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The maximum value of the expression 1/(s...

The maximum value of the expression `1/(sin^2theta+3sinthetacostheta+5cos^2theta)`

Text Solution

Verified by Experts

The correct Answer is:
2
Let ` f ( theta ) = ( 1)/( sin ^(2) theta + 3 sin theta cos theta + 5 cos ^(2) theta ) `
Again let, ` g ( theta ) = sin ^(2) theta + 3 sin theta cos theta + 5 cos ^(2) theta `
` " " = ( 1 - cos 2 theta )/( 2 ) + 5(( 1 + cos 2 theta )/(2)) + (3)/(2) sin 2 theta `
`" " = 3 + 2 cos 2 theta + (3)/(2) sin 2 theta `
`therefore " " g ( theta ) _(min) = 3 - sqrt ( 4 + ( 9)/( 4)) `
`" " = 3 - ( 5)/(2) = (1)/(2)`
` therefore ` Maximum value of ` f ( theta ) = (1)/( 1//2) = 2 `
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