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Question

Let A be the area bounded by the curve y=(tanx)n for n>2 and the lines x=0 and y=0 and x=π4 then the range of A10 is

A
(120,116)
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B
(118,116)
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C
(114,110)
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D
(122,118)
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Solution

The correct option is D (122,118)
y=(tanx)n

0<tanx<1

0<x<π4

0<(tanx)n+1<(tanx)n

An+1<An

Now, for n=2

An+An+2=π40((tanx)n+(tanx)n+2)dx

An+An+2=π40(tanx)n(1+tan2x)dx

An+An+2=π40(tanx)nsec2xdx

An+An+2=[(tanx)n+1n+1]π40

An+An+2=1n+1

Now,

An+2<An+1<An

An+2+An<2An

1n+1<2An

An>12n+2

Also, by symmetry, we can write

An+An<An+An2

2An<1n1

An<12n2

Therefore, 12n+2<An<12n2

Substitute n=10

122<An<118

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