Q120 / 180 · KCET 2021 · Mathematics

If the area of the parallelogram with $\mathbf{a}$ and $\mathbf{b}$ as two adjacent sides is 15 sq units, then the area of the parallelogram having $3 \mathbf{a}+2 \mathbf{b}$ and $\mathbf{a}+3 \mathbf{b}$ as two adjacent sides in sq units is

  • A
    45
  • B
    75
  • Correct answer
    C
    105
  • D
    120

Solution

Area of parallelogram having $\mathbf{a}$ and $\mathbf{b}$ as its adjacent sides is 15 sq units. $\Rightarrow \quad|\mathbf{a} \times \mathbf{b}|=15$ Area of parallelogram having $(3 \mathbf{a}+2 \mathbf{b})$ and $(\mathbf{a}+3 \mathbf{b})$ as two adjacent side $=|(3 \mathbf{a}+2 \mathbf{b}) \times(\mathbf{a}+3 \mathbf{b})|$ $=|3 \mathbf{a} \times \mathbf{a}+2 \mathbf{b} \times \mathbf{a}+9 \mathbf{a} \times \mathbf{b}+6 \mathbf{b} \times \mathbf{b}|$ $=|2 \mathbf{b} \times \mathbf{a}+9 \mathbf{a} \times \mathbf{b}|$ $[\therefore \mathbf{a} \times \mathbf{a}=\mathbf{b} \times \mathbf{b}=0]$ $=7|\mathbf{a} \times \mathbf{b}|$ $[\therefore \mathbf{b} \times \mathbf{a}=-\mathbf{a} \times \mathbf{b}]$ $=7 \times 15$ $=105$

Hence, the correct answer is option: 'C'.

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