Before You Begin

Question #40679

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all real numbers $x$ for which the function $P(X=k) = (24x^4 - 62x^3 + 27x^2 + 2x - 1) x^{k-1}, \quad k = 1, 2, 3, \dots$ represents a valid probability mass function of a discrete random variable $X$ . Find the sum of all elements in $S$ .

Options

A.
$\frac{5}{6}$
B.
$\frac{7}{12}$
C.
$\frac{31}{12}$
D.
$\frac{17}{6}$

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

Infinite GP Convergence|Probability Mass Function (PMF)|Closed vs Open Intervals

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous definition of a Probability Mass Function (PMF) intertwined with the convergence criteria of an infinite Geometric Progression.

The primary trap lies in blindly applying the infinite GP sum formula $a/(1-r) = 1$ and summing all roots of the resulting polynomial (yielding 31/12).

A secondary trap occurs when students remember the convergence domain $|r| < 1$ and exclude $x=2$ , but forget the fundamental probability axiom $P(X=k) \ge 0$ .

This leads them to include $x = -1/4$ , yielding the trap answer 7/12.

The negative ratio causes alternating signs in the GP, producing negative probabilities, which is a lethal collision of algebraic roots and probability axioms.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Infinite GP Convergence
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free