Match List-I with List-II
| List-I | List-II | ||
| (A) | \(\displaystyle\frac{d^2y}{dx^2}\) at x = 1, when \(y = x^4 + 2x\) | (I) | 2 |
| (B) | \(\displaystyle\frac{dy}{dx}\) at t = 2, when \(x = t^2,\ y = t^3\) | (II) | 3 |
| (C) | \(\displaystyle\frac{dy}{dx}\) at the point (2, 1), when \(x^2 - y^2 = 3\) | (III) | 10 |
| (D) | Marginal cost at x = 4, when the total cost is \(C(x) = x^2 + 2x + 50\) | (IV) | 12 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A)-(IV), (B)-(II), (C)-(I), (D)-(III)
(A) \(\frac{d^2y}{dx^2}\) at x = 1, when \(y = x^4 + 2x\)
This matches (IV) 12.
\( \displaystyle \frac{dy}{dx} = 4x^3 + 2 \)
\( \displaystyle \frac{d^2y}{dx^2} = 12x^2 \)
\(\text{At } x = 1: \ 12\)
(B) \(\frac{dy}{dx}\) at t = 2, when \(x = t^2,\ y = t^3\)
This matches (II) 3. In parametric form:
\( \displaystyle \frac{dy}{dx} = \frac{3t^2}{2t} = \frac{3t}{2} \)
\( \displaystyle \text{At } t = 2: \ \frac{6}{2} = 3 \)
(C) \(\frac{dy}{dx}\) at (2, 1), when \(x^2 - y^2 = 3\)
This matches (I) 2. By implicit differentiation:
\( \displaystyle 2x - 2y\frac{dy}{dx} = 0 \)
\( \displaystyle \frac{dy}{dx} = \frac{x}{y} = \frac{2}{1} = 2 \)
(D) Marginal cost at x = 4, when \(C(x) = x^2 + 2x + 50\)
This matches (III) 10.
\(C'(x) = 2x + 2\)
\(C'(4) = 8 + 2 = 10\)