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Question #40296

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all distinct coefficients in the expansion of $(x_1 + x_2 + x_3 + x_4)^4$ . Let $A$ be the set of all polynomials of the form $P(x) = x^4 - ax^3 + bx^2 - cx + d$ such that $a, b, c, d \in S$ and the equation $P(x) = 0$ has roots of even multiplicity. If $K$ is the set of all distinct coefficients $(a, b, c, d)$ that appear in the polynomials of $A$ , find the number of proper subsets of $K$ .

Options

A.
7
B.
15
C.
31
D.
63

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Subset vs. Proper Subset|Multinomial Theorem|Even Multiplicity Roots

Why This Question Is Tricky

Common Mistakes

This question tests the integration of the Multinomial Theorem with polynomial root properties, specifically focusing on the constraint of even multiplicity roots forcing the polynomial to be a perfect square.

The trap lies in the final combinatorics step: students often calculate the total number of subsets ( $2^n$ ) and forget to subtract 1 for the 'proper' subset condition, or they fail to correctly identify the distinct elements of the coefficient set $S$ generated by the multinomial expansion.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Subset vs. Proper Subset
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free