CUET · MATHS · PYQ PAPER 2025

A function \(f: R \rightarrow\{x \in R:-1< x<1\}\) is defined as \(f(x)=\frac{x}{1+|x|}\), then \(f\) is :

A neither one-one nor onto
B one-one only
C onto only
D both one-one and onto

Verified Solution

Answer & Solution

Correct Answer

(D) both one-one and onto

Step-by-step Solution

Detailed explanation

For one-one: For \(x>0\), \(f'(x) = \frac{d}{dx}\left(\frac{x}{1+x}\right) = \frac{1}{(1+x)^2} > 0\). For \(x 0\). If \(x_1>0, x_2\(f\) is strictly increasing over its domain \(R\), thus \(f\) is one-one. For onto: Let \(y \in (-1,1)\). We find \(x\) such that \(f(x)=y\). If…

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 (where \(c\) is an arbitrary constant)

List-I	List-II
(A) \(\int \tan x d x\)	(I) \(\log \|\sec x+\tan x\|+c\)
(B) \(\int \cot x d x\)	(II) \(\log \|\sec x\|+c\)
(C) \(\int \sec x d x\)	(III) \(\log \|\sin x\|+c\)
(D) \(\int \operatorname{cosec} x d x\)	(IV) \(\log \|\operatorname{cosec} x-\cot x\|+c\)

Choose the correct answer from the options given below :

CUET 2025 Easy

Explore more questions on app

From CUET

Explore more questions on app