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3.6 · Q3

Q.The demand function for a commodity is p=10−2xp = 10 - 2x. Find the consumers' surplus for

(i) p=2p = 2
(ii) p=6p = 6.
Arunachal CbseNCERTSubjective· 3mImportance★★★★★
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✓ Free question

For the linear demand p=10−2xp=10-2x, find x0x_0 from each given price, then apply CS=∫0x0p dx−p0x0CS=\int_0^{x_0}p\,dx-p_0x_0: ₹16\text{₹}16 at p=2p=2 and ₹4\text{₹}4 at p=6p=6.

Consumers' surplus: CS=∫0x0p dx−p0x0\displaystyle CS=\int_0^{x_0}p\,dx-p_0x_0; here p=10−2xp=10-2x so ∫0x0(10−2x) dx=[10x−x2]0x0\int_0^{x_0}(10-2x)\,dx=\big[10x-x^2\big]_0^{x_0}.

(i) p0=2p_0=2

  1. 2=10−2x⇒x0=42=10-2x\Rightarrow x_0=4.
  2. ∫04(10−2x) dx=[10x−x2]04=40−16=24\int_0^4(10-2x)\,dx=\big[10x-x^2\big]_0^4=40-16=24.
  3. CS=24−(2)(4)=24−8=16CS=24-(2)(4)=24-8=16.

(ii) p0=6p_0=6

  1. 6=10−2x⇒x0=26=10-2x\Rightarrow x_0=2.
  2. ∫02(10−2x) dx=[10x−x2]02=20−4=16\int_0^2(10-2x)\,dx=\big[10x-x^2\big]_0^2=20-4=16.
  3. CS=16−(6)(2)=16−12=4CS=16-(6)(2)=16-12=4.
✓Final answer

  1. p=2⇒CS=₹16p=2\Rightarrow CS=\text{₹}16;
  2. p=6⇒CS=₹4p=6\Rightarrow CS=\text{₹}4.

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