NIMCET 2024 — Mathematics PYQ

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Question

NIMCET 2024 — Mathematics PYQ

NIMCET | Mathematics | 2024

The value of $m$ for which volume of the parallelopiped is 4 cubic units whose three edges are represented by $a = m i + j + k$ , $b = i - j + k$ , $c = i + 2 j - k$ is

Choose the correct answer:

Explanation

Solution:

The volume of a parallelepiped with adjacent edges $\mathbf{a}, \mathbf{b}, \mathbf{c}$ is given by the magnitude of their scalar triple product:

$V = |[\mathbf{a} \text{ } \mathbf{b} \text{ } \mathbf{c}]| = 4$

The scalar triple product is calculated using the determinant:

\begin{vmatrix} m & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 2 & -1 \end{vmatrix} = \pm 4

Expanding along the first row:

$m[(-1)(-1) - (1)(2)] - 1[(1)(-1) - (1)(1)] + 1[(1)(2) - (-1)(1)] = \pm 4$

$m[1 - 2] - 1[-1 - 1] + 1[2 + 1] = \pm 4$

$-m + 2 + 3 = \pm 4$

$5 - m = \pm 4$

Case 1:

$5 - m = 4$

$m = 1$

Case 2:

$5 - m = -4$

$m = 9$

Comparing with the given options, $m = 1$ is the correct value.

Correct Option: (D)

Choose the correct answer:

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