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Question #40403

Difficulty: 8/10 | 🟢 Free

The Question

Let $N$ be the exponent of the highest power of $5$ that divides $970!$ . Let $\theta_k$ ( $k \in \mathbb{N}$ ) be the sequence of positive solutions to the equation $\tan \theta = \sqrt{3}$ , arranged in strictly increasing order. Consider the polynomial $P(x) = \prod_{k=1}^{N} \left( 1 + 2\cos(\theta_k) x \right)$ If $C$ is the coefficient of $x^{100}$ in $P(x)$ , find the exponent of the highest power of $3$ that divides $|C|$ .

Options

A.
1
B.
2
C.
3
D.
4

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

General Solution Principal Values|Coefficient of x^r in Expansion|Exponent of Prime in n!

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of number theory, trigonometric general solutions, and binomial expansions.

The primary trap lies in the evaluation of the trigonometric product.

Students often incorrectly assume that the periodic nature of $\tan \theta = \sqrt{3}$ yields a constant cosine value, leading to the erroneous polynomial $P(x) = (1+x)^{240}$ and an incorrect exponent of 1.

A secondary trap is embedded in Legendre's formula for $N$ ; forgetting the $5^4 = 625$ term yields $N=239$ , which cascades into an incorrect binomial coefficient $\binom{119}{50}$ and an exponent of 3.

The problem forces rigorous tracking of alternating signs and precise combinatorial arithmetic.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	General Solution Principal Values
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free