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Question #40641

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the sum of the numerically greatest terms in the binomial expansion of $\left( \sqrt{1+\sin 2\theta} + \sqrt{1-\sin 2\theta} \right)^{14}$ , where $\theta \in \left( \frac{3\pi}{4}, \pi \right)$ and $\tan \theta = -\frac{1}{5}$ . Find the value of $\frac{13^7 \cdot S}{1001 \cdot 2^{12} \cdot 3^8}$ .

Options

A.
0
B.
6
C.
12
D.
24

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

Numerically Greatest Term|Trigonometric Identical Substitutions|Symmetry and Periodicity|Square Root and Modulus Logic

Why This Question Is Tricky

Common Mistakes

This question tests the concept of the Numerically Greatest Term (NGT) combined with trigonometric identities and a critical modulus trap.

The core trap lies in the simplification of $\sqrt{1 \pm \sin 2\theta}$ to $|\sin \theta \pm \cos \theta|$ .

Students who ignore the modulus will incorrectly evaluate $a = -4/\sqrt{26}$ , leading to $T_{10}$ being negative.

This results in the sum of the two numerically greatest terms being exactly 0 (Option A).

The domain $\theta \in (3\pi/4, \pi)$ is specifically chosen to force $\sin \theta + \cos \theta < 0$ , activating this trap.

Furthermore, if a student fails to recognize that an integer index $m$ implies two equal NGTs, they will only sum one term, yielding 6 (Option B).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Numerically Greatest Term
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free