Question:
If the volume of the tetrahedron whose coterminous edges are given by the vectors \(\overset{̄}{a} = -2\hat{i}+3\hat{j}-3\hat{k}\), \(\overset{̄}{b} = 4\hat{i}+5\hat{j}+(λ-10)\hat{k}\), \(\overset{̄}{c} = 6\hat{i}+2\hat{j}-3\hat{k}\) is 11 cubic units, then the sum of the possible values of \(λ\) is \(\ldots\)
If the volume of the tetrahedron whose coterminous edges are given by the vectors \(\overset{̄}{a} = -2\hat{i}+3\hat{j}-3\hat{k}\), \(\overset{̄}{b} = 4\hat{i}+5\hat{j}+(λ-10)\hat{k}\), \(\overset{̄}{c} = 6\hat{i}+2\hat{j}-3\hat{k}\) is 11 cubic units, then the sum of the possible values of \(λ\) is \(\ldots\)
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Volume = |scalar triple product| / 6. Evaluate the determinant in terms of lambda.
Updated On: Oct 1, 2026
- \(7\)
- \(8\)
- \(1\)
- \(6\)
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept:
The volume of a tetrahedron with coterminous edges \(\bar a, \bar b, \bar c\) is \(\dfrac16\left|[\bar a\,\bar b\,\bar c]\right|\).
Step 2: Key Formula or Approach:
\[ [\bar a\,\bar b\,\bar c] = \begin{vmatrix}-2 & 3 & -3\\ 4 & 5 & \lambda-10\\ 6 & 2 & -3\end{vmatrix} \]
Step 3: Detailed Explanation:
Expand along the first row:
\[ -2\left[5(-3) - 2(\lambda-10)\right] - 3\left[4(-3) - 6(\lambda-10)\right] + (-3)\left[4\cdot2 - 5\cdot6\right] \]
\[ = -2(5 - 2\lambda) - 3(48 - 6\lambda) - 3(-22) \]
\[ = -10 + 4\lambda - 144 + 18\lambda + 66 = 22\lambda - 88 \]
The volume is 11, so \(\left|22\lambda - 88\right| = 66\).
\[ 22\lambda - 88 = 66 \Rightarrow \lambda = 7 \]
\[ 22\lambda - 88 = -66 \Rightarrow \lambda = 1 \]
Sum of the possible values: \(7 + 1 = 8\).
Option (A) 7 and (C) 1 are single roots, and (D) 6 does not come from either case.
Final Answer:
The sum of the possible values of \(\lambda\) is 8, option (B). \[ \boxed{8 \text{ (B)}} \]
The volume of a tetrahedron with coterminous edges \(\bar a, \bar b, \bar c\) is \(\dfrac16\left|[\bar a\,\bar b\,\bar c]\right|\).
Step 2: Key Formula or Approach:
\[ [\bar a\,\bar b\,\bar c] = \begin{vmatrix}-2 & 3 & -3\\ 4 & 5 & \lambda-10\\ 6 & 2 & -3\end{vmatrix} \]
Step 3: Detailed Explanation:
Expand along the first row:
\[ -2\left[5(-3) - 2(\lambda-10)\right] - 3\left[4(-3) - 6(\lambda-10)\right] + (-3)\left[4\cdot2 - 5\cdot6\right] \]
\[ = -2(5 - 2\lambda) - 3(48 - 6\lambda) - 3(-22) \]
\[ = -10 + 4\lambda - 144 + 18\lambda + 66 = 22\lambda - 88 \]
The volume is 11, so \(\left|22\lambda - 88\right| = 66\).
\[ 22\lambda - 88 = 66 \Rightarrow \lambda = 7 \]
\[ 22\lambda - 88 = -66 \Rightarrow \lambda = 1 \]
Sum of the possible values: \(7 + 1 = 8\).
Option (A) 7 and (C) 1 are single roots, and (D) 6 does not come from either case.
Final Answer:
The sum of the possible values of \(\lambda\) is 8, option (B). \[ \boxed{8 \text{ (B)}} \]
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