CUET · MATHS · PYQ PAPER 2025

Let f : R → R be defined as f(x) = [x], where [x] denotes the greatest integer less than or equal to x. Then which of the following statements are correct?
(A) f is one-one but not onto
(B) f is not onto
(C) f is not one-one
(D) f is one-one and onto
Choose the correct answer from the options given below:

A (B) and (C) only
B (A), (B) and (C) only
C (B), (C) and (D) only
D (C) and (D) only

Verified Solution

Answer & Solution

Correct Answer

(A) (B) and (C) only

Step-by-step Solution

Detailed explanation

\(f(x_1) = [x_1]\), \(f(x_2) = [x_2]\) For \(x_1 = 1.2, x_2 = 1.5\), \(f(x_1) = 1, f(x_2) = 1\). \(f(x_1) = f(x_2)\) but \(x_1 \neq x_2\). Thus, f is not one-one. (C) is correct. The range of \(f(x) = [x]\) is \(\mathbb{Z}\). The codomain is \(\mathbb{