Disaccharide molecules that contain 1 4) glycosidic linkage are
(A) sucrose and maltose (B) sucrose and isomaltose
(C) maltose and isomaltose (D) lactose and cellobiose
Options:
(A) sucrose and maltose
(B) sucrose and isomaltose
(C) maltose and isomaltose
(D) lactose and cellobiose
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2016 Q.10)**
**1. Governing Concept & Question Classification**:
- **Section**: General Aptitude (GA)
- **Question Type**: **MCQ** (2 Marks)
- **Topic**: Functions & Analytical Geometry (Logarithmic & Exponential Inverse Identities)
**2. Detailed Step-by-Step Derivation**:
- Given the function:
$y = \ln\left( \left| e^{\left|\sin(|x|)\right|} \right| \right) \quad \text{for } |x| < 2\pi$
- Simplify step-by-step using elementary algebra:
1. The exponential function $e^u$ is strictly positive for any real number $u$, so $\left| e^u \right| = e^u$.
2. The natural logarithm and exponential functions are mutual inverses:
$\ln(e^u) = u$
3. Therefore:
$y = \ln\left( e^{\left|\sin(|x|)\right|} \right) = \left|\sin(|x|)\right|$
4. Since the sine function is an odd function ($\sin(-x) = -\sin x$), taking the absolute value gives:
$|\sin(|x|)| = |\sin x|$
- The function simplifies to:
$y = |\sin x|$
- Characteristics of $y = |\sin x|$ over $-2\pi < x < 2\pi$:
- Periodic with period $\pi$.
- $y \ge 0$ everywhere (non-negative, rectified sinusoidal wave).
- Peaks of height $1$ at $x = \pm \pi/2, \pm 3\pi/2$.
- Sharp cusp minima at $y = 0$ at $x = 0, \pm \pi, \pm 2\pi$.
- This represents an **absolute value periodic sinusoidal wave $|\sin x|$ with cusp minima at multiples of $\pi$** (Curve C).
- **Option-by-Option Evaluation**:
- **Option (A) [INCORRECT]**: Distractor choice.
- **Option (B) [INCORRECT]**: The wave is undamped with constant amplitude of 1.
- **Option (C) [CORRECT]**: Precisely describes the graph of $y = |\sin x|$ with periodic cusps at multiples of $\pi$.
- **Option (D) [INCORRECT]**: The function is positive everywhere, not inverted.
**3. Final Answer / Verification Summary**:
- **Official GATE Answer Key**: **C**
- **Scoring Criteria**: +2.0 Marks (Negative marking: -0.67 marks).
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