KVPY 2019 SB SX Exam Previous Papers

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Question : 86 of 120

Marks: +1, -0

Let

J = ∫↖{1}↙{0}x/{1+x^8}dx

. Consider the following assertions :
I.

J >1/4

II.

J >π/8

Then

only I is true
only II is true
both I and III are true
neither I nor II is true

Solution:

x^ 8 < x^4 \as x ∈ (0, 1)

⇒

x/{1+x^8}>x/{1+x^4}

⇒

J > ∫↖{1}↙{0}{xdx}/{1+x^4}

⇒

J > 1/2∫↖{1}↙{0}{2xdx}/ {1+x^4}=1/2∫↖{1}↙{0}{ dt}/ {1+t^2}

={\tan^{-1}1}/2=π/8

⇒

J=π/8

Also

x^8 < 1 ⇒1 + x^8 < 2

⇒

x/{1+x^8}>x/2⇒J=∫↖{1}↙{0}x/2dx=1/4

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