Step 1: Elimination with Alcoholic KOH \begin{itemize} \item Tert-butyl bromide \((\text{CH}_3)_3\text{CBr}\) undergoes an \(E2\) elimination with alcoholic KOH to form 2-methylpropene \(\big(\text{CH}_2=C(\text{CH}_3)_2\big)\). \end{itemize}
Step 2: Addition of HBr to the Alkene \begin{itemize} \item 2-Methylpropene reacts with HBr via Markovnikov addition, giving back the tertiary bromide, \((\text{CH}_3)_3\text{CBr}\). \item The reaction pathway results in the same tertiary haloalkane as the starting material. \end{itemize} Hence, the major product \((P)\) is \(\text{tert-butyl bromide}\), \((\text{CH}_3)_3\text{CBr}\).
If \( \sqrt{5} - i\sqrt{15} = r(\cos\theta + i\sin\theta), -\pi < \theta < \pi, \) then
\[ r^2(\sec\theta + 3\csc^2\theta) = \]
The system of simultaneous linear equations :
\[ \begin{array}{rcl} x - 2y + 3z &=& 4 \\ 2x + 3y + z &=& 6 \\ 3x + y - 2z &=& 7 \end{array} \]
Calculate the determinant of the matrix: