NIMCET 2008 Question Paper with solutions

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Question : 4 of 120

Marks: +1, -0

The maximum value of

(\cos α_{1})(\cos α_{2}) . . .(\cos α_{n})

where

0 ≤ α_{1}, α_{2}, α_{n} ≤ \;{π}/{2}

and

(\cot α_{1})(\cot α_{2}) . . .(\cot α_{n})=1

is

${1}/{2^{n / 2}}$
${1}/{2^{n}}$
${1}/{2 n}$
1

Solution:

(\cot α_{1})(\cot α_{2}) . . .(\cot α_{n})=1

⇒(\cos α_{1})(\cos α_{2}) . . .(\cos α_{n})

=(\sin α_{1})(\sin α_{2}) . . .(\sin α_{n})

.....(i)
LHS and RHS will he maximum with equal values
it

α_{1}=α_{2}=. . .=α_{n}={π}/{4}

As,

\cos {π}/{4}=\sin \;{π}/{4}

\Rightarrow

Maximum value of

(\cos α_{1})(\cos α_{2}) . . .(\cos α_{n})

=(\;{1}/{√{2}})(\;{1}/{√{2}}) .....({1}/{√{2}}) n \;\text " times "\;={1}/{2^{n{/}2}}

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