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DETERMINANTS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS1 Mark
Question
Let $\text{A}=\begin{vmatrix}1&\sin\theta&1\\-\sin\theta&1&\sin\theta\\-1&-\sin\theta&1\end{vmatrix},$ where $0\leq\theta\leq2\pi.$ Then:
  1. $\text{Det (A)}=0$
  2. $\text{Det (A)}\in(2,\infty)$
  3. $\text{Det (A)}\in(2,4)$
  4. $\text{Det (A)}\in[2,4]$
✓

Answer

  1. $\text{Det (A)}\in[2,4]$

Solution:

$\begin{vmatrix}1&\sin\theta&1\\-\sin\theta&1&\sin\theta\\-1&-\sin\theta&1\end{vmatrix}$

$=\begin{vmatrix}1&\sin\theta&2\\-\sin\theta&1&\sin\theta\\-1&-\sin\theta&1\end{vmatrix}$ [Applying C3 → C3 + C1]

$=2\times\begin{vmatrix}-\sin\theta&1\\-1&-\sin\theta\end{vmatrix}$ [Expanding along C3]

$=2(\sin^2\theta+1)$

Given, $0\leq\theta\leq2\pi$

$-1\leq\sin\theta\leq1$

$0\leq\sin^2\theta\leq1$

$|\text{A}|=2(\sin^2\theta+1)$

$|\text{A}|=2\times1=2$ $[\theta=0]$

$|\text{A}|=2\times2=4$ $[\theta=2\pi]$

$\text{Det (A)}\in[2,4]$

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