NIMCET 2025 — Mathematics PYQ

To find the value of $\lambda$, we use the geometric property of the scalar triple product.

Step 1: Formula for the Volume of a Parallelepiped

The volume $V$ of a parallelepiped formed by three co-terminal vector edges $\vec{a}$, $\vec{b}$, and $\vec{c}$ is given by the absolute value of their scalar triple product:

$$V=|[\vec{a}\vec{b}\vec{c}]|$$

The scalar triple product $[\vec{a}\vec{b}\vec{c}]$ is evaluated using the determinant of the matrix formed by the components of the vectors:

$$[\vec{a}\vec{b}\vec{c}]=\left|\begin{array}{ccc}2& amp;-3& amp;4\\ 1& amp;2& amp;-1\\ 3& amp;1& amp;\lambda \end{array}\right|$$

Step 2: Evaluate the Determinant

Expanding the determinant along the first row:

$$[\vec{a}\vec{b}\vec{c}]=2\left|\begin{array}{cc}2& amp;-1\\ 1& amp;\lambda \end{array}\right|-(-3)\left|\begin{array}{cc}1& amp;-1\\ 3& amp;\lambda \end{array}\right|+4\left|\begin{array}{cc}1& amp;2\\ 3& amp;1\end{array}\right|$$

Calculating each $2\times 2$ determinant:

$$=2(2\lambda -(-1))+3(\lambda -(-3))+4(1-6)$$

$$=2(2\lambda +1)+3(\lambda +3)+4(-5)$$

Simplifying the terms:

$$=4\lambda +2+3\lambda +9-20$$

$$=7\lambda -9$$

Step 3: Solve for $\lambda$

Since the volume of the parallelepiped is given as $5$ units:

$$|7\lambda -9|=5$$

This absolute value equation gives two cases:

• Case 1:

  $$7\lambda -9=5$$

  $$7\lambda =14$$

  $$\lambda =2$$

• Case 2:

  $$7\lambda -9=-5$$

  $$7\lambda =4$$

  $$\lambda =\frac{4}{7}$$

Correct Answer

The values of $\lambda$ are Option A:

$$\lambda =2\text{or}\lambda =\frac{4}{7}$$