CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q384 Marksapplications_of_derivatives
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Question 38 (Applications Of Derivatives)
An online delivery company in a city has 5000 subscribers and collects annual subscription fees of ₹300 per subscriber for unlimited free deliveries. The company wishes to increase the annual subscription fee. It is predicted that for every increase of ₹1, ten subscribers will discontinue. Assume that the company increased the annual fee by ₹x.
Part (1) 1 Marks
How many subscribers will discontinue after an increase of ₹x in annual fee?
Answer: 10x
- For each ₹1 increase, 10 leave. So for ₹x increase, 10x leave.
- 10x
Part (2) 1 Marks
If R(x) denotes the total revenue collected after the increase of ₹x in subscription fee, express R(x) as a function of x.
Answer: R(x) = (5000−10x)(300+x) = −10x² + 2000x + 1500000
- New subscribers = 5000 − 10x, new fee = 300 + x. Revenue = (5000−10x)(300+x) = −10x² + 2000x + 1500000.
- R(x) = (5000−10x)(300+x) = −10x² + 2000x + 1500000
Part (3) 2 Marks
Find the value of x for which R(x) is maximum.
Answer: x = 100
- R'(x) = 2000 − 20x. Set R'(x)=0 ⇒ x=100.
- R''(x) = −20 < 0 at x=100, so R(x) is maximum at x=100.
- x = 100
Part (4) 2 Marks
Find the sub-intervals of (0, 5000) in which R(x) is increasing and decreasing.
Answer: Increasing on (0,100], decreasing on [100,5000).
- R'(x) = −20x + 2000. Set R'(x)=0 ⇒ x=100.
- For x<100, R'(x)>0 ⇒ increasing; for x>100, R'(x)<0 ⇒ decreasing.
- Increasing on (0,100], decreasing on [100,5000).