Subtopic: Fundamentals of Biomolecules, enzyme kinetics
Question Statement:
The value of the series 1+ sin x + cos2 x + sin3 x + ⋯ at x =
𝜋
4 is ________.
Options:
(A) 1
√2+1
(B) √2
√2+1
(C) 1
√2-1
(D) √2
√2-1
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2024 Q.29)**
**1. Governing Concept**:
- **Topic**: Engineering Mathematics & Infinite Geometric Series
- The series is given as:
$S = 1 + \sin x + \cos^2 x + \sin^3 x + \cos^4 x + \cdots$
At $x = \frac{\pi}{4}$:
$\sin\left(\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$
- Therefore, every term is an exponent of $r = \frac{1}{\sqrt{2}}$:
$S = 1 + \frac{1}{\sqrt{2}} + \left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^3 + \cdots$
- This is an infinite geometric series with first term $a = 1$ and common ratio $r = \frac{1}{\sqrt{2}} < 1$:
$S = \frac{a}{1 - r} = \frac{1}{1 - \frac{1}{\sqrt{2}}} = \frac{\sqrt{2}}{\sqrt{2} - 1}$
**2. Option Analysis**:
- **Option (A) is INCORRECT**: Denominator sign error and numerator inverted.
- **Option (B) is INCORRECT**: Plus sign in denominator corresponds to alternating series.
- **Option (C) is INCORRECT**: Missing $\sqrt{2}$ in numerator.
- **Option (D) [√2 / (√2 - 1)] is CORRECT**: Matches $\frac{1}{1 - 1/\sqrt{2}} = \frac{\sqrt{2}}{\sqrt{2} - 1}$.
**3. Final Answer**:
- **Correct Choice**: **D**
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