# Class 12 Maths MCQ – Three Dimensional Geometry

This set of Class 12 Maths Chapter 11 Multiple Choice Questions & Answers (MCQs) focuses on “Three Dimensional Geometry”. These MCQs are created based on the latest CBSE syllabus and the NCERT curriculum, offering valuable assistance for exam preparation.

1. If a, b, c are the direction ratios of the line and l, m, n are the direction cosines of the line, then which of the following is true?
a) $$\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=μ$$
b) $$\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=μ-1$$
c) $$\frac{l}{a}=\frac{m}{c}=\frac{n}{b}=μ$$
d) $$\frac{l}{a}=\frac{n+1}{b}=\frac{n}{c}=μ$$

Explanation: For a given line, if a, b, c are the direction ratios and l, m, n are the direction cosines of the line then
a=μl, b=μm, c=μn
Or we can say that,
$$\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=μ$$, where μ is a constant.

2. If a line makes an angle of 120°, 45°, 30° with the positive x, y, z-axis respectively then find the direction cosines.
a) l=$$\frac{1}{2}, \,m=\frac{1}{\sqrt{2}}, \,n=\frac{\sqrt{3}}{2}$$
b) l=-$$\frac{1}{2}, \,m=-\frac{1}{\sqrt{2}}, \,n=-\frac{\sqrt{3}}{2}$$
c) l=-$$\frac{1}{2}, \,m=\frac{1}{\sqrt{2}}, \,n=\frac{\sqrt{3}}{2}$$
d) l=$$0, \,m=\frac{1}{\sqrt{2}}, \,n=\frac{\sqrt{3}}{2}$$

Explanation: Let l, m, n be the direction cosines of the line.
We know that, if α, β, γ are the angles that the line makes with the x, y, z- axis respectively, then
l=cos⁡α
m=cos⁡β
n=cos⁡γ
∴l=cos⁡120°, m=cos⁡45°, n=cos⁡30°
Hence, $$l=-\frac{1}{2}, \,m=\frac{1}{\sqrt{2}}, \,n=\frac{\sqrt{3}}{2}$$

3. If a line has direction ratios 2, -3, 7 then find the direction cosines.
a) l=$$\frac{2}{\sqrt{62}},m=-\frac{7}{\sqrt{62}},n=\frac{7}{\sqrt{62}}$$
b) l=$$\frac{2}{\sqrt{6}},m=-\frac{3}{\sqrt{6}},n=\frac{7}{\sqrt{6}}$$
c) l=-$$\frac{2}{\sqrt{62}},m=-\frac{3}{\sqrt{62}},n=-\frac{7}{\sqrt{62}}$$
d) l=$$\frac{2}{\sqrt{62}},m=-\frac{3}{\sqrt{62}},n=\frac{7}{\sqrt{62}}$$

Explanation: For a given line, if a, b, c are the direction ratios and l, m, n are the direction cosines of the line then
l=±$$\frac{a}{\sqrt{a^2+b^2+c^2}}$$
m=±$$\frac{b}{\sqrt{a^2+b^2+c^2}}$$
n=±$$\frac{c}{\sqrt{a^2+b^2+c^2}}$$
∴l=$$\frac{2}{\sqrt{2^2+(-3)^2+7^2}}, \,m=-\frac{3}{\sqrt{2^2+(-3)^2+7^2}}, \,n=\frac{7}{\sqrt{2^2+(-3)^2+7^2}}$$
Hence, l=$$\frac{2}{\sqrt{62}}, \,m=-\frac{3}{\sqrt{62}}, \,n=\frac{7}{\sqrt{62}}$$.

