Cesium chloride density gradient centrifugation is commonly used for the separation of DNA
molecules. The buoyant density, ρ, of a double stranded Cs+DNA is given by the equation
ρ = 1.66 + 0.098XG+C
where XG+C denotes
(A) total number of G and C (B) mole fraction of G+C
(C) number of GC repeats (D) ratio of G+C to A+T content
Options:
(A) total number of G and C
(B) mole fraction of G+C
(C) number of GC repeats
(D) ratio of G+C to A+T content
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2016 Q.09)**
**1. Governing Concept & Question Classification**:
- **Section**: General Aptitude (GA)
- **Question Type**: **MCQ** (2 Marks)
- **Topic**: Abstract Algebra (Binary Operations & Identity Element)
**2. Detailed Step-by-Step Derivation**:
- The binary operation $\square$ is defined on real numbers as:
$a \mathbin{\square} b = ab + (a + b)$
- An identity element $x$ of this operation must satisfy the condition:
$a \mathbin{\square} x = a \quad \text{for all } a \in \mathbb{R}$
- Substitute $b = x$ into the definition:
$ax + a + x = a$
- Subtract $a$ from both sides:
$ax + x = 0$
$x(a + 1) = 0$
- For this equation to hold true for **any** real number $a$ (for all $a \in \mathbb{R}$), we must have:
$x = 0$
- Verification: Check whether $a \mathbin{\square} 0 = a$ and $0 \mathbin{\square} a = a$:
$a \mathbin{\square} 0 = a(0) + a + 0 = a$
$0 \mathbin{\square} a = 0(a) + 0 + a = a$
Both sides confirm that $x = 0$ is the unique identity element.
- **Option-by-Option Evaluation**:
- **Option (A) [CORRECT]**: $0$ satisfies $a \mathbin{\square} 0 = a$ for all real numbers $a$.
- **Option (B) [INCORRECT]**: If $x = 1$, $a \mathbin{\square} 1 = a(1) + a + 1 = 2a + 1 \neq a$.
- **Option (C) [INCORRECT]**: If $x = 2$, $a \mathbin{\square} 2 = 3a + 2 \neq a$.
- **Option (D) [INCORRECT]**: 10 is an arbitrary distractor.
**3. Final Answer / Verification Summary**:
- **Official GATE Answer Key**: **A**
- **Scoring Criteria**: +2.0 Marks (Negative marking: -0.67 marks).
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