Match List I with List II:

  List - I List - II
(I)

The number of solutions of the equation where  is

(P) 4
(II)  equals (Q) 3
(III) The number of solutions of (x, y), which satisfy |y| = sin x and y = cos-1(cos x), where -2π ≤ x ≤ 2π is (R) 2
(IV) For the equation , the number of real solutions is (S) 1
    (T) 0

 

  1. (I) → T, (II) → S, (III) → S, (IV) → Q 
  2. (I) → S, (II) → Q, (III) → R, (IV) → S 
  3. (I) → T, (II) → P, (III) → Q, (IV) → T 
  4. (I) → S, (II) → T, (III) → S, (IV) → Q 

Answer (Detailed Solution Below)

Option 3 : (I) → T, (II) → P, (III) → Q, (IV) → T 

Detailed Solution

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Calculation

(I)

But for all values of x, is not defined.

Hence, no solution exist.

(I) → T

(II)

=

=

=

=

=

=

=

=

=

(II) → P

(III) In [0, π], |y| = sin x, y = cos-1(cos x) = x

In [π, 2π], |y| = sin x, y = cos-1(cos(2π - x)) = 2π - x

In [-π, 0], |y| = sin x, y = cos-1(cos(-x)) = -x

In [-2π, -π], |y| = sin x, y = cos-1(cos(2π + x)) = 2π + x

Plotting the graph, we have

Total 3 solutions

(III) → Q

(IV) Given equation is

On squaring both sides, we get

∴ But x = 0 does not satisfy the given equation.

So, the number of real solutions is zero.

(IV) → T 

Hence option 3 is correct

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