A) \[\frac{5}{2}\sqrt{3}sq\text{ }unit\]
B) \[10\sqrt{3}sq\text{ }unit\]
C) \[5\sqrt{\frac{3}{2}}sq\text{ }unit\]
D) \[\frac{3}{2}sq\text{ }unit\]
E) \[\frac{7}{2}sq\text{ }unit\]
Correct Answer: A
Solution :
Since P, Q, R, S are in cyclic order \[\Rightarrow \]\[\overset{\to }{\mathop{PR}}\,\]and\[\overset{\to }{\mathop{QS}}\,\]are in diagonals. \[\therefore \] Area of quadrilateral \[=\frac{1}{2}|\overset{\to }{\mathop{PR}}\,\times \overset{\to }{\mathop{QS}}\,|\] \[=\frac{1}{2}\left| \begin{matrix} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ 2 & -1 & 1 \\ -1 & 3 & 2 \\ \end{matrix} \right|\] \[=\frac{1}{2}|\hat{i}(5)-\hat{j}(5)+\hat{k}(5)|\] \[=\frac{1}{2}\sqrt{{{5}^{2}}+{{5}^{2}}+{{5}^{2}}}=\frac{5}{2}\sqrt{3}\]sq unit
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