CUET · MATHS · PYQ PAPER 2025

The general solution of the differential equation \(\frac{d y}{d x}+y \tan x=\sec x\)

A \(y \sec x-\tan x=c\), where \(c\) is an arbitrary constant
B \(y \sec x+\tan x=c\), where \(c\) is an arbitrary constant
C \(y \tan x-\sec x=c\), where \(c\) is an arbitrary constant
D \(y \tan x+\sec x=c\), where \(c\) is an arbitrary constant

Verified Solution

Answer & Solution

Correct Answer

(A) \(y \sec x-\tan x=c\), where \(c\) is an arbitrary constant

Step-by-step Solution

Detailed explanation

\( \text{IF} = e^{\int \tan x dx} = e^{\ln |\sec x|} = \sec x \) \( y \cdot \sec x = \int (\sec x \cdot \sec x) dx + c \) \( y \sec x = \int \sec^2 x dx + c \) \( y \sec x = \tan x + c \) \( y \sec x - \tan x = c \)

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

The direction ratios of the perpendicular from the point \((1,0,1)\) to the plane \(x-y+z=0\) are :

CUET 2023 Hard

Match List-I with List-II

List I (Function)	List II (Derivative with respect to ' x ')
(A) \(f(x)=x^x\)	(I) \(a x^{a-1}\)
(B) \(f(x)=a^x\)	(II) 0
(C) \(f(x)=a^a\)	(III) \(a^x \log _e a\)
(D) \(f(x)=x^a\)	(IV) \(x^x\left(1+\log _e x\right)\)

Choose the correct answer from the options given below:

CUET 2025 Medium

Match List - I with List - II.

LIST I Differential equation	LIST II Integrating Factor (I.F)
A. \(\frac{d y}{d x}-\frac{y}{x}=2 x\)	I. \(x^2-1\)
B. \(\frac{d y}{d x}+\left(\frac{2 x}{x^2-1}\right) y=\frac{2}{\left(x^2-1\right)^2}\)	II. \(\sqrt{1-x^2}\)
C. \(\frac{d y}{d x}-\left(\frac{x}{1-x^2}\right) y=\frac{1}{1-x^2}\)	III. \(\frac{1}{x}\)
D. \(\frac{d y}{d x}+\frac{2 x y}{1+x^2}=\frac{\cot x}{1+x^2}\)	IV. \(1+x^2\)

Choose the correct answer from the options given below:

CUET 2023 Hard

Match List-I with List-II

List-I	List-II
(A) \(\int \frac{d x}{x^2-16}\)	(I) \(\frac{1}{8} \log \left\|\frac{4+x}{4-x}\right\|+c\)
(B) \(\int \frac{d x}{x^2+16}\)	(II) \(\log \left\|x+\sqrt{x^2-16}\right\|+c\)
(C) \(\int \frac{d x}{16-x^2}\)	(III) \(\frac{1}{8} \log \left\|\frac{x-4}{x+4}\right\|+c\)
(D) \(\int \frac{d x}{\sqrt{x^2-16}}\)	(IV) \(\frac{1}{4} \tan ^{-1}\left(\frac{x}{4}\right)+c\)

Choose the Correct answer from the options given below :

CUET 2025 Medium

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