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Question #74453

Difficulty: 8/10 | 🟢 Free

The Question

Let $\\vec{u}(x), \\vec{v}(x),$ and $\\vec{w}(x)$ be three variable vectors such that their scalar triple product is given by:\n $[\\vec{u}(x), \\vec{v}(x), \\vec{w}(x)] = \\frac{1}{14} \\begin{vmatrix} 1 & 1 & 1 \\\\ x & 13 & 27 \\\\ x^2 & 169 & 729 \\end{vmatrix} - 24 \\text{sgn}(x - 20)$ \nwhere $\\text{sgn}(\\cdot)$ denotes the signum function. An objective function representing the accumulated volume is defined as $f(x) = \\int_{0}^{x} [\\vec{u}(t), \\vec{v}(t), \\vec{w}(t)] \\, dt$ . If $x_1$ and $x_2$ are the points of local maximum and local minimum of $f(x)$ respectively in the interval $(0, 40)$ , find the value of $x_1^2 + (x_2 - 20)^2$ .

Options

A.
298
B.
218
C.
351
D.
322

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Coplanarity Condition|Factor Theorem in Determinants|Signum Function Discontinuity|Objective Function Gradient

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 handle piecewise gradients defined by signum functions within an integral objective function.

The primary trap lies in the evaluation of the scalar triple product determinant using the Factor Theorem, which yields a continuous polynomial $x^2 - 40x + 351$ .

Students often ignore the domain restrictions imposed by the $\\text{sgn}(x-20)$ term, incorrectly assuming the critical points of the objective function $f(x)$ are simply the roots of the polynomial (13 and 27).

Furthermore, the discontinuity at $x=20$ creates a piecewise derivative; students must correctly reject the extraneous roots ( $x=25$ for the left branch and $x=20-\\sqrt{73}$ for the right branch) by strictly checking the boundary conditions of each interval.

Finally, they must verify that the non-differentiable point $x=20$ is not an extremum since the gradient does not change sign across the jump discontinuity.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Coplanarity Condition
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free