Question
Download Solution PDFMatch List I with List II:
List - I | List - II | |
(I) |
The number of solutions of the equation |
(P) 4 |
(II) | (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 |
(S) 1 |
(T) 0 |
Answer (Detailed Solution Below)
Option 3 : (I) → T, (II) → P, (III) → Q, (IV) → T
Detailed Solution
Download Solution PDFCalculation
(I)
⇒
But for all values of x,
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