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Question #40650

Difficulty: 8/10 | 🟢 Free

The Question

Let $x, y, z$ be the roots of the equation $24t^3 - 26t^2 + 9t - 1 = 0$ . For a real number $t$ such that $|t| < 1$ , define the vector: $\vec{v}_t = \sum_{r=0}^{\infty} (-t)^r \left[ \binom{-1}{r} \hat{i} + \binom{-2}{r} \hat{j} + \binom{-3}{r} \hat{k} \right]$ Find the volume of the parallelepiped whose coterminous edges are the vectors $\vec{v}_x, \vec{v}_y$ , and $\vec{v}_z$ .

Options

A.
$\frac{2}{9}$
B.
0
C.
$\frac{4}{9}$
D.
$\frac{1}{18}$

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

Binomial Theorem for Negative/Fractional Indices|Scalar Triple Product Volume Logic|Collinearity and Coplanarity

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of the negative binomial theorem within vector algebra, specifically targeting a coplanarity trap.

Students often incorrectly assume the vectors are coplanar by truncating the infinite series prematurely.

For instance, if truncated at $r=1$ , the vectors become linear combinations of $\hat{i}+\hat{j}+\hat{k}$ and $\hat{i}+2\hat{j}+3\hat{k}$ , making any three such vectors coplanar (yielding a volume of 0).

By recognizing the infinite sum as $(1-t)^{-n}$ , the scalar triple product resolves to a non-zero Vandermonde determinant, rewarding students who respect the infinite series nature of the components.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Binomial Theorem for Negative/Fractional Indices
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free