If $\Delta=\left|\begin{array}{ccc}1 & \cos \theta & 1 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos…

If $\Delta=\left|\begin{array}{ccc}1 & \cos \theta & 1 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1\end{array}\right|$, then $\Delta$ lies in the interval
  1. [2, 4]
  2. (2, 4)
  3. [1, 4]
  4. [– 1, 1]

Solution

Given that, $ \begin{aligned} \Delta & =\left|\begin{array}{ccc} 1 & \cos \theta & 1 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1 \end{array}\right| \\ \Delta & \left.=\left|\begin{array}{ccc} 1 & \cos \theta & 1 \\ -\cos \theta & 1 & \cos \theta \\ 0 & 0 & 2 \end{array}\right| \quad \text { [Applying } R_3 \rightarrow R_3+R_1\right] \end{aligned} $ Now, on expanding along $R_3$, we get $ \Delta=2\left(1+\cos ^2 \theta\right) $ As we know, $ \begin{aligned} & 0 \leq \cos ^2 \theta \leq 1 \\ & 1 \leq 1+\cos ^2 \theta \leq 2 \\ & \Rightarrow \quad 2 \leq 2\left(1+\cos ^2 \theta\right) \leq 4 \end{aligned} $ Thus, $\Delta$ lies in the interval $[2,4]$

Asked in: AP EAMCET 2017 (26 Apr Shift 1)

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