Q65 / 160 · AP EAMCET 2023 (15 May Shift 2) · Mathematics

In $\triangle \mathrm{ABC}$, if $\sin ^2 \mathrm{~B}=\sin \mathrm{C}$ and $3 \cos ^2 \mathrm{~B}=2 \cos ^2 \mathrm{C}$, then $\triangle \mathrm{ABC}$ is

  • A
    a right angled triangle
  • B
    an isosceles triangle
  • C
    an equilateral triangle
  • Correct answer
    D
    a scalene triangle

Solution

Given $2 \cos ^2 C=3 \cos ^2 B$ $\Rightarrow 2-2 \sin ^2 C=3\left(1-\sin ^2 B\right)$ $\begin{aligned} & \Rightarrow 2-2 \sin ^2 C=3-3 \sin C \\ & \Rightarrow 2 \sin ^2 C-3 \sin C+1=0 \\ & \Rightarrow(2 \sin C-1)(\sin C-1) \\ & \Rightarrow \sin C=\frac{1}{2}, 1 \Rightarrow C=\frac{\pi}{6} \text { or } \frac{\pi}{2} \end{aligned}$ So $\triangle A B C$ is a scalene triangle.

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

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