(a) The supply function of a commodity is $100p = (x + 20)^2$. Find the Producer's Surplus (PS), when the market price is ₹ 25. OR (b) Find : $$\int \dfrac{2x^2 + 1}{x^2 - 3x + 2}\,dx$$

> [!TLDR] > (a) At market price ₹25 the supply quantity is $x_0=30$; PS $= x_0p_0-\int_0^{x_0}p\,dx = 750-390 = \mathbf{₹360}$. > (b) After polynomial division and partial fractions the integral is $2x-3\ln|x-1|+9\ln|x-2|+C$. > [!FORMULA] > **(a)** Producer's Surplus: $\;PS = x_0\,p_0 - \displaystyle\int_0^{x_0} p\,dx$, where $p$ is the supply price as a function of quantity $x$, $x_0$ is the quantity supplied at the market price, and $p_0$ is the market price. > **(b)** Partial fractions: if $\dfrac{N(x)}{(x-a)(x-b)}=\dfrac{A}{x-a}+\dfrac{B}{x-b}$, then $\displaystyle\int\frac{dx}{x-a}=\ln|x-a|+C$. **Part (a)** 1. The supply function is $100p=(x+20)^2$, i.e. $p=\dfrac{(x+20)^2}{100}$. 2. Find the quantity supplied at the market price $p_0=25$: $\;100(25)=(x+20)^2\Rightarrow (x+20)^2=2500\Rightarrow x+20=50\Rightarrow x_0=30$. 3. Compute $\displaystyle\int_0^{30} p\,dx = \int_0^{30}\frac{(x+20)^2}{100}\,dx = \frac{1}{100}\left[\frac{(x+20)^3}{3}\right]_0^{30} = \frac{1}{300}\big[(50)^3-(20)^3\big]$. 4. Evaluate: $\dfrac{1}{300}\big[125000-8000\big]=\dfrac{117000}{300}=390$. 5. Revenue at market price: $x_0p_0 = 30\times 25 = 750$. 6. Therefore $PS = 750 - 390 = 360$. **Part (b)** …

(a) The supply function of a commodity is 100p = (x + 20)². Find the Producer's Surplus (PS), when the market

Question

100p = (x + 20)^2

(OR)

(b) Find :

\int \dfrac{2x^2 + 1}{x^2 - 3x + 2}\,dx

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Concept understanding — Consumer Surplus

Consumer Surplus: The Extra You Get for Free

Imagine you walk into a shop to buy a phone. You were willing to pay up to ₹30,000 for it — that's the maximum you'd give. But the shopkeeper has priced it at ₹22,000. You pay ₹22,000 and walk out with the phone. How much did you gain from this deal?

You gained ₹8,000. That's the difference between what you were willing to pay (₹30,000) and what you actually paid (₹22,000). That extra ₹8,000 is your consumer surplus.

Note

Consumer surplus is the benefit a buyer receives when the market price is lower than the maximum price they would have been willing to pay.

The Intuition: Why It Exists

Markets don't charge each person their personal maximum. Instead, they set a single price for everyone. If you value the good more than that price, you get a "bonus" — you paid less than it was worth to you. If you value it less, you simply don't buy.

Think of it like this: you'd have been happy paying ₹30,000. Paying only ₹22,000 feels like getting ₹8,000 for free. That feeling of "extra value" is consumer surplus.

The Precise Statement

For a single unit, consumer surplus is:

\text{Consumer Surplus} = \text{Maximum Willingness to Pay} - \text{Actual Price Paid}

For the whole market, it's the sum of all individual surpluses. Graphically, it's the area below the demand curve and above the market price, up to the quantity bought.

\text{Consumer Surplus} = \int_{0}^{Q^*} (P_{\text{demand}}(Q) - P^*) \, dQ

A Simple Example

Suppose the demand for apples is:

Price (₹)	Quantity Demanded
10	1
8	2
6	3
4	4

If the market price is ₹6, then: …

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At market price ₹25 the supply quantity is $x_0=30$ ; PS $= x_0p_0-\int_0^{x_0}p\,dx = 750-390 = \mathbf{₹360}$ .
After polynomial division and partial fractions the integral is $2x-3\ln|x-1|+9\ln|x-2|+C$ .

\;PS = x_0\,p_0 - \displaystyle\int_0^{x_0} p\,dx

Part (a)

The supply function is $100p=(x+20)^2$ , i.e. $p=\dfrac{(x+20)^2}{100}$ .
Find the quantity supplied at the market price $p_0=25$ : $\;100(25)=(x+20)^2\Rightarrow (x+20)^2=2500\Rightarrow x+20=50\Rightarrow x_0=30$ .
Compute $\displaystyle\int_0^{30} p\,dx = \int_0^{30}\frac{(x+20)^2}{100}\,dx = \frac{1}{100}\left[\frac{(x+20)^3}{3}\right]_0^{30} = \frac{1}{300}\big[(50)^3-(20)^3\big]$ .
Evaluate: $\dfrac{1}{300}\big[125000-8000\big]=\dfrac{117000}{300}=390$ .
Revenue at market price: $x_0p_0 = 30\times 25 = 750$ .
Therefore $PS = 750 - 390 = 360$ .

Part (b) …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2022Set 465/BAB/43 marks
$100p = (x + 20)^2$
(OR)
$\int \dfrac{2x^2 + 1}{x^2 - 3x + 2}\,dx$
›Reveal solutionSolution
At market price ₹25 the supply quantity is $x_0=30$ ; PS $= x_0p_0-\int_0^{x_0}p\,dx = 750-390 = \mathbf{₹360}$ .

After polynomial division and partial fractions the integral is $2x-3\ln|x-1|+9\ln|x-2|+C$ .

$\;PS = x_0\,p_0 - \displaystyle\int_0^{x_0} p\,dx$
Part (a)

The supply function is $100p=(x+20)^2$ , i.e. $p=\dfrac{(x+20)^2}{100}$ .

Find the quantity supplied at the market price $p_0=25$ : $\;100(25)=(x+20)^2\Rightarrow (x+20)^2=2500\Rightarrow x+20=50\Rightarrow x_0=30$ .

Compute $\displaystyle\int_0^{30} p\,dx = \int_0^{30}\frac{(x+20)^2}{100}\,dx = \frac{1}{100}\left[\frac{(x+20)^3}{3}\right]_0^{30} = \frac{1}{300}\big[(50)^3-(20)^3\big]$ .

Evaluate: $\dfrac{1}{300}\big[125000-8000\big]=\dfrac{117000}{300}=390$ .

Revenue at market price: $x_0p_0 = 30\times 25 = 750$ .

Therefore $PS = 750 - 390 = 360$ .

Part (b) …
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Q.(a) The supply function of a commodity is 100p=(x+20)2100p = (x + 20)^2100p=(x+20)2. Find the Producer's Surplus (PS), when the market price is ₹ 25.

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Q.(a) The supply function of a commodity is $100p = (x + 20)^2$ . Find the Producer's Surplus (PS), when the market price is ₹ 25.

Unlock everything free for 14 days