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Question #41623

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a}_r = \left( \frac{10 - r}{r(r+1)(r+2)} - 1 \right) \hat{i} + r\hat{j} + r^2\hat{k}$ and $\vec{b}_r = \hat{i}$ . Let $M_r$ be the $3 \times 3$ matrix representing the linear transformation $T_r(\vec{x}) = \vec{x} + \vec{a}_r \times (\vec{x} \times \vec{b}_r)$ for any vector $\vec{x} \in \mathbb{R}^3$ . If $N_r = \text{adj}(M_r)$ , find the value of $\sum_{r=1}^{20} \sqrt[4]{\det(N_r)}$ .

Options

A.
$\frac{475}{231}$
B.
$\frac{470}{231}$
C.
$\frac{485}{231}$
D.
$\frac{465}{231}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Vector Triple Product Expansion|Adjoint-Determinant Relation|V-N Method (Partial Fractions)|Modulus Function Symmetry

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of Vector Triple Products, Matrix Determinant properties, and the Method of Differences (V-N method).

The primary trap lies in the modulus function symmetry.

Students often correctly identify the telescoping sequence $V_r - V_{r+1}$ but blindly sum it from $r=1$ to $20$ without realizing that the determinant's fourth root enforces an absolute value.

Because the general term $\frac{10-r}{r(r+1)(r+2)}$ becomes negative after $r=10$ , failing to split the summation at the root of the numerator leads directly to the trap answer 470/231.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Vector Triple Product Expansion
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free