JEE Mathematics 2023 Question with Solution

Given: $\vec{a} = 2\hat{i} + 7\hat{j} - \hat{k}$, $\vec{b} = 3\hat{i} + 5\hat{k}$, $\vec{c} = \hat{i} - \hat{j} + 2\hat{k}$. Vector $\vec{d}$ is perpendicular to both $\vec{a}$ and $\vec{b}$, and $\vec{c} \cdot \vec{d} = 12$.

Find: $\left( \hat{i} + \hat{j} - \hat{k} \right) \cdot \left( \vec{c} \times \vec{d} \right)$.

Since $\vec{d}$ is perpendicular to both $\vec{a}$ and $\vec{b}$, take

$$\vec{d} = \lambda (\vec{a} \times \vec{b})$$

Now compute

$$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 7 & -1 \\ 3 & 0 & 5 \end{vmatrix}$$ $$= \hat{i}(7 \times 5 - (-1) \times 0) - \hat{j}(2 \times 5 - (-1) \times 3) + \hat{k}(2 \times 0 - 7 \times 3)$$ $$= 35\hat{i} - 13\hat{j} - 21\hat{k}$$

Hence,

$$\vec{d} = \lambda (35\hat{i} - 13\hat{j} - 21\hat{k})$$

Using $\vec{c} \cdot \vec{d} = 12$,

$$(\hat{i} - \hat{j} + 2\hat{k}) \cdot \lambda (35\hat{i} - 13\hat{j} - 21\hat{k}) = 12$$ $$\lambda (35 + 13 - 42) = 12$$ $$6\lambda = 12$$ $$\lambda = 2$$

Therefore,

$$\vec{d} = 70\hat{i} - 26\hat{j} - 42\hat{k}$$

Now compute

$$\vec{c} \times \vec{d} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -1 & 2 \\ 70 & -26 & -42 \end{vmatrix}$$ $$= \hat{i}((-1)(-42) - 2(-26)) - \hat{j}(1(-42) - 2(70)) + \hat{k}(1(-26) - (-1)(70))$$ $$= 94\hat{i} + 182\hat{j} + 44\hat{k}$$

Then,

$$(\hat{i} + \hat{j} - \hat{k}) \cdot (\vec{c} \times \vec{d}) = 94 + 182 - 44 = 232$$

This computed value is $232$, which does not match any option. The provided solution concludes with $44$, and the provided correct answer also marks option D. Therefore, there is a discrepancy in the provided working, but the accepted answer on the page is D.

Given: The page solution states The Correct Option is C, but the answer key marks option D = $44$.

Check the working:

1. $\vec{a} \times \vec{b} = 35\hat{i} - 13\hat{j} - 21\hat{k}$ is correct.
2. From $\vec{c} \cdot \vec{d} = 12$, we get $\lambda = 2$, so $\vec{d} = 70\hat{i} - 26\hat{j} - 42\hat{k}$.
3. Then $\vec{c} \times \vec{d} = 94\hat{i} + 182\hat{j} + 44\hat{k}$.
4. Hence

$$(\hat{i} + \hat{j} - \hat{k}) \cdot (\vec{c} \times \vec{d}) = 94 + 182 - 44 = 232$$

So the extracted working supports neither C = $48$ nor D = $44$.

Because the listed options do not contain the computed value and the answer key explicitly gives (4) $44$, the final recorded answer is D while noting the inconsistency.