Match each statement in List-I with the differential equation it gives in List-II (k is a positive constant).
| List-I | List-II | ||
| (A) | A population P grows at a rate proportional to its size. | (I) | \(\displaystyle\frac{dy}{dx} = xy\) |
| (B) | The temperature T of a body falls at a rate proportional to the excess of T over the room temperature 25°C. | (II) | \(\displaystyle\frac{dT}{dt} = -k(T - 25)\) |
| (C) | The slope of the tangent at any point (x, y) of a curve is twice the abscissa of the point. | (III) | \(\displaystyle\frac{dy}{dx} = 2x\) |
| (D) | The slope of the tangent at any point (x, y) of a curve is the product of the coordinates of the point. | (IV) | \(\displaystyle\frac{dP}{dt} = kP\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(IV), (B)-(II), (C)-(III), (D)-(I)
A rate is a derivative with respect to time, and the slope of the tangent at (x, y) is \(\frac{dy}{dx}\).
(A) A population grows at a rate proportional to its size
This matches (IV) \(\frac{dP}{dt} = kP\): the rate \(\frac{dP}{dt}\) is a constant times P, and it is positive because P grows.
(B) The temperature falls at a rate proportional to the excess over 25°C
This matches (II) \(\frac{dT}{dt} = -k(T - 25)\): the excess is T − 25, and the minus sign shows that T decreases.
(C) Slope equal to twice the abscissa
This matches (III) \(\frac{dy}{dx} = 2x\): the abscissa of (x, y) is x.
(D) Slope equal to the product of the coordinates
This matches (I) \(\frac{dy}{dx} = xy\).