NIMCET 2013 — Mathematics PYQ

Concept:

For three vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ to be coplanar, their scalar triple product must be zero. This is equivalent to saying that the determinant formed by their components is equal to zero:

Step 1: Identify the components of the given vectors.

• $\vec{a}=2\hat{i}-\hat{j}+\hat{k}⟹(2,-1,1)$

• $\vec{b}=\hat{i}+2\hat{j}-3\hat{k}⟹(1,2,-3)$

• $\vec{c}=3\hat{i}+\lambda \hat{j}+5\hat{k}⟹(3,\lambda ,5)$

Step 2: Set up the determinant and solve for $\lambda$.

Step 3: Expand the determinant along the first row.

Step 4: Simplify the equation.

Final Answer:

The value of $\lambda$ is $-4$, which corresponds to Option 4.