Match the LIST-I with LIST-II, LIST-I (Differential equation), LIST-II (Degree), A., (dy/dx)^2 + 1/(dy/dx) = 5, I., 2, B., d^3y/dx^3 = \3 + (dy/dx)^2\^(3/2), II., 4, C., d^2y/dx^2 = (dy/dx)^(5/4), III., 1, D., yd^2x/dy^2 = y^2 + 1, IV., 3, Choose the correct answer from the options given below: Option 1: A-II, B-IV, C-I, D-III Option 2: A-IV, B-I, C-II, D-III Option 3: A-III, B-I, C-IV, D-II Option 4: A-I, B-IV, C-II, D-III

Answer: A-IV, B-I, C-II, D-III. Solution: Clear fractions and radicals first. A: multiplying by dy/dx gives a cube of the first derivative, degree 3 (IV). B: squaring gives (d^3y/dx^3)^2, degree 2 (I). C: raising to the fourth power gives (d^2y/dx^2)^4, degree 4 (II). D: already polynomial and linear in the highest derivative, degree 1 (III).

Applied Mathematics

Calculus

Medium

applied

Match the LIST-I with LIST-II

LIST-I (Differential equation) LIST-II (Degree)
A. $\left(\dfrac{dy}{dx}\right)^2 + \dfrac{1}{\dfrac{dy}{dx}} = 5$ I. $2$
B. $\dfrac{d^3y}{dx^3} = \left\{3 + \left(\dfrac{dy}{dx}\right)^2\right\}^{3/2}$ II. $4$
C. $\dfrac{d^2y}{dx^2} = \left(\dfrac{dy}{dx}\right)^{5/4}$ III. $1$
D. $y\dfrac{d^2x}{dy^2} = y^2 + 1$ IV. $3$

Choose the correct answer from the options given below:

	LIST-I (Differential equation)		LIST-II (Degree)
A.	$\left(\dfrac{dy}{dx}\right)^2 + \dfrac{1}{\dfrac{dy}{dx}} = 5$	I.	$2$
B.	$\dfrac{d^3y}{dx^3} = \left\{3 + \left(\dfrac{dy}{dx}\right)^2\right\}^{3/2}$	II.	$4$
C.	$\dfrac{d^2y}{dx^2} = \left(\dfrac{dy}{dx}\right)^{5/4}$	III.	$1$
D.	$y\dfrac{d^2x}{dy^2} = y^2 + 1$	IV.	$3$

A-II, B-IV, C-I, D-III

A-IV, B-I, C-II, D-III

A-III, B-I, C-IV, D-II

A-I, B-IV, C-II, D-III

✅ Correct Option: 2

Clear fractions and radicals first.

A: multiplying by $\frac{dy}{dx}$ gives a cube of the first derivative, degree $3$ (IV).

B: squaring gives $\left(\frac{d^3y}{dx^3}\right)^2$ , degree $2$ (I).

C: raising to the fourth power gives $\left(\frac{d^2y}{dx^2}\right)^4$ , degree $4$ (II).

D: already polynomial and linear in the highest derivative, degree $1$ (III).

2023: 23 May Shift 3

For a manufacturer, total cost function is given by $C = \frac{x^2}{25} + 2x$ . Which of the following are correct ?

A. 2.6 is the marginal cost when 5 units are produced.

B. $\frac{2x}{25} + 2$ is the marginal cost function.

C. $\frac{x}{25} + 2$ is the average cost function.

D. 2.4 is the marginal cost when 5 units are produced.

E. $\frac{x}{25} + 1$ is the average cost function.

Choose the correct answer from the options given below:

2023: 30 May Shift 3

If $C(x) = ax^2 - bx - c$ represents the total cost function then the slope of the tangent to the marginal cost curve at the point $(x, y)$ is :

2023: 30 May Shift 3

Let $f : R \to R$ be defined such that $f(x) = 16x^2 - 16x + 12$

(A) Maximum value of $f(x)$ is 8

(B) Minimum value of $f(x)$ is 8

(D) No maximum value of $f(x)$

Choose the correct answer from the options given below :

2026: 29 May Shift 2

The marginal cost (MC) of producing x units of an item in a factory is given by $MC=0.006x^2-0.001x+4$ .

If the fixed cost is ₹1000, then the cost of producing 200 units will be :

2022: 25 May Shift 1

If the cost function $C(x)$ of producing $x$ units of a commodity is given as $C(x) = x^3 - 60x^2 + 13x + 50$ , then the level of output for which the marginal cost is minimum is

2023: 7 June Shift 1

If $t = e^{2x}$ and $\log_e t^2$ , then $\frac{d^2y}{dx^2}$ is :

Solution

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