4z – 6 = 0 from the origin is A. Here, A refers to

(a) 6

(b) -6

(c) \(-\frac{6}{\sqrt{29}}\)

(d) \(\frac{6}{\sqrt{29}}\)

Answer:

(b) -6

Question 64.

Find the length of perpendicular from origin to the plane \(r \cdot(3 \hat{i}-4 \hat{j}-12 \hat{k})+39=0\)

(a) 1

(b) 3

(c) \(\frac{1}{7}\)

(d) None of these

Answer:

(b) 3

Question 65.

The distance of the origin from the plane through the points (1, 1, 0), (1, 2, 1) and (-2, 2, -1) is

(a) \(\frac{3}{\sqrt{11}}\)

(b) \(\frac{5}{\sqrt{22}}\)

(c) 3

(d) \(\frac{4}{\sqrt{22}}\)

Answer:

(b) \(\frac{5}{\sqrt{22}}\)

Question 66.

The angle θ between the line r = a + λb is given by

Answer:

(a) \(\sin ^{-1}\left(\frac{\tilde{h}_{\hat{\pi}}^{\pi}}{|\vec{b}|}\right)\)

Question 67.

Answer:

(a) \(\sin ^{-1}\left(\frac{2 \sqrt{2}}{3}\right)\)

Question 68.

The angle between the straight line \(\frac{x-1}{2}=\frac{y+3}{-1}=\frac{z-5}{2}\) and the plane 4x – 2y + 4z = 9 is

(a) 60°

(b) 90°

(c) 45°

(d) 30°

Answer:

(b) 90°

Question 69.

Distance of the point (α, β, γ) from y-axis is

(a) β

(b) |β|

(c) |β| + |γ|

(d) \(\sqrt{\alpha^{2}+\gamma^{2}}\)

Answer:

(d) \(\sqrt{\alpha^{2}+\gamma^{2}}\)

Question 70.

The distance of the plane \(r \cdot\left(\frac{2}{7} \hat{i}+\frac{3}{7} \hat{j}-\frac{6}{7} \hat{k}\right)=1\) from the origin is

(a) 1

(b) 7

(c) \(\frac{1}{7}\)

(d) None of these

Answer:

(a) 1

Question 71.

Answer:

(d) \(\frac{\sqrt{2}}{10}\)

Question 72.

The reflection of the point (α, β, γ) in the xy-plane is

(a) (α, β, 0)

(b) (0, 0, γ)

(c) (-α, -β, -γ)

(d) (α, β, -y)

Answer:

(d) (α, β, -y)

Question 73.

The area of the quadrilateral ABCD, where A(0, 4, 1), B(2, 3, -1), C(4, 5, 0) and D(2, 6, 2), is equal to

(a) 9 sq. units

(b) 18 sq. units

(c) 27 sq. units

(d) 81 sq. units

Answer:

(a) 9 sq. units

Question 74.

The locus represented by xy + yz = 0 is

(a) A pair of perpendicular lines

(b) A pair of parallel lines

(c) A pair of parallel planes

(d) A pair of perpendicular planes

Answer:

(d) A pair of perpendicular planes