Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Provide solution for RD Sharma Maths Class 12 Chapter 26 Directions Cosines and Direction Ratios Exercise Multiple Choice Question, question 13.

Provide solution for RD Sharma Maths Class 12 Chapter 26 Directions Cosines and Direction Ratios Exercise Multiple Choice Question, question 13.

Answers (1)

Final Answer:

(c) \frac{4}{3}

Hint: Use direction cosines

Given: Line makes angles

\alpha ,\beta ,\gamma ,\delta

with diagonals of cube.

To Find:

\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma+\cos ^{2} \delta

Solution: Let us consider a cube as shown

Direction ratio of OP are (a-0,a-0,a-0)

=(a,a,a)

Direction cosine of OP are

\begin{aligned} &\left(\frac{a}{\sqrt{a^{2}+a^{2}+a^{2}}}, \frac{a}{\sqrt{a^{2}+a^{2}+a^{2}}}, \frac{a}{\sqrt{a^{2}+a^{2}+a^{2}}}\right) \\ &\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right) \end{aligned}

Similarly, direction cosine of all diagonals

\begin{aligned} &O P=\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right) \\ &B N=\left(\frac{-1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right) \\ &A M=\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{-1}{\sqrt{3}}\right) \\ &C L=\left(\frac{1}{\sqrt{3}}, \frac{-1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right) \end{aligned}

Let (l,m,n) are the direction cosine of line which is inclined at angle

\alpha ,\beta ,\gamma ,\delta with diagonals. Now, using dot product

\begin{aligned} &\cos \alpha=\frac{l+m+n}{\sqrt{3}} \\ &\cos \beta=\frac{-l+m+n}{\sqrt{3}} \\ &\cos \gamma=\frac{l+m-n}{\sqrt{3}} \\ &\cos \delta=\frac{l-m+n}{\sqrt{3}} \end{aligned}

Squaring and adding above equations, we get

\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma+\cos ^{2} \delta=\frac{1}{3}\left[(l+m+n)^{2}+(-l+m+n)^{2}+(l+m-n)^{2}+(l-m+n)^{2}\right]\

\begin{aligned} &=\frac{1}{3}\left[4\left(l^{2}+m^{2}+n^{2}\right)\right] \\ &=\frac{4}{3} \end{aligned}

Therefore, value of

\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma+\cos ^{2} \delta=\frac{4}{3}

Hence option(c) is correct.

Posted by

infoexpert24

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

Board Exam Time Table 2025 Live: UP Board exams from February 24; CBSE, BSEB date sheet soon
anam-khan

CBSE Date Sheet 2025 Live: Class 10, 12 board exam dates soon; syllabus, sample papers
anam-khan

Extra time in CBSE exams for students with type 1 diabetes: Kerala SHRC seeks report on plea

Latest Question