Question Statement:
Which one of the following is INCORRECT about protein structures?
Options:
-
(A) A protein fold is stabilized by favorable non-covalent interactions
-
(B) All parts of a fold can be classified as helices, strands or turns
-
(C) Two non-covalent atoms cannot be closer than the sum of their van der Waals radii
-
(D) The peptide bond is nearly planar
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2018 Q.09)**
**1. Governing Concept & Question Classification**:
- **Section**: General Aptitude (GA)
- **Question Type**: **MCQ** (2 Marks)
- **Topic**: Algebra (Symmetric Polynomials & Quadratic Equations)
**2. Detailed Step-by-Step Derivation**:
- Given the quadratic equation:
$x^2 + x - 1 = 0 \implies x^2 - 1 = -x$
- Dividing both sides by $x$ (since $x
eq 0$):
$x - rac{1}{x} = -1$
- Squaring both sides:
$\left(x - rac{1}{x}
ight)^2 = (-1)^2 \implies x^2 - 2 + rac{1}{x^2} = 1$
$x^2 + rac{1}{x^2} = 3$
- Squaring both sides once more to find $x^4 + rac{1}{x^4}$:
$\left(x^2 + rac{1}{x^2}
ight)^2 = 3^2 \implies x^4 + 2 + rac{1}{x^4} = 9$
$x^4 + rac{1}{x^4} = 9 - 2 = 7$
- **Option-by-Option Evaluation**:
- **Option (A) [INCORRECT]**: 1 is obtained by erroneous sign operations.
- **Option (B) [INCORRECT]**: 5 is obtained if $-2$ is subtracted from 3 instead of squaring and subtracting 2.
- **Option (C) [CORRECT]**: 7 is the exact algebraic value ($3^2 - 2 = 7$).
- **Option (D) [INCORRECT]**: 9 omits subtracting the cross-term $2 \cdot x^2 \cdot (1/x^2)$.
**3. Final Answer / Verification Summary**:
- **Official GATE Answer Key**: **C**
- **Scoring Criteria**: +2.0 Marks (Negative marking: -0.67 marks).
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