Q.64 The value of determinant A given below is __________.
A =
| 5 | 16 | 81 |
| 0 | 2 | 2 |
| 0 | 0 | 16 |
V =aS/b + S + S2/c
Final Answer: S = 3
Given Equation
V =aS/b + S + S2/c
Given constants:
- a = 4
- b = 1
- c = 9
Substituting values:
V(S) =4S/1 + S + S2/9
Step 1: Simplify the Expression
Multiply numerator and denominator by 9 to remove the fraction:
V(S) =36S/9 + 9S + S2
Step 2: Differentiate V(S)
Let:
- f(S) = 36S
- g(S) = S2 + 9S + 9
Using the quotient rule:
dV/dS =f′(S)g(S) − f(S)g′(S)/g(S)2
dV/dS = 36(S2 + 9S + 9) − 36S(2S + 9)/(S2 + 9S + 9)2
Step 3: Simplify the Numerator
36(S2 + 9S + 9) = 36S2 + 324S + 324
36S(2S + 9) = 72S2 + 324S
Subtracting:
(36S2 + 324S + 324) − (72S2 + 324S) = −36S2 + 324
Therefore:
dV/dS = 324 − 36S2/ (S2 + 9S + 9)2
Step 4: Find the Maximum
For maximum V, set the numerator equal to zero:
324 − 36S2 = 0
36S2 = 324
S2 = 9
S = ±3
Step 5: Select the Physical Solution
Since substrate concentration cannot be negative:
S = 3
Final Answer
S = 3
Explanation of Possible Distractors
| Value | Why Incorrect |
|---|---|
| −3 | Negative substrate concentration is not physically meaningful |
| 1 or 2 | Do not satisfy dV/dS = 0 |
| 9 | Confuses S2 = 9 with S = 9 |
Conclusion
The equation
V = aS / (b + S + S2/c)
reaches its maximum value at
S = 3
for a = 4, b = 1, and c = 9.
