Match the following List-I with List -IIList-IList-II(I)The number of solutions of x4−2x2sin2πx2+1=0 is(P)4(II)The number of solutions of 16(x2+1)+π2=|tanx|+8πx, x∈(−π2,3π2),x≠π2 is(Q)2(III)The number of pairs (x,y) satisfying the equation sinx+siny=sin(x+y) and |x|+|y|=1 is(R)1(IV)The number of integral values of x in [0,6] satisfying sinx+12sin2x+13sin3x>0 is(S)3 (T)6

(I) Here, x2=sin2πx2=1(II) |tanx|=(4x−π)2+16(III) x+y=0 or x=0⇒(x,y)=(±12,±12),(0,±1),(±1,0)(IV) The given equation becomes, (sinx)[(1+cosx)+(1+cosx)2+3cos2x]3>0⇒x∈(0,π)

Match the following List-I with List -IIList-IList-II(I)The number of solutions of x4−2x2sin2πx2+1=0 is(P)4(II)The number of solutions of 16(x2+1)+π2=|tanx|+8πx, x∈(−π2,3π2),x≠π2 is(Q)2(III)The number of pairs (x,y) satisfying the equation sinx+siny=sin(x+y) and |x|+|y|=1 is(R)1(IV)The number of integral values of x in [0,6] satisfying sinx+12sin2x+13sin3x>0 is(S)3 (T)6

(I) Here, x2=sin2πx2=1(II) |tanx|=(4x−π)2+16(III) x+y=0 or x=0⇒(x,y)=(±12,±12),(0,±1),(±1,0)(IV) The given equation becomes, (sinx)[(1+cosx)+(1+cosx)2+3cos2x]3>0⇒x∈(0,π)

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Match the following List-I with List -II
List-I
List-II
(I)
The number of solutions of $x^{4} - 2 x^{2} \sin^{2} \frac{π x}{2} + 1 = 0$ is
(P)
4
(II)
The number of solutions of $16 (x^{2} + 1) + π^{2} = | \tan x | + 8 π x,$ $x \in (- \frac{π}{2}, \frac{3 π}{2}), x \neq \frac{π}{2}$ is
(Q)
2
(III)
The number of pairs $(x, y)$ satisfying the equation $\sin x + \sin y = \sin (x + y) a n d | x | + | y | = 1 i s$
(R)
1
(IV)
The number of integral values of x in $[0, 6]$ satisfying $\sin x + \frac{1}{2} \sin 2 x + \frac{1}{3} \sin 3 x > 0$ is
(S)
3

(T)
6

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$(I) \to (S); (I I) \to (P); (I I I) \to (T); (I V) \to (Q)$

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

$(I) \to (Q); (I I) \to (P); (I I I) \to (T); (I V) \to (S)$

$(I) \to (Q); (I I) \to (T); (I I I) \to (P); (I V) \to (S)$

answer is C.

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Detailed Solution

(I) Here, $x^{2} = \sin^{2} \frac{π x}{2} = 1$
(II) $| \tan x | = {(4 x - π)}^{2} + 16$
(III) $x + y = 0 o r x = 0 \Rightarrow (x, y) = (\pm \frac{1}{2}, \pm \frac{1}{2}), (0, \pm 1), (\pm 1, 0)$
(IV) The given equation becomes, $\frac{(\sin x) [(1 + \cos x) + {(1 + \cos x)}^{2} + 3 \cos^{2} x]}{3} > 0$

$\Rightarrow x \in (0, π)$

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