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The following question consist of two statements, one labelled as the 'Assertion (A)' and the other as 'Reason $(R)^{\prime}$. You are to examine these two statements carefully and select the answer. Assertion(A): If $ < 1, \mathrm{~m}, \mathrm{n}>$ are direction cosines of a line, there can be a line whose direction cosines are
$\left\langle\sqrt{\frac{1^{2}+m^{2}}{2}}, \sqrt{\frac{m^{2}+n^{2}}{2}}, \sqrt{\frac{n^{2}+1^{2}}{2},}\right\rangle .$
Reason(R): The sum of direction cosines of a line is unity.
Options:
$\left\langle\sqrt{\frac{1^{2}+m^{2}}{2}}, \sqrt{\frac{m^{2}+n^{2}}{2}}, \sqrt{\frac{n^{2}+1^{2}}{2},}\right\rangle .$
Reason(R): The sum of direction cosines of a line is unity.
Solution:
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Verified Answer
The correct answer is:
$\mathrm{A}$ is true but $\mathrm{R}$ is false.
Sum of directions cosines of a line i.e. $\ell+m+n \neq 1$. So, $\mathrm{R}$ is false Since sum of squares of direction cosines is unity
$=\left(\sqrt{\frac{\ell^{2}+\mathrm{m}^{2}}{2}}\right)^{2}+\left(\sqrt{\frac{\mathrm{m}^{2}+\mathrm{n}^{2}}{2}}\right)^{2}+\left(\sqrt{\frac{\mathrm{n}^{2}+\ell^{2}}{2}}\right)^{2}$
$=\frac{\ell^{2}+\mathrm{m}^{2}}{2}+\frac{\mathrm{m}^{2}+\mathrm{n}^{2}}{2}+\frac{\mathrm{n}^{2}+\ell^{2}}{2}$
$\Rightarrow \frac{2\left(\ell^{2}+\mathrm{m}^{2}+\mathrm{n}^{2}\right)}{2}=\ell^{2}+\mathrm{m}^{2}+\mathrm{n}^{2}=1$
Hence, assertion A is true.
So, $A$ is true but $R$ is false.
$=\left(\sqrt{\frac{\ell^{2}+\mathrm{m}^{2}}{2}}\right)^{2}+\left(\sqrt{\frac{\mathrm{m}^{2}+\mathrm{n}^{2}}{2}}\right)^{2}+\left(\sqrt{\frac{\mathrm{n}^{2}+\ell^{2}}{2}}\right)^{2}$
$=\frac{\ell^{2}+\mathrm{m}^{2}}{2}+\frac{\mathrm{m}^{2}+\mathrm{n}^{2}}{2}+\frac{\mathrm{n}^{2}+\ell^{2}}{2}$
$\Rightarrow \frac{2\left(\ell^{2}+\mathrm{m}^{2}+\mathrm{n}^{2}\right)}{2}=\ell^{2}+\mathrm{m}^{2}+\mathrm{n}^{2}=1$
Hence, assertion A is true.
So, $A$ is true but $R$ is false.
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