JEE Main 2024 — 9 April, Shift 1 — Question 10

Area of parallelogram having sides $\overrightarrow{\mathrm{OA}} \& \overrightarrow{\mathrm{OC}}=|\overrightarrow{\mathrm{OA}} \times \overrightarrow{\mathrm{OC}}|=|2 \overrightarrow{\mathrm{a}} \times 3 \overrightarrow{\mathrm{~b}}|=15$

$6|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|=15$

$\Rightarrow|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|=\frac{5}{2}$

Area of quadrilateral $\mathrm{OABC}=\frac{1}{2}\left|\overrightarrow{\mathrm{~d}}_{1} \times \overrightarrow{\mathrm{d}}_{2}\right|$

$=\frac{1}{2}|\overrightarrow{\mathrm{AC}} \times \overrightarrow{\mathrm{OB}}|=\frac{1}{2}|(3 \overrightarrow{\mathrm{~b}}-2 \overrightarrow{\mathrm{a}}) \times(6 \overrightarrow{\mathrm{a}}+5 \overrightarrow{\mathrm{~b}})|$

$=\frac{1}{2}|18 \overrightarrow{\mathrm{~b}} \times \overrightarrow{\mathrm{a}}-10 \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|=14|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|$

$=14 \times \frac{5}{2}=35$