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Question #41516

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all integers $x \in [1, 60]$ satisfying the equation $\frac{2^{x+1}}{\pi} \int_0^{\pi/2} \sin^x \theta \, d\theta = \sum_{r=0}^{\lfloor x/2 \rfloor} \binom{\lfloor x/2 \rfloor}{r}^2$ Find the sum of all elements in $S$ .

Options

A.
930
B.
1830
C.
900
D.
1860

Check Your Answer

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

Reduction Formulae|Vandermonde's Identity|Greatest Binomial Coefficient

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of Wallis' reduction formula and Vandermonde's identity, while setting a parity trap.

The core trap exploits the common student oversight of ignoring the domain constraints of Wallis' formula, where the presence of $\pi$ strictly depends on the parity of the exponent.

Students who blindly equate the integral's coefficient to the binomial sum without checking for odd/even cases will incorrectly assume the identity holds for all integers, leading them to sum all numbers in the interval (yielding the distractor 1830).

By forcing the evaluation of an integral that must match an integer binomial coefficient, the problem demands the recognition that the irrationality of the odd case acts as a strict boundary constraint, restricting valid solutions exclusively to even integers.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Reduction Formulae
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free