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STD 12 - 11. three dimension geometry — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 11. three dimension geometry4 Marks
MCQ
Let $\alpha$ be the angle between the lines whose direction cosines satisfy the equations $l+m-n=0$ and $l^{2}+m^{2}-n^{2}=0 .$ Then the value of $\sin ^{4} \alpha+\cos ^{4} \alpha$ is
  • A
    $\frac{3}{4}$
  • B
    $\frac{3}{8}$
  • ✓
    $\frac{5}{8}$
  • D
    $\frac{1}{2}$
✓

Answer

Correct option: C.
$\frac{5}{8}$
c
$n =\ell+ m$

Now, $\ell^{2}+ m ^{2}= n ^{2}=(\ell+ m )^{2}$

$\Rightarrow 2 \ell m =0$

If $\ell=0 \Rightarrow m = n =\pm \frac{1}{\sqrt{2}}$

And, If $m=0 \Rightarrow n=\ell=\pm \frac{1}{\sqrt{2}}$

So, direction cosines of two lines are

$\left(0, \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$ and $\left(\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}}\right)$

Thus, $\cos \alpha=\frac{1}{2} \Rightarrow \alpha=\frac{\pi}{3}$

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