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  • 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

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