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Line and Plane — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsLine and Plane2 Marks
MCQ
The acute angle between the plane $2 x+3 y-z+7=0$ and $X$-axis is
  • A
    $\cos ^{-1}\left(\frac{2}{\sqrt{14}}\right)$
  • B
    $\cos ^{-1}\left(\frac{-2}{\sqrt{14}}\right)$
  • ✓
    $\sin ^{-1}\left(\frac{2}{\sqrt{14}}\right)$
  • D
    $\sin ^{-1}\left(\frac{-2}{\sqrt{14}}\right)$
✓

Answer

Correct option: C.
$\sin ^{-1}\left(\frac{2}{\sqrt{14}}\right)$
(C)
The d.r.s. of normal to the plane are $2,3,-1$ The d.r.s. of X -axis are $1,0,0$
$\therefore$ the angle between the plane and X -axis is
$\sin \theta=\frac{a a_1+b b_1+c c_1}{\sqrt{a^2+b^2+c^2} \cdot \sqrt{a_1^2+b_1^2+c_1^2}}$
$\Rightarrow \sin \theta=\frac{2(1)+0+0}{\sqrt{4+9+1} \cdot \sqrt{1}}$
$\Rightarrow \sin \theta=\frac{2}{\sqrt{14}}$
$\Rightarrow \theta=\sin ^{-1}\left(\frac{2}{\sqrt{14}}\right)$

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