✓ Board-verified
Mathematics · 2026 · 4 marks
CBSE 2026 · Region 1 · Set 1 · Q36
An online delivery company in a city has 5000 subscribers and collects annual subscription fees of ₹ 300 per subscriber for unlimited free…
An online delivery company in a city has $\displaystyle 5000$ subscribers and collects annual subscription fees of ₹ $\displaystyle 300$ per subscriber for unlimited free deliveries.
The company wishes to increase the annual subscription fee. It is predicted that, for every increase of ₹$\displaystyle 1$, ten subscribers will discontinue. Assume that the company increased the annual fee by $\displaystyle ₹ x$.
Based on the given information, answer the following questions :
(i)
How many subscribers will discontinue after an increase of $\displaystyle ₹ x$ in annual fee?
(ii)
If $\displaystyle \mathrm{R}(x)$ denotes the total revenue collected after the increase of $\displaystyle ₹ x$ in subscription fee, express $\displaystyle \mathrm{R}(x)$ as a function of $\displaystyle x$.
(iii)
(a)
Find the value of $\displaystyle x$ for which $\displaystyle \mathrm{R}(x)$ is maximum.
OR
(b)
Find the sub-intervals of $\displaystyle (0,5000)$ in which $\displaystyle \mathrm{R}(x)$ is increasing and decreasing.
Official answer
x = $\displaystyle 100$
Marking-scheme solution
(i)
$\displaystyle 10 x$
(ii)
$\displaystyle R(x)=(5000-10 x)(300+x)$
(iii)
(a)
$\displaystyle R^{\prime}(x)=2000-20 x$
$\displaystyle R^{\prime}(x)=0 \Rightarrow x=100$
$\displaystyle R^{\prime \prime}(x)=-20<0$ at $\displaystyle x=100$
At $\displaystyle x=100, R(x)$ is maximum
OR
(b)
$\displaystyle R^{\prime}(x)=2000-20 x$
$\displaystyle R^{\prime}(x)=0 \Rightarrow x=100$
In $\displaystyle (0,100), R^{\prime}(x)>0$. Therefore, $\displaystyle R(x)$ is increasing in $\displaystyle (0,100)$ or $\displaystyle (0,100]$
In $\displaystyle (100,5000), R^{\prime}(x)<0$. Therefore, $\displaystyle R(x)$ is decreasing in $\displaystyle (100,5000)$ or $\displaystyle [100,5000)$
All Application of Derivatives questions2026 question paperNCERT solutions: Application of DerivativesPractice Application of Derivatives →
More from Application of Derivatives
- Overspeeding increases fuel consumption and decreases fuel economy as a result of tyre rolling friction and air resistance. While vehicles reach…2024
- Find the local maxima and local minima of the function f(x)=8/3 x^3-12 x^2+18 x+5.2025
- The least value of f(x)=x^3-12 x, x ∈[0,3] is2026
- Let f(x) be a continuous function on [a, b] and differentiable on (a, b). Then, this function f(x) is strictly increasing in (a, b) if2024
- Show that the function f given by f(x)= sin x+ cos x, is strictly decreasing in the interval ((π)/4, (5 π)/4).2024
- A cylindrical tank is being filled with sand at a rate of 314 m^3 / h. If the radius of the tank is 10 m, then the height of sand in the tank…2026
- Find the sub-interval of (0, ∞) in which f(x)=x^2 e^-x is increasing.2026
- A carpenter needs to make a wooden cuboidal box, closed from all sides, which has a square base and fixed volume. Since he is short of the paint…2025
CBSE Class 12 Mathematics past-paper question from the 2026 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.