NIMCET 2011 Question Paper with solutions

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Question : 42 of 120

Marks: +1, -0

If 'a' is a positive integer, then the number of values satisfying

∫↙{0}↖{π / 2}\{a^{2}(\;{\cos 3 x}/{4}+\;{3}/{4} \cos x) +a \sin x-20 \cos x\} d x ≤ \;{-a^{2}}/{3}

is

only one
two
three
four

Solution:

Given,

∫↙{0}↖{π / 2}\{a^{2}( {\cos 3 x}/{4}+ {3}/{4} \cos x) +a \sin x-20 \cos x\} d x ≤- {a^{2}}/{3}

⇒[a^{2}( {\sin 3 x}/{12}+ {3}/{4} \sin x)-a \cos x-20 \sin x]_{0}^{π 2} ≤- {a^{2}}/{3}

⇒[a^{2}(- {1}/{12}+\;{3}/{4}-0-0)-a(0-1)-20(1-0)] ≤- {a^{2}}/{3}

⇒ {8 a^{2}}/{12}+a-20 ≤- {a^{2}}/{3}

⇒ a^{2}+a-20 ≤ 0

⇒ -5 ≤ a ≤ 4

Since,

a

is a positive integer, hence four values of a

(i, θ, 1,2,3,4)

exist.

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