A crossing was performed between the genotypes DdEeFfgg and ddEeFfGg. Assuming that the
allelic pairs of all genes assort independently, the proportion of progeny having the genotype
ddeeffgg is expected to be ______________%.
Options:
(A) 2
2
2
s
s
a
(B) 2
2
a
s
a
(C) 2
2
s
s
a
(D) 2
2
s
s
a
GATE 2016 Biotechnology (BT)
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2016 Q.34)**
**1. Governing Concept & Question Classification**:
- **Section**: BT
- **Question Type**: **MCQ** (1 Mark)
- **Topic**: Engineering Mathematics (Laplace Transforms of Trigonometric Functions)
**2. Detailed Step-by-Step Derivation**:
- The unilateral Laplace transform of a time-domain function $f(t)$ is defined as:
$\mathcal{L}\{f(t)\} = F(s) = \int_0^\infty f(t) e^{-st} dt$
- For the harmonic function $f(t) = \cos(at)$:
- Express $\cos(at)$ using Euler's formula:
$\cos(at) = \frac{e^{iat} + e^{-iat}}{2}$
- Applying linearity of the Laplace transform:
$\mathcal{L}\{\cos(at)\} = \frac{1}{2} \left[ \mathcal{L}\{e^{iat}\} + \mathcal{L}\{e^{-iat}\} \right] = \frac{1}{2} \left[ \frac{1}{s - ia} + \frac{1}{s + ia} \right]$
- Taking a common denominator:
$F(s) = \frac{1}{2} \left[ \frac{(s + ia) + (s - ia)}{(s - ia)(s + ia)} \right] = \frac{1}{2} \left[ \frac{2s}{s^2 - (i^2 a^2)} \right] = \frac{s}{s^2 + a^2} \quad (\text{for } s > 0)$
- **Option-by-Option Evaluation**:
- **Option (A) [INCORRECT]**: $\frac{a}{s^2 + a^2}$ is the Laplace transform of $\sin(at)$.
- **Option (B) [INCORRECT]**: $\frac{a}{s^2 - a^2}$ is the Laplace transform of $\sinh(at)$.
- **Option (C) [CORRECT]**: $\frac{s}{s^2 + a^2}$ is the exact Laplace transform of $\cos(at)$.
- **Option (D) [INCORRECT]**: $\frac{s}{s^2 - a^2}$ is the Laplace transform of $\cosh(at)$.
**3. Final Answer / Verification Summary**:
- **Official GATE Answer Key**: **C**
- **Scoring Criteria**: +1.0 Mark (Negative marking: -0.33 marks).
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