MHTCET 18 Apr 2026 Shift 1 Paper

Concept:The scalar triple product $[\bar{a}\ \bar{b}\ \bar{c}]$ is the determinant of the matrix formed by the components of the three vectors. To find dependence, we simplify this determinant and see whether $x$ or $y$ remain in the final value.Explanation:Write each vector in component form.$\bar{a} = \hat{i} - \hat{k}$, so $\bar{a} = (1, 0, -1)$.$\bar{b} = x\hat{i} + \hat{j} + (1-x)\hat{k}$, so $\bar{b} = (x, 1, 1-x)$.$\bar{c} = y\hat{i} + x\hat{j} + (1+x-y)\hat{k}$, so $\bar{c} = (y, x, 1+x-y)$.Form the determinant:$$[\bar{a}\ \bar{b}\ \bar{c}] = \begin{vmatrix} 1 & 0 & -1 \\ x & 1 & 1-x \\ y & x & 1+x-y \end{vmatrix}$$Expand along the first row:$$[\bar{a}\\ \bar{b}\\ \bar{c}] = 1\begin{vmatrix} 1 & 1-x \\ x & 1+x-y \end{vmatrix} + (-1)\begin{vmatrix} x & 1 \\ y & x \end{vmatrix}$$Simplify the minors:$$= \left[(1)(1+x-y) - x(1-x)\right] - \left[x^2 - y\right]$$$$= (1+x-y-x+x^2) - x^2 + y$$$$= 1$$The value is always $1$, a constant independent of both $x$ and $y$.Answer:The scalar triple product depends on neither $x$ nor $y$.Therefore, the correct option is B. neither x nor y.