CUET · MATHS · PYQ PAPER 2023

The function \(f: R \rightarrow R , f(x)=x^2\) is

A injective but not surjective
B surjective but not injective
C injective as well as surjective
D neither injective nor surjective

Verified Solution

Answer & Solution

Correct Answer

(D) neither injective nor surjective

Step-by-step Solution

Detailed explanation

For injectivity: \(f(-1) = f(1) = 1\), but \(-1 \neq 1\). Thus, \(f\) is not injective. For surjectivity: For any \(y Therefore, \(f\) is neither injective nor surjective.

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.

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(A) \(\int_0^3[x] d x\)	(I) \(\frac{1}{2}\)
(B) \(\int_0^1[2 x] d x\)	(II) 1
(C) \(\int_0^1[3 x] d x\)	(III) \(\frac{3}{2}\)
(D) \(\int_0^1[4 x] d x\)	(IV) 3