4. Find the direction cosines of the line passing through two points (4, -5, -6) and (-1, 2, 8).
a) $$\frac{5}{270},\frac{7}{\sqrt{270}},\frac{14}{\sqrt{270}}$$
b) –$$\frac{7}{\sqrt{270}}, \frac{7}{\sqrt{270}},\frac{7}{\sqrt{270}}$$
c) –$$\frac{5}{\sqrt{270}}, \frac{7}{\sqrt{270}},\frac{14}{\sqrt{270}}$$
d) –$$\frac{5}{\sqrt{20}}, \frac{7}{\sqrt{720}},\frac{14}{\sqrt{270}}$$

Explanation: The direction cosines of two lines passing through two points is given by:
$$\frac{x_2-x_1}{PQ}, \frac{y_2-y_1}{PQ}, \frac{z_2-z_1}{PQ}$$
and $$PQ = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$$
In the given problem we have, P(4,-5,-6) and Q(-1,2,8)
∴$$PQ = \sqrt{(-1-4)^2+(2+5)^2+(8+6)^2}$$
$$=\sqrt{25+49+196}=\sqrt{270}$$
Hence, the direction ratios are $$l=\frac{(-1-4)}{\sqrt{270}}=-\frac{5}{\sqrt{270}}$$
$$m=\frac{(2+5)}{\sqrt{270}}=\frac{7}{\sqrt{270}}$$
$$n=\frac{(8+6)}{\sqrt{270}}=\frac{14}{\sqrt{270}}$$.

5. The direction ratios of the line segment joining $$P(x_1,y_1,z_1)$$ and $$Q(x_2,y_2,z_2)$$ is given by _______, ____________ and __________
a) $$x_2+x_1,y_2+y_1,z_2+z_1$$
b) $$x_2-x_1,y_2+y_1,z_2-z_1$$
c) $$x_2-x_1,y_2-y_1,z_2-z_1$$
d) $$x_2+x_1,y_2-y_1,z_2+z_1$$

Explanation: Let a.,b,c be the direction ratios of the line segment PQ.
Then, the direction ratios of the line segment joining $$P(x_1,y_1,z_1)$$ and $$Q(x_2,y_2,z_2)$$ is given by
$$a=x_2-x_1$$
$$b=y_2-y_1$$
$$c=z_2-z_1$$
Note: Join free Sanfoundry classes at Telegram or Youtube

6. Find the direction cosines of the line passing through two points P(-6,7,3) and Q(3,-2,5).
a) –$$\frac{2}{\sqrt{166}},\frac{-9}{\sqrt{166}},\frac{2}{\sqrt{166}}$$
b) –$$\frac{9}{\sqrt{166}},\frac{-7}{\sqrt{166}},\frac{2}{\sqrt{166}}$$
c) –$$\frac{9}{\sqrt{66}},\frac{-9}{\sqrt{66}},\frac{2}{\sqrt{66}}$$
d) –$$\frac{9}{\sqrt{166}},\frac{-9}{\sqrt{166}},\frac{2}{\sqrt{166}}$$

Explanation: The direction cosines of two lines passing through two points is given by:
$$\frac{x_2-x_1}{PQ},\frac{y_2-y_1}{PQ},\frac{z_2-z_1}{PQ}$$
and $$PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$$
In the given problem we have, P(-6,7,3) and Q(3,-2,5)
∴$$PQ=\sqrt{(3+6)^2+(-2-7)^2+(5-3)^2}$$
=$$\sqrt{81+81+4}=\sqrt{166}$$
Hence, the direction ratios are $$l=\frac{-6-3}{\sqrt{166}}=-\frac{9}{\sqrt{166}}$$
m=$$\frac{-2-7}{\sqrt{166}}=\frac{-9}{\sqrt{166}}$$
n=$$\frac{5-3}{\sqrt{166}}=\frac{2}{\sqrt{166}}$$

