KVPY 2020-21 SB SX Exam Previous Papers

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Question : 87 of 120

Marks: +1, -0

For

\θ \∈ [0, \π]

, let

f(\θ)=\sin (\cos \θ)

and

{g}(\θ)=\cos (\sin \θ) .

Let

{a}=\max ↙{0 \≤ \θ \≤ \π} {f}(\θ), {b}=\min↙{0 \≤ \θ \≤ \π}

{f}(\θ), {c}=\max ↙{0 \≤ \θ \≤ \π} {g}(\θ)

and

{d}=\min↙{0 \≤ \θ \≤ \π} {g}(\θ) .

The correct inequalities satisfied by

{a}, {b}, {c}, {d}

are

b < d < c < a
d < b < a < c
b < d < a < c
b < a < d < c

Solution:

f(\θ)=\sin (\cos \θ)

g(\θ)=\cos (\sin \θ)

f'(\θ)=\cos (\cos \θ)(-\sin \θ) < 0 \∀\θ \∈ [0, \π]

\∴f(\θ)

decreases monotonically

\∴{a}=\max f(\θ)=f(0)=\sin 1

{b}=\min f(\θ)=f(\π)=-\sin 1

g'(\θ)=-\sin (\sin \θ) \cos \θ

g(\θ)=1 ; g(\π)=1 ; g({\π}/{2})=\cos 1

\∴c=\max g(\θ)=1

d=\min g(\θ)=\cos 1

$\∴b

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