Question

Assertion :If $A,B$ and $C$ are the angles of a triangle and $\left|\begin{array}{ccc}1& 1& 1\\ 1+\sin A& 1+\sin B& 1+\sin C\\ \sin A+{\sin }^{2}A& \sin B+{\sin }^{2}B& \sin C+{\sin }^{2}C\end{array}\right|=0$, then triangle may not be equilateral Reason: If any two rows of a determinant are the same, then the value of that determinant is zero

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$\left|\begin{array}{ccc}1& 1& 1\\ 1+\sin A& 1+\sin B& 1+\sin C\\ \sin A+{\sin }^{2}A& \sin B+{\sin }^{2}B& \sin C+{\sin }^{2}C\end{array}\right|=0$
$\Rightarrow \left|\begin{array}{ccc}1& 1& 1\\ \sin A& \sin B& \sin C\\ {\sin }^{2}A& {\sin }^{2}B& {\sin }^{2}C\end{array}\right|=0$
if triangle is an equilateral then $\sin A=\sin B=\sin C$
then determinant is zero, as two rows are identical(proportional).
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion