KVPY SX SB Exam 2012 Question Paper

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Question : 84 of 120

Marks: +1, -0

The sum of all

x∈[0 ,π]

which satisfy the equation

\sin x+1/2\cos x=\sin^2(x+π/4)

is

$π/6$
${5π}/6$
$π$
$2π$

Solution:

\sin x+1/2\cos x=\sin^2(x+π/4)

\sin x+1/2\cos x

=1/2[1-\cos (π/2+2x)]

\sin x+1/2\cos x=1/2(1+\sin 2x)

2\sin x+\cos x=1+\sin x\cos x

2\sin x\cos x-2\sin x

+1-\cos x=0

(1-\cos x)-2\sin x(1-\cos x)=0

(1-\cos x)(1-2\sin x)=0

\table 1-\cos x=0,1-2\sin x=0;\cos x=1,\sin x=1/2;x=0,x=π/6{,}{5π}/6

sum

=0+π/6+{5π}/6=π

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