NIMCET 2015 — Mathematics PYQ

The scalar triple product $[\vec{a}\vec{b}\vec{c}]$ is calculated using the determinant of the components of the three vectors.

Given vectors:

• $\vec{a}=1\hat{i}+0\hat{j}-1\hat{k}$

• $\vec{b}=x\hat{i}+1\hat{j}+(1-x)\hat{k}$

• $\vec{c}=y\hat{i}+x\hat{j}+(1+x-y)\hat{k}$

2. Setting up the Determinant

3. Expanding the Determinant

We expand along the first row:

Now, solve the $2\times 2$ determinants:

4. Simplifying the Expression

Combine the like terms:

• $x-x=0$

• $-y+y=0$

• $x^2-x^2=0$

Conclusion

The value of the scalar triple product is a constant ($1$). Since the final result does not contain $x$ or $y$, the value depends on neither $x$ nor $y$.

Correct Option: (a)