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Question #41725

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} 3p - 2 & \tan \theta \\ -\cot \theta & q \end{pmatrix}$ , where $p \in (0, 1)$ , $q \in \mathbb{R}$ , and $\theta \in \left(0, \frac{\pi}{2}\right)$ . It is given that $A$ is a nilpotent matrix of index 2. If $p$ represents the probability of success in a single trial of a binomial experiment, find the probability of at least 4 successes in 6 independent trials.

Options

A.
$\frac{73}{729}$
B.
$\frac{64}{729}$
C.
$\frac{31}{243}$
D.
$\frac{20}{243}$

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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 Distribution Tail-Ends|Nilpotent Matrix Logic|Matrix Equality|Implicit to Explicit Conversion

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of linear algebra and probability.

The primary trap lies in the implicit-to-explicit conversion of the matrix parameters.

Students often attempt to manually compute $A^2 = O$ and solve the resulting system of non-linear equations, which is algebraically heavy and prone to error.

The elegant approach requires recognizing that for any $2 \times 2$ nilpotent matrix of index 2, both the trace and determinant must strictly be zero.

A secondary trap is the boundary condition of probability: solving the determinant equation yields two roots ( $p=1$ and $p=1/3$ ), and failing to reject $p=1$ (since $p \in (0,1)$ is given) leads to a degenerate binomial distribution and an incorrect tail-end probability.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Binomial Distribution Tail-Ends
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free