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A. The volume of the parallelopiped determined by vectors $\to a, \to b$ and $\to c$ is 2. Then, the volume of the parallelopiped determined by vectors $2 (\to a \times \to b), 3 (\to b \times \to c)$ and $(\to c \times \to a)$ is 1. $100$
B. The volume of the parallelopiped determined by vectors $\to a, \to b$ and $\to c$ is 5. Then, the volume of the parallelopiped determined by vectors $3 (\to a + \to b), (\to b + \to c)$ and $2 (\to c + \to a)$ is 2. $30$
C. The area of a triangle with adjacent sides determined by vectors $\to a$ and $\to b$ is 20. Then, the area of the triangle with adjacent sides determined by vectors $(2 \to a + 3 \to b)$ and $(\to a - \to b)$ is 3. $24$
D. The area of a parallelogram with adjacent sides determined by vectors $\to a$ and $\to b$ is $30$ . Then, the area of the parallelogram with adjacent sides determined by vectors $(\to a + \to b)$ and $\to a$ is 4. $60$

A

A - 3, B - 4, C - 2, D - 1

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B

A - 3, B - 4, C - 1, D - 2

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C

A - 3, B - 1, C - 4, D - 2

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D

A - 1, B - 4, C - 2, D - 3

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Solution

The correct option is B $A - 3, B - 4, C - 1, D - 2$
$A \to [\to a \to b \to c] = 2$
$[2 (\to a \times \to b) 3 (\to b \times \to c) \to c \times \to a] = 6 [\to a \to b \to c]^{2} = 6 \times 4 = 24$
$B \to [\to a \to b \to c] = 5$
$6 [\to a + \to b \to b + \to c \to c + \to a] = 12 [\to a \to b \to c] = 60$
$C \to \frac{1}{2} | \to a \times \to b | = 20$
$\frac{1}{2} | (2 \to a + 3 \to b) \times (\to a - \to b) |$
$\frac{1}{2} | - 2 (\to a \times \to b) - 3 (\to a \times \to b) |$
$\frac{5}{2} \times 40 = 100$
$D \to | \to a \times \to b | = 30$
$\Rightarrow | (\to a + \to b) \times \to a | = | \to b \times \to a | = 30$

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