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Question #41040

Difficulty: 8/10 | 🟢 Free

The Question

Let $L$ be the horizontal asymptote of the curve $y = \frac{3000t^3 + 2t^2 - 7}{t^3 - t + 4}$ as $t \to \infty$ . Two boxes contain mathematical expressions. Box I contains the expression $\left(x^2 + \frac{1}{x}\right)^{15}$ and Box II contains the expression $\left(x^3 - \frac{2}{x^2}\right)^{10}$ . A fair coin is tossed. If it shows heads, Box I is selected; if tails, Box II is selected. Then, an integer $r$ is chosen at random from the set $S = \{0, 1, 2, \dots, 30\}$ . What is the probability that the coefficient of $x^r$ in the expression of the selected box is strictly greater than $L$ ?

Options

A.
$\frac{4}{31}$
B.
$\frac{9}{62}$
C.
$\frac{5}{31}$
D.
$\frac{27}{62}$

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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

Multiplication Theorem|Coefficient of x^r in Expansion|The Empty Set Trap|Rational Function Asymptotes

Why This Question Is Tricky

Common Mistakes

This question tests the Multiplication Theorem of probability intertwined with binomial expansion coefficients.

The primary trap is the 'Empty Set Trap': students often assume that for any integer r within the minimum and maximum possible degrees, a non-zero coefficient exists.

They might erroneously treat the degree equation (e.g., 30-3k = r) as continuous and count all integers r in that interval, failing to realize that r must be generated strictly by an integer k.

A secondary trap involves ignoring the boundary condition r ∈ S, specifically missing that r = -10 for Box II falls outside the sample space, or incorrectly including negative coefficients by taking the absolute value.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Multiplication Theorem
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free