If $[\bar{a} \bar{b} \bar{c}]=3$, then the volume of the parallelopiped with $2 \bar{a}+\bar{b}, 2…

If $[\bar{a} \bar{b} \bar{c}]=3$, then the volume of the parallelopiped with $2 \bar{a}+\bar{b}, 2 \bar{b}+\bar{c}, 2 \bar{c}+\bar{a}$ as coterminus edges is
  1. 22 cubic units
  2. 15 cubic units
  3. 27 cubic units
  4. 25 cubic units

Solution

Volume of parallelepiped $\begin{array}{l} =(2 \bar{a}+\bar{b}) \cdot[(2 \bar{b}+\bar{c}) \times(2 \bar{c}+\bar{a})] \\ =(2 \bar{a}+\bar{b}) \cdot[(4 \bar{b} \times \bar{c})+(2 \bar{b} \times \bar{a})+(2 \bar{c} \times \bar{c})+(\bar{c} \times \bar{a})] \\ =[8 \bar{a} \cdot(\bar{b} \times \bar{c})]+[4 \bar{a} \cdot(\bar{b} \times \bar{a})]+[2 \bar{a} \cdot(\bar{c} \times \bar{a})]+[4 \bar{b} \cdot(\bar{b} \times \bar{c})]+[2 \bar{b} \cdot(\bar{b} \times \bar{a})]+[\bar{b} \cdot(\bar{c} \times \bar{a})] \\ =[8 \bar{a} \cdot(\bar{b} \times \bar{c})]+0+[\bar{b} \cdot(\bar{c} \times \bar{a})] \\ =8[\bar{a} \cdot(\bar{b} \times \bar{c})]+[\bar{a} \cdot(\bar{b} \times \bar{c})]=9 \bar{a} \cdot(\bar{b} \times \bar{c}) \\ =9[\bar{a} \quad \bar{b} \quad \bar{c}]=9(3)=27 \end{array}$

Asked in: MHT CET 2020 (14 Oct Shift 2)

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