Applied Mathematics

Calculus

Medium

applied

The demand function for a commodity is $p = 45 + x - x^2$ , then the consumer's surplus at equilibrium price $p_0 = 25$ will be:

$\dfrac{125}{3}$

$\dfrac{425}{6}$

$\dfrac{225}{2}$

$\dfrac{925}{3}$

✅ Correct Option: 2

Setting $25 = 45 + x - x^2$ gives $x^2 - x - 20 = 0$ , so $x_0 = 5$ .

Consumer surplus $= \int_0^5 (45+x-x^2)dx - 25(5) = 225 + 12.5 - \frac{125}{3} - 125 = \frac{425}{6}$ .

2026: 18 May Shift 1

Which of the following statements are true?

(A) The break-even point is the level of production where the revenue from sales is equal to the cost of production and marketing.

(B) If R is the total revenue collected by a company when it sells $x$ units of a product at a price of p per unit, then R is given by $R=px$ .

(C) Let $C=C(x)$ be the total cost of producing and marketing $x$ units of a product, then marginal cost, $MC=\frac{dC}{dx}$ .

(D) If AC is the average cost, then $AC=\frac{x}{C}$ .

Choose the correct answer from the options given below:

2023: 11 Aug Shift 1

The degree of the differential equation $\left(1 - \left(\frac{dy}{dx}\right)^2\right)^{3/2} = k \frac{d^2 y}{dx^2}$ is :

2023: 11 Aug Shift 1

If $x = 2at, y = at^2$ , where 'a' is a constant, then $\frac{d^2 y}{dx^2}$ at $x = 2$ is :

2023: 22 May Shift 3

Match List - I with List - II.

	List - I		List - II
(A)	If $y = x^2 - 8$ and $\frac{dy}{dx} = 0$ , then $x = ?$	(I)	1
(B)	If $p(x) = 3x + 1$ , then $R(x)$ at $x = 2$	(II)	0
(C)	If $y = x^3$ , then $\frac{dy}{dx}$ at $x = -1$	(III)	14
(D)	If $C(x) = 100 + 5x$ , $R(x) = 102 + 3x$ , then break-even point	(IV)	3

Choose the correct answer from the options given below :

2023: 11 Aug Shift 1

The order and degree Of the differential equation $\frac{d^2 y}{dx^2} + 2 e^{-x} = 0$ , respectively are

2026: 14 May Shift 2

The relation between 'marginal cost (MC)' and 'average cost (AC)' of producing 'x' units of a product is

The demand function for a commodity is p=45+x−x2p = 45 + x - x^2p=45+x−x2, then the consumer's surplus at equilibrium price p0=25p_0 = 25p0​=25 will be: