JEE Mathematics 2023 Question with Solution

Given: $\mathbf{c} = (2\mathbf{a} \times \mathbf{b}) - 3\mathbf{b}$, $|\mathbf{b}| = 4$.

Find: $\mathbf{b} \cdot \mathbf{c}$.

Using the given expression,

$$\mathbf{b} \cdot \mathbf{c} = \mathbf{b} \cdot \left((2\mathbf{a} \times \mathbf{b}) - 3\mathbf{b}\right)$$

Expanding the dot product,

$$\mathbf{b} \cdot \mathbf{c} = \mathbf{b} \cdot (2\mathbf{a} \times \mathbf{b}) - 3(\mathbf{b} \cdot \mathbf{b})$$

Now $\mathbf{b} \cdot (\mathbf{a} \times \mathbf{b}) = 0$ because $\mathbf{a} \times \mathbf{b}$ is perpendicular to $\mathbf{b}$. Also,

$$\mathbf{b} \cdot \mathbf{b} = |\mathbf{b}|^2 = 16$$

Therefore,

$$\mathbf{b} \cdot \mathbf{c} = -3 \times 16 = -48$$

Thus, the value of $\mathbf{b} \cdot \mathbf{c}$ is $-48$. The correct option is B.

Given: $\mathbf{c} = 2(\mathbf{a} \times \mathbf{b}) - 3\mathbf{b}$.

Find: $\mathbf{b} \cdot \mathbf{c}$.

Notice that the term $\mathbf{a} \times \mathbf{b}$ is automatically perpendicular to both $\mathbf{a}$ and $\mathbf{b}$. So when dotted with $\mathbf{b}$, that entire part vanishes.

$$\mathbf{b} \cdot \mathbf{c} = \mathbf{b} \cdot \left(2(\mathbf{a} \times \mathbf{b}) - 3\mathbf{b}\right) = 0 - 3|\mathbf{b}|^2$$

Since $|\mathbf{b}| = 4$,

$$\mathbf{b} \cdot \mathbf{c} = -3(16) = -48$$

Therefore, the correct option is B.