Subtopic: Fundamentals of Biomolecules, enzyme kinetics
Question Statement:
The value of k, for which the linear equations 2x + 3y = 6 and 4x + 6y = 3k
have at least one solution, is _____________.
(Answer in integer)
Official Answer Key:
4 to 4
Step-by-Step Solution & Explanation
### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2025 Q.34)**
**1. Governing Concept & Definition**:
- **Topic**: Engineering Mathematics & Linear Systems of Equations
- For a system of two linear equations in two variables:
$a_1 x + b_1 y = c_1$
$a_2 x + b_2 y = c_2$
to be consistent (have at least one solution), the coefficient ratios must satisfy:
- If lines are parallel with infinite solutions: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$
- If $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$, lines are parallel and disjoint (no solution).
**2. Detailed Step-by-Step Calculation**:
- **Given System**:
1. $2x + 3y = 6$
2. $4x + 6y = 3k$
- **Compare coefficients**:
$\frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2}$
$\frac{b_1}{b_2} = \frac{3}{6} = \frac{1}{2}$
- Because the left-hand side coefficients are in the ratio $1/2$, the equations represent parallel lines.
- For the system to have at least one solution (in fact, infinitely many solutions coincident with each other):
$\frac{c_1}{c_2} = \frac{6}{3k} = \frac{1}{2}$
$3k = 6 \times 2 = 12 \implies k = \frac{12}{3} = 4$
**3. Final Answer**:
- **Calculated Value**: **4**
- **Official GATE Accepted Range**: **4 to 4**
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.