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  • State True or False for the statements The maximum value of 1 1 1 1plussintheta 1 1 1 1pluscostheta is 1/2.

State True or False for the statements

The maximum value of  \left|\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1+\sin \theta & 1 \\ 1 & 1 & 1+\cos \theta \end{array}\right| is 1/2.

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\\ \begin{array}{l} \Delta=\left|\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1+\sin \theta & 1 \\ 1 & 1 & 1+\cos \theta \end{array}\right| \\ \text { Apply, } \mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1} \text { and } \mathrm{R}_{3} \rightarrow \mathrm{R}_{3}-\mathrm{R}_{1} \\ \Rightarrow \Delta=\left|\begin{array}{ccc} 1 & 1 & 1 \\ 0 & \sin \theta & 0 \\ 0 & 0 & \cos \theta \end{array}\right| \end{array}

\\ \vspace{\baselineskip}= cos $ \theta $ . sin $ \theta $ \\ \\ \vspace{\baselineskip}\text{Multiply and divide by 2},\\ \vspace{\baselineskip} = 1/2 (2sin $ \theta $ cos $ \theta $ )\\ \vspace{\baselineskip} \text{We know}, 2 sin $ \theta $ cos $ \theta $ = sin 2$ \theta $ \\ \\ \vspace{\baselineskip}= 1/2 (sin 2$ \theta $ )\\ \\

Since the maximum value of sin 2$ \theta $ is 1, $ \theta $ = 45$ ^{\circ} $ .\\


\\ \\ \vspace{\baselineskip}$ \therefore $ $ \Delta $ = 1/2 (sin 2(45$ ^{\circ} $ ))\\ \\ \vspace{\baselineskip}= 1/2 sin 90$ ^{\circ} $ \\ \\ \vspace{\baselineskip}= 1/2 (1)\\ \\ \vspace{\baselineskip}$ \therefore $ $ \Delta $ = 1/2\\ \vspace{\baselineskip}

Thus the given statement is true.

Posted by

infoexpert22

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