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(P) If $2 \sin x, \sin 2 x & 2 \cos x$ are in Arithmetic Progression, then the value of $| \sin x + \cos x |$ is (1) $\frac{\sqrt{5} - 1}{2}$
(Q) If the angles of a triangle are in Arithmetic Progression with the common difference equal to $\frac{1}{3}$ of the greatest angle, then ratio of the greatest side to second greater side is (2) $\frac{2}{\sqrt{3}}$
(R) Let the side AB, BC, CD & DA of a cyclic quadrilateral ABCD are in Geometric Progression with common ratio $\sqrt{3}$ , then the value of $\frac{B D}{A B}$ is (3) $2 \sqrt{5}$
(S) Let a,b,c be in Arithmetic Progression & $a^{2}, b^{2}, c^{2}$ be in Harmonic Progression. If $a \neq c$ , then $a : c$ is (4) $\sqrt{3} - 2$

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(P) – (1), (Q) – (4), (R) – (3), (S) – (2)

(P) – (3), (Q) – (4), (R) – (1), (S) – (2)

(P) – (1), (Q) – (2), (R) – (3), (S) – (4)

(P) – (3), (Q) – (2), (R) – (1), (S) – (4)

answer is C.

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Detailed Solution

(P) $2 \sin x, \sin 2 x & 2 \cos x$ are in A.P., hence $\sin 2 x = \sin x + \cos x$
Now Let $\sin x + \cos x = t$ , then $\sin 2 x = t^{2} - 1$ , then the above equation gives
$t^{2} - t - 1 = 0$ or $t = \sin x + \cos x = \frac{1 - \sqrt{5}}{2}$
(Q) Let $θ$ be the greatest angle, then angles will be $\frac{π}{3} - θ, \frac{π}{3}, \frac{π}{3} + θ$
$θ = \frac{1}{3} (\frac{π}{3} + θ) \Rightarrow θ = \frac{π}{6}$
Hence angles are $\frac{π}{6}, \frac{π}{3}, \frac{π}{2}$
Now, $\frac{a}{\sin \frac{π}{6}} = \frac{b}{\sin \frac{π}{3}} = \frac{c}{\sin \frac{π}{2}} \Rightarrow \frac{a}{1} = \frac{b}{\sqrt{3}} = \frac{c}{2}$

(R) Let $A B = a, B C = \sqrt{3} a, C D = 3 a & A D = 3 \sqrt{3} a$
Now $\cos A + \cos C = 0 \Rightarrow \frac{a^{2} + 27 a^{2} - B D^{2}}{3 \sqrt{3} a^{2}} + \frac{3 a^{2} + 9 a^{2} - B D^{2}}{3 \sqrt{3} a^{2}} = 0$
$\Rightarrow B D^{2} = 20 a^{2} o r \frac{B D}{A B} = 2 \sqrt{5} (S) a + c = 2 b & b^{2} = \frac{2 a^{2} c^{2}}{a^{2} + c^{2}} \Rightarrow {(a + c)}^{2} = \frac{8 a^{2} c^{2}}{a^{2} + c^{2}} \Rightarrow {(a + c)}^{4} - 2 a c {(a + c)}^{2} - 8 a^{2} c^{2} = 0$

$\Rightarrow \frac{{(a + c)}^{2}}{a c} = - 8$

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