KVPY 2020-21 SB SX Exam Previous Papers

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Question : 4 of 120

Marks: +1, -0

Let

{\ℝ}

be the set of all real numbers and

f(x)=\sin ^{10} x(\cos ^{8} x+\cos ^{4} x+\cos ^{2} x+1)

for

{x} \∈ {\ℝ}

. Let

{S}=\{\λ\∈ {\ℝ} \|

there exits a point

{c} \∈ (0,2 \π)

with

f'({c})=\λ{f}({c})\}

Then

S = ℝ
S = {0}
S = [0, 2π]
S is a finite set having more than oneelement

Solution:

Let

g(x)=f(x) e^{-\λx} ; x \∈ [0,2 \π]

so,

g(0)=g(2 \π)=0

(as

f(0)=f(2 \π)=0)

Thus,

\exists {c} \∈ (0,2 \π)

such that

{{g}'}({c})=0

\⇒f'({c})=\λ f({c}) \∀\λ\∈ {\ℝ}

⇒{S}={\ℝ}

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