Q39 / 90 · KVPY 2011 (SB/SX) · Mathematics

In triangle $A B C$, we are given that $3 \sin A+4 \cos B=6$ and $4 \sin B+3 \cos A=1$. Then the measure of the angle $C$ is -

Correct answer
A
$30^{\circ}$
B
$150^{\circ}$
C
$60^{\circ}$
D
$75^{\circ}$

Solution

Square \& add both equations $\begin{array}{r} 9+16+24 \sin (\mathrm{A}+\mathrm{B})=37 \\ \sin (\mathrm{A}+\mathrm{B})=\frac{1}{2} \Rightarrow \mathrm{A}+\mathrm{B}=\frac{\pi}{6} \Rightarrow C=\frac{5 \pi}{6} \text { (wrong) } \\ \Rightarrow A+B=\frac{5 \pi}{6} \Rightarrow C=\frac{\pi}{6} \\ \text { because } \mathrm{C}=\frac{5 \pi}{6} \end{array}$ does not follow equation $3 \sin A+4 \cos B=6$

Hence, the correct answer is option: 'A'.

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