Question

Column I		Column II
A	The acute angle which the line of intersection of the planes $2x+y+z=0$ and $x+y+2z=0$ makes with the positive $x$-axis, is	P	${\tan }^{-1}\sqrt{2}$
B	One corner of a rectangular sheet of paper of width $1m$ 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
$V=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 1& 1\\ 1& 1& 2\end{array}\right|=\hat{i}(2-1)-\hat{j}(4-1)+\hat{k}(2-1)=\hat{i}-3\hat{j}+\hat{k}$
$u=\hat{i};$
$\therefore \cos \theta =\frac{1}{\sqrt{11}}={\tan }^{-1}\sqrt{10}\text{ Ans. }\Rightarrow (S)$
(B) acute angle $={\cos }^{-1}\frac{1}{\sqrt{3}}\Rightarrow {\tan }^{-1}\sqrt{2}\Rightarrow$ (P)
(C) ${\tan }^{-1}(2\sqrt{2})$ $\Rightarrow$(R)