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Question #73941

Difficulty: 8/10 | 🟢 Free

The Question

Consider three lines in $\mathbb{R}^3$ given by: $L_1: \vec{r} = 119\hat{i} + t(\hat{j} + \hat{k})$ $L_2: \vec{r} = 239\hat{j} + t\hat{k}$ $L_3: \vec{r} = \hat{k} + t\hat{i}$ Let $A, B,$ and $C$ be points on $L_1, L_2,$ and $L_3$ respectively, corresponding to the same parametric value $t \in \mathbb{R}$ . Let $O$ be the origin. If the scalar triple product $S(t) = [\vec{OA}, \vec{OB}, \vec{OC}]$ attains a local minimum at $t = t_0 > 0$ , find the value of $3t_0 - 239$ .

Options

A.
239
B.
119
C.
478
D.
-239

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Scalar Triple Product Volume Logic|Parametric form pitfalls|Family of Lines|Triangular/Diagonal Matrix Shortcut

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the student's ability to correctly formulate and optimize the volume of a tetrahedron whose vertices lie on a parametrically defined family of lines.

The primary trap is the 'Triangular Matrix Shortcut': at $t=0$ , the matrix of vertices is upper triangular, and students often mistakenly assume that the parametric variable $t$ acts as a static coordinate offset, leading them to evaluate the determinant only at $t=0$ (yielding a constant volume of 28441).

By making the scalar triple product a cubic polynomial $S(t) = t^3 - 239t^2 + 28441$ , the problem forces students to expand the full determinant and apply calculus to find the true local minimum.

The use of prime numbers 119 and 239 acts as a secondary anti-guessing trap, ensuring that students who attempt to bypass the algebraic expansion cannot easily guess the roots.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Scalar Triple Product Volume Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free