Question

If $\left|\begin{array}{ccc}1& sin\theta & 1\\ -sin\theta & 1& sin\theta \\ -1& -sin\theta & 1\end{array}\right|$ then,

• A. $\Delta =0$
• B. $\Delta \in (0,\infty )$
• C. $\Delta \in [-1,2]$
• D. $\Delta \in [2,4]$

Answer 1

Answer: (D)

$\Delta \in [2,4]$

Let $\Delta =\left|\begin{array}{ccc}1& sin\theta & 1\\ -sin\theta & 1& sin\theta \\ -1& -sin\theta & 1\end{array}\right|$
$=1(1+si{n}^2\theta )-sin\theta \left(0\right)+1(si{n}^2\theta +1)$
$=2(1+si{n}^2\theta )$
$\because 0\le si{n}^2\theta \le 1$
$\Rightarrow 1\le 1+si{n}^2\theta \le 2$
$\Rightarrow 2\le 2(1+si{n}^2\theta )\le 4$
$\Rightarrow 2\le \Delta \le 4$

$\Rightarrow \Delta \in [2,4]$