The difference between maximum and ...
Question

# The difference between maximum and minimum value of the expression ( y=1+2 sin x+3 cos ^{2} x ) is(A) ( frac{16}{3} )(B) ( frac{13}{3} )(C) 7(D) 8

JEE/Engineering Exams
Maths
Solution
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( begin{aligned} y &=1+2 sin x+3 cos ^{2} x=1+2 sin x+3-3 sin ^{2} x &=4+2 sin x-3 sin ^{2} x &=-left(3 sin ^{2} x-2 sin xright)+4 &=-left(3 sin ^{2} x-2 sin x+frac{1}{3}right)+4+frac{1}{3} &=-left(sqrt{3} sin x frac{1}{sqrt{3}}right)^{2}+frac{13}{3} &=frac{-(3 sin x-1)^{2}+13}{3} &-1 leqslant sin x leqslant 1 Rightarrow-3 leqslant 3 sin x leqslant 3 -4 & leqslant 3 sin x-1 leq 2 & 0 leqslant(3 sin x-1)^{2} leqslant 16 -16 & leqslant-(3 sin x-1)^{2} leqslant 0 -3 & leqslant-(3 sin x-1)^{2}+13 leqslant 13 -1 & leqslant frac{-(3 sin x-1)^{2}+13}{3} leqslant frac{13}{3} &=frac{13}{3}-(-1)=frac{16}{3} & text { difion }(A) end{aligned} )

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