KVPY SX SB Exam 2013 Question Paper

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Question : 8 of 120

Marks: +1, -0

The angles

α, β, γ

of a triangle satisfy the equations

2\sinα + 3\cosβ = 3√2

and

3\sinβ + 2\cosα = 1

. Then angle

γ

equals

150°
120°
60°
30°

Solution:

2\sinα + 3\cos\β = 3√2

....(1)

3\sin\β + 2\cosα = 1

....(2)
sum of squares of equation (1) and (2)

4 + 9 + 12 \sin (\α + \β) = 19

\sin(\α+\β)=1/2

⇒\α+\β=150°\or30°

If

α + \β = 30°

⇒ \β= 30 - \α

put in equation (1) and (2)
we get

7\sin\α+3√3\cos\α=6√2

7\cos\α-3√3\sin α=2

\cosα={7+9√6}/37 =0.8 < √3/2

\cosα < \cos 30°

∵\α > 30°

∴\α+\β≠30°

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