Question

Column I		Column II
A	The acute angle which the line of intersection of the planes $2 x + y + z =0$ and $x + y +2 z =0$ makes with the positive $x$-axis, is	P	$\tan ^{-1} \sqrt{2}$
B	One corner of a rectangular sheet of paper of width $1 m$ is folded over so as to just reach on the opposite edge of the sheet. If $\theta$ is the acute angle which the line along the minimum length of the crease, makes with the width of the paper, then $\theta$ equals	Q	$\tan ^{-1} \sqrt{3}$
C	The acute angle between the two plane faces of a regular tetrahedron is	R	$\tan ^{-1} \sqrt{8}$
		S	$\tan ^{-1} \sqrt{10}$

• A. (A) S; (B) P; (C) P
• B. (A) S; (B) P; (C) Q
• C. (A) S; (B) P; (C) R
• D. (A) S; (B) P; (C) S

Answer 1

Answer: (C)

(A) Vector $V $along the line of intersection of the given planes is
$V =\begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 2 & 1 & 1 \\ 1 & 1 & 2\end{vmatrix}=\hat{ i }(2-1)-\hat{ j }(4-1)+\hat{ k }(2-1)=\hat{ i }-3 \hat{ j }+\hat{ k }$
$ { u } =\hat{ i } ; $
$ \therefore \cos \theta=\frac{1}{\sqrt{11}}=\tan ^{-1} \sqrt{10} \text { Ans. } \Rightarrow (S)$
(B) acute angle $=\cos ^{-1} \frac{1}{\sqrt{3}} \quad \Rightarrow \quad \tan ^{-1} \sqrt{2} \Rightarrow$ (P)
(C) $\tan ^{-1}(2 \sqrt{2})$ $\Rightarrow$(R)