7. If the direction ratios of a line are 5, 4, -7 respectively, then find the direction cosines.
a) $$\frac{75}{\sqrt{90}},\frac{4}{\sqrt{90}},\frac{5}{\sqrt{90}}$$
b) $$\frac{5}{\sqrt{90}},\frac{4}{\sqrt{90}},-\frac{7}{\sqrt{90}}$$
c) $$\frac{5}{\sqrt{70}},\frac{4}{\sqrt{70}},-\frac{7}{\sqrt{70}}$$
d) $$\frac{3}{\sqrt{90}},\frac{4}{\sqrt{90}},-\frac{5}{\sqrt{90}}$$

Explanation: If a,b,c are the direction ratios and l,m,n are the direction cosines respectively for a given line, then the direction cosines in terms of the direction ratios can be expressed as
l=±$$\frac{a}{\sqrt{a^2+b^2+c^2}}$$
m=±$$\frac{b}{\sqrt{a^2+b^2+c^2}}$$
n=±$$\frac{c}{\sqrt{a^2+b^2+c^2}}$$
Given that, a=5, b=4, c=-7
l=$$\frac{5}{\sqrt{(5^2+4^2+(-7)^2)}}=\frac{5}{\sqrt{(25+16+49)}}=\frac{5}{\sqrt{90}}$$
m=$$\frac{4}{\sqrt{(5^2+4^2+(-7)^2)}}=\frac{4}{\sqrt{90}}$$
n=-$$\frac{7}{\sqrt{(5^2+4^2+(-7)^2)}}=-\frac{7}{\sqrt{90}}$$

8. If a, b, c are the direction ratios of the line and l, m, n are the direction cosines of the line, then which of the following is incorrect?
a) $$\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=k$$
b) l2+m2+n2=1
c) k=±$$\frac{1}{\sqrt{(a^2+b^2+c^2)}}$$
d) l2-m2=n2-1

Explanation: Given that, a, b, c are the direction ratios of the line and l, m, n are the direction cosines of the line,
$$\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=k$$ and l2+m2+n2=1
⇒l=ak, m=bk, n=ck
(ak)2+(bk)2+(ck)2=1
k2 (a2+b2+c2)=1
k2=$$\frac{1}{a^2+b^2+c^2}$$
∴k=±$$\frac{1}{\sqrt{(a^2+b^2+c^2)}}$$
Hence, l2-m2=n2-1 is incorrect.

9. If the direction cosines of the line are $$\frac{1}{2},-\frac{\sqrt{3}}{2}$$,x respectively, then find the value of x.
a) 1
b) 0
c) $$\frac{\sqrt{3}}{2}$$
d) $$\frac{1}{2}$$

Explanation: If the direction cosines of a line are l,m,n respectively, then
l2+m2+n2=1
∴$$\frac{1}{2}^2+(\frac{\sqrt{3}}{2})^2+x^2=1$$
x2=$$1-\frac{1}{4}-\frac{3}{4}$$
x2=0
⇒x=0

10. If a line makes an angle of 60°, 150°, 45° with the positive x, y, z-axis respectively, find its direction cosines.
a) –$$\frac{1}{2},-\frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}}$$
b) –$$\frac{1}{2},-\frac{\sqrt{3}}{2},-\frac{1}{\sqrt{2}}$$
c) $$\frac{1}{2},-\frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}}$$
d) $$\frac{1}{2},\frac{\sqrt{3}}{2},-\frac{1}{\sqrt{2}}$$

Explanation: Let l, m, n be the direction cosines of the line.
We know that, if α, β, γ are the angles that the line makes with the x, y, z-axis respectively, then
l=cos⁡α=cos⁡60°=$$\frac{1}{2}$$
m=cos⁡β=cos⁡150°=-$$\frac{\sqrt{3}}{2}$$
n=cos⁡γ=cos⁡45°=$$\frac{1}{\sqrt{2}}$$.

More MCQs on Class 12 Maths Chapter 11:

To practice all chapters and topics of class 12 Mathematics, here is complete set of 1000+ Multiple Choice Questions and Answers.

If you find a mistake in question / option / answer, kindly take a screenshot and email to [email protected]