Match List I with List II and select the correct answer using the code given below the lists :…

Match List I with List II and select the correct answer using the code given below the lists :
\(\begin{array}{|c|c|c|c|}
\hline & \text { List - I } & & \text { List - II } \\
\hline \text { A. } & \left(\frac{1}{y^2}\left(\frac{\cos \left(\tan ^{-1} y\right)+y \sin \left(\tan ^{-1} y\right)}{\cot \left(\sin ^{-1} y\right)+\tan \left(\sin ^{-1} y\right)}\right)^2+y^4\right)^{1 / 2} \text { takes value } & \text { P. } & \frac{1}{2} \sqrt{\frac{5}{3}} \\
\hline \text { B. } & \begin{array}{l}
\text { If } \cos x+\cos y+\cos z=0=\sin x+\sin y+\sin z \text { then possible } \\
\text { value of } \cos \frac{x-y}{2} \text { is }
\end{array} & \text { Q. } & \sqrt{2} \\
\hline \text { C. } & \begin{array}{l}
\text { If } \cos \left(\frac{\pi}{4}-x\right) \cos 2 x+\sin x \sin 2 x \sec x=\cos x \sin 2 x \sec x+ \\
\cos \left(\frac{\pi}{4}+x\right) \cos 2 x \text { then possible value of sec } x \text { is }
\end{array} & \mathrm{R} \text {. } & \frac{1}{2} \\
\hline \text { D. } & \begin{array}{l}
\text { If } \cot \left(\sin ^{-1} \sqrt{1-x^2}\right)=\sin \left(\tan ^{-1}(x \sqrt{6})\right), x \neq 0, \text { then } \\
\text { possible value of } x \text { is }
\end{array} & \text { S. } & 1 \\
\hline
\end{array}\)
  1. a-r;b-p;c-s;d-q;
  2. a-p;b-q;c-r;d-s;
  3. a-p;b-r;c-s;d-q;
  4. a-s;b-r;c-q;d-p;

Solution

P - 4 ; Q - 3 ; R - 2 or 4 ; S - 1
P 1 y 2 cos tan -1 y + y sin tan -1 y cot sin -1 y + tan sin -1 y 2 + y 4 1 / 2
                  = 1 y 2 1 1 + y 2 + y . y 1 + y 2 1 - y 2 y + y 1 - y 2 2 + y 4 1 / 2
                = y 2 y 2 1 - y 2 + y 4
               = 1

4
Q cos x + cos y = - cos z
sin x + sin y = - sin z Square and add
2 + 2 cos x - y = 1
⇒ cos x - y = - 1 / 2
⇒ 2 cos 2 x - y 2 - 1 = - 1 / 2 , ⇒ cos x - y 2 = 1 / 2
3
R cos 2 x cos π 4 - x - cos π 4 + x + 2 sin 2 x = 2 sin x cos x
cos 2 x 2 sin x + 2 sin 2 x = 2 sin x cos x
2 sin x cos 2 x + 2 sin x - 2 cos x = 0
Either sin x = 0 OR cos 2 x - sin 2 x = 2 cos x - sin x
sec x = 1    OR    cos x = sin x
                     ⇒ sec x = 2
2 OR 4
S cot sin -1 1 - x 2 = sin tan -1 x 6

x 1 - x 2 = x 6 1 + 6 x 2
⇒ 1 + 6 x 2 = 6 - 6 x 2
⇒ 1 2 x 2 = 5
⇒ x = 5 1 2 = = 1 2 5 3
1

Asked in: JEE Advanced 2013 (Paper 2)

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