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Question #73520

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a} = \hat{i}$ , $\vec{b} = x\hat{i} + \hat{j}$ , and $\vec{c} = (x^2 - 3x + 1)\hat{i} + \hat{k}$ be three vectors for $x \in \mathbb{R}$ . Let $\vec{a}', \vec{b}', \vec{c}'$ be the reciprocal system of vectors of $\vec{a}, \vec{b}, \vec{c}$ . Let $f(x) = |\vec{a}'|^2 + |\vec{b}'|^2 + |\vec{c}'|^2$ . If $x_1$ and $x_2$ (where $x_1 < x_2$ ) are the $x$ -coordinates of the points of inflection of the curve $y = f(x)$ , find the value of the scalar triple product $[\vec{a}'(x_1), \vec{a}'(x_2), \vec{a}'(x_1 + x_2)]$ .

Options

A.
2
B.
4
C.
-2
D.
0

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

Reciprocal System of Vectors|Geometric Interpretation of Cross Product|Concavity Analysis

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 problem tests the intersection of vector algebra and differential calculus.

The primary trap lies in the assumption that a constant scalar triple product (volume) implies constant reciprocal vectors or constant cross-product magnitudes.

Students who rely on rote memorization might assume $f(x)$ is constant and fail to find the inflection points.

By forcing the explicit computation of the reciprocal basis $\vec{a}'(x)$ , the question transitions into a concavity analysis of a quartic polynomial.

Finally, evaluating the scalar triple product of the reciprocal vectors at the inflection points rewards students who recognize the underlying Vandermonde determinant structure, bypassing tedious arithmetic.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Reciprocal System of Vectors
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free