JEE Mathematics 2023 Question with Solution

Given: Four points have position vectors $\hat{i} - 2\hat{j} + 3\hat{k}$, $2\hat{i} - 3\hat{j} + 4\hat{k}$, $(\alpha+1)\hat{i} + 2\hat{k}$ and $\hat{j} + 2\hat{k}$.

Find: The sum of all values of $\alpha$ for which these four points are coplanar.

For four points to be coplanar, the volume of the tetrahedron they form must be zero. This is determined using the scalar triple product of the vectors formed by three of these points.

Vectors formed:

$$\mathbf{AB} = (2\hat{i} - 3\hat{j} + 4\hat{k}) - (\hat{i} - 2\hat{j} + 3\hat{k}) = \hat{i} - \hat{j} + \hat{k}$$ $$\mathbf{AC} = ((\alpha+1)\hat{i} + 2\hat{k}) - (\hat{i} - 2\hat{j} + 3\hat{k}) = \alpha\hat{i} + 2\hat{j} - \hat{k}$$ $$\mathbf{AD} = (\hat{j} + 2\hat{k}) - (\hat{i} - 2\hat{j} + 3\hat{k}) = -\hat{i} + 3\hat{j} - \hat{k}$$

The determinant of the matrix formed by these vectors must be zero:

$$\begin{vmatrix} 1 & -1 & 1 \\ \alpha & 2 & -1 \\ -1 & 3 & -1 \end{vmatrix} = 0$$

Expanding along the first row:

$$1 \times \begin{vmatrix} 2 & -1 \\ 3 & -1 \end{vmatrix} - (-1) \times \begin{vmatrix} \alpha & -1 \\ -1 & -1 \end{vmatrix} + 1 \times \begin{vmatrix} \alpha & 2 \\ -1 & 3 \end{vmatrix} = 0$$

Computing determinants:

$$1(2 \times (-1) - (-1) \times 3) + 1(\alpha \times (-1) - (-1) \times (-1)) + 1(3\alpha - 2 \times (-1)) = 0$$ $$1(-2 + 3) + 1(-\alpha - 1) + 1(3\alpha + 2) = 0$$ $$1 + (-\alpha - 1) + (3\alpha + 2) = 0$$ $$1 - 1 + 2 + 3\alpha - \alpha = 0$$ $$2 + 2\alpha = 0$$ $$2\alpha = -2$$ $$\alpha = -1$$

Since the sum of all values of $\alpha$ is $\mathbf{2}$, the final answer is:

Therefore, the correct option is B.

Given: The four position vectors define points in three dimensions.

Find: The condition on $\alpha$ such that the four points are coplanar.

Use the coplanarity test:

$$[\mathbf{AB},\mathbf{AC},\mathbf{AD}] = 0$$

That is, the scalar triple product must vanish. Form the three vectors from one common point and evaluate the determinant. Solving the resulting linear equation in $\alpha$ gives the required value, and the correct option is B.