CUET · MATHS · PYQ PAPER 2025

The differential equation of the family of curves \(y=A e^{3 x}+B e^{-3 x}\) where A and B are arbitrary constants, is

A \(\frac{d^2 y}{d x^2}+9 y=0\)
B \(\frac{d^2 y}{d x^2}+3 y=0\)
C \(\frac{d^2 y}{d x^2}=9 y\)
D \(\frac{d^2 y}{d x^2}=3 y\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\frac{d^2 y}{d x^2}=9 y\)

Step-by-step Solution

Detailed explanation

\(y=A e^{3 x}+B e^{-3 x}\) \(\frac{d y}{d x}=3 A e^{3 x}-3 B e^{-3 x}\) \(\frac{d^2 y}{d x^2}=9 A e^{3 x}+9 B e^{-3 x}\) \(\frac{d^2 y}{d x^2}=9(A e^{3 x}+B e^{-3 x})\) \(\frac{d^2 y}{d x^2}=9 y\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

In the context of linear programming, Match List I with List II

List - I	List - II
(A) Z = 250x +75y	I. non-trivial constraints
(B) \(5 x+y \leq 100, x-y \leq 4\)	II. feasible solution
(C) \(x, y \geq 0\)	III. objective function
(D) points within and on the boundary of shaded region	IV. Trivial constraints

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