CUET · MATHS · PYQ PAPER 2025

The differential equation representing the family of curves \(y=A x+\frac{B}{x}, x \neq 0\), where A and B are arbitrary constants, is given by

A \(x^2 \frac{d^2 y}{d x^2}+x \frac{d y}{d x}+y=0\)
B \(8 x^2 \frac{d^2 y}{d x^2}-x \frac{d y}{d x}-y=0\)
C \(x^2 \frac{d^2 y}{d x^2}-x \frac{d y}{d x}+y=0\)
D \(x^2 \frac{d^2 y}{d x^2}+x \frac{d y}{d x}-y=0\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(x^2 \frac{d^2 y}{d x^2}+x \frac{d y}{d x}-y=0\)

Step-by-step Solution

Detailed explanation

\(y = A x + \frac{B}{x} \implies xy = Ax^2 + B\) \(\frac{d}{dx}(xy) = \frac{d}{dx}(Ax^2 + B) \implies y + x\frac{dy}{dx} = 2Ax\) \(\frac{d}{dx}(y + x\frac{dy}{dx}) = \frac{d}{dx}(2Ax) \implies 2\frac{dy}{dx} + x\frac{d^2y}{dx^2} = 2A\)…

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

Match \(List - I\) with \(List - II\)

\(List - I\)	\(List - II\)
General solution	Differential Equation
(A) \(y=c x\)	(I) \(y^{\prime}=\frac{1}{2 \sqrt{x}}-1\)
(B) \(y=e^{-3 x}+c\)	(II) \(y^{\prime}+3 y=0\)
(C) \(x+y=\sqrt{x}+c\)	(III) \(y^{\prime}=\frac{y^2}{1-x y}(x y \neq 1)\)
(D) \(x y=\log y+c\)	(IV) \(x y^{\prime}=y\)

(c is an arbitrary constant and \(y^{\prime}=\frac{d y}{d x}\) )
Choose the correct answer from the options given below:

CUET 2023 Hard

If \(\vec{a}=2 \hat{i}-\hat{j}+3 \hat{k}\) and \(\vec{b}=2 \hat{i}+2 \hat{j}+\hat{k}\) then
Match List-I with List-II

List-I	List-II
(A) Projection of \(\vec{a}\) on \(\vec{b}\) is	(I) \(-7 \hat{i}+4 \hat{j}+6 \hat{k}\)
(B) \(\vec{a} \times \vec{b}\) is	(II) \(\frac{1}{\sqrt{101}}(-7 \hat{i}+4 \hat{j}+6 \hat{k})\)
(C) Unit vector along \(\vec{a}+\vec{b}\) is	(III) \(\frac{5}{3}\)
(D) Unit vector perpendicular to both \(\vec{a}\) and \(\vec{b}\) is	(IV) \(\frac{1}{\sqrt{33}}(4 \hat{i}+\hat{j}+4 \hat{k})\)

Choose the correct answer from the options given below :

CUET 2025 Easy

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X	0	1	2	3
P(X)	0.2	0.1	0.3	0.4

X	0	1	2	3	4
P(X)	0.1	K	2K	2K	K

List-I	List-II
(A) Output always opposite of input	(I) NOR
(B) Output 1 only when all inputs are 0	(II) NAND
(C) Output 0 only when all inputs are 1	(III) OR
(D) Output 1 when all inputs are 1	(IV) NOT