Let t1=(sin−1x)sin−1x,t2=(sin−1x)cos−1x,t3=(cos−1x)sin−1x,t4=(cos−1x)cos−1x Match the entries of list –I with the entries of list –II List – I List – II(I)x ∈ (0, cos 1)(P)t1>t2>t4>t3(II)x∈(cos1,12)(Q)t4>t3>t1>t2(III)x∈(12,sin1)(R)t2>t1>t4>t3(IV)x∈(sin1,1)(S)t3>t4>t1>t2 (T)t1>t4>t3>t2Which of the following is the only correct combination.

Answer: I → Q, II → Q, III → S, IV → P.Reason (sketch):Let a=sin⁡−1x, c=cos⁡−1x=π2−aa=\sin^{-1}x,\ c=\cos^{-1}x=\tfrac{\pi}{2}-aa=sin−1x, c=cos−1x=2π−a.t1=aa, t2=ac, t3=ca, t4=cct_1=a^a,\ t_2=a^c,\ t_3=c^a,\ t_4=c^ct1=aa, t2=ac, t3=ca, t4=cc.For x∈(0,cos⁡1)x\in(0,\cos1)x∈(0,cos1) and x∈(cos⁡1,12)x\in(\cos1,\tfrac12)x∈(cos1,21): a t3>t1>t2t_4>t_3>t_1>t_2t4>t3>t1>t2 ⇒ Q.For x∈(12,sin⁡1)x\in(\tfrac12,\sin1)x∈(21,sin1): a t1>t4>t2t_3>t_1>t_4>t_2t3>t1>t4>t2 ⇒ S.For x∈(sin⁡1,1)x\in(\sin1,1)x∈(sin1,1): 1 t2>t4>t3t_1>t_2>t_4>t_3t1>t2>t4>t3 ⇒ P.

Let t1=(sin−1x)sin−1x,t2=(sin−1x)cos−1x,t3=(cos−1x)sin−1x,t4=(cos−1x)cos−1x Match the entries of list –I with the entries of list –II List – I List – II(I)x ∈ (0, cos 1)(P)t1>t2>t4>t3(II)x∈(cos1,12)(Q)t4>t3>t1>t2(III)x∈(12,sin1)(R)t2>t1>t4>t3(IV)x∈(sin1,1)(S)t3>t4>t1>t2 (T)t1>t4>t3>t2Which of the following is the only correct combination.

Answer: I → Q, II → Q, III → S, IV → P.Reason (sketch):Let a=sin⁡−1x, c=cos⁡−1x=π2−aa=\sin^{-1}x,\ c=\cos^{-1}x=\tfrac{\pi}{2}-aa=sin−1x, c=cos−1x=2π−a.t1=aa, t2=ac, t3=ca, t4=cct_1=a^a,\ t_2=a^c,\ t_3=c^a,\ t_4=c^ct1=aa, t2=ac, t3=ca, t4=cc.For x∈(0,cos⁡1)x\in(0,\cos1)x∈(0,cos1) and x∈(cos⁡1,12)x\in(\cos1,\tfrac12)x∈(cos1,21): a t3>t1>t2t_4>t_3>t_1>t_2t4>t3>t1>t2 ⇒ Q.For x∈(12,sin⁡1)x\in(\tfrac12,\sin1)x∈(21,sin1): a t1>t4>t2t_3>t_1>t_4>t_2t3>t1>t4>t2 ⇒ S.For x∈(sin⁡1,1)x\in(\sin1,1)x∈(sin1,1): 1 t2>t4>t3t_1>t_2>t_4>t_3t1>t2>t4>t3 ⇒ P.

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Let $t_{1} = {(\sin^{- 1} x)}^{\sin^{- 1} x}, t_{2} = {(\sin^{- 1} x)}^{\cos^{- 1} x}, t_{3} = {(\cos^{- 1} x)}^{\sin^{- 1} x}, t_{4} = {(\cos^{- 1} x)}^{\cos^{- 1} x}$
Match the entries of list –I with the entries of list –II

List – I List – II
(I) x $\in$ (0, cos 1) (P) $t_{1} > t_{2} > t_{4} > t_{3}$
(II) $x \in (\cos 1, \frac{1}{\sqrt{2}})$ (Q) $t_{4} > t_{3} > t_{1} > t_{2}$
(III) $x \in (\frac{1}{\sqrt{2}}, \sin 1)$ (R) $t_{2} > t_{1} > t_{4} > t_{3}$
(IV) $x \in (\sin 1, 1)$ (S) $t_{3} > t_{4} > t_{1} > t_{2}$
(T) $t_{1} > t_{4} > t_{3} > t_{2}$
Which of the following is the only correct combination.

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$I \to P; I I \to Q; I I I \to R; I V \to S$

$I \to Q; I I \to S; I I I \to R; I V \to P$

$I \to T; I I \to S; I I I \to R; I V \to P$

$I \to Q; I I \to S; I I I \to T; I V \to R$

answer is B.

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Detailed Solution

Answer: I → Q, II → Q, III → S, IV → P.

Reason (sketch):
Let $a=\sin^{-1}x,\ c=\cos^{-1}x=\tfrac{\pi}{2}-a$ $a$ $=$ $sin$ $-1$ $x$ $,$ $c$ $=$ $cos$ $-1$ $x$ $=$ $2$ $π$ $$ $-$ $a$ .
$t_1=a^a,\ t_2=a^c,\ t_3=c^a,\ t_4=c^c$ $t$ $1$ $$ $=$ $a$ $a$ $,$ $t$ $2$ $$ $=$ $a$ $c$ $,$ $t$ $3$ $$ $=$ $c$ $a$ $,$ $t$ $4$ $$ $=$ $c$ $c$ .

For $x\in(0,\cos1)$ $x$ $\in$ $($ $0$ $,$ $cos$ $1$ $)$ and $x\in(\cos1,\tfrac12)$ $x$ $\in$ $($ $cos$ $1$ $,$ $21$ $$ $)$ : $a<1<c$ $a$ $<$ $1$ $<$ $c$ ⇒ $t_4>t_3>t_1>t_2$ $t$ $4$ $$ $>$ $t$ $3$ $$ $>$ $t$ $1$ $$ $>$ $t$ $2$ $$ ⇒ Q.
For $x\in(\tfrac12,\sin1)$ $x$ $\in$ $($ $21$ $$ $,$ $sin$ $1$ $)$ : $a<1,\ c<1$ $a$ $<$ $1$ $,$ $c$ $<$ $1$ (near end) ⇒ $t_3>t_1>t_4>t_2$ $t$ $3$ $$ $>$ $t$ $1$ $$ $>$ $t$ $4$ $$ $>$ $t$ $2$ $$ ⇒ S.
For $x\in(\sin1,1)$ $x$ $\in$ $($ $sin$ $1$ $,$ $1$ $)$ : $1<a,\ c<1$ $1$ $<$ $a$ $,$ $c$ $<$ $1$ ⇒ $t_1>t_2>t_4>t_3$ $t$ $1$ $$ $>$ $t$ $2$ $$ $>$ $t$ $4$ $$ $>$ $t$ $3$ $$ ⇒ P.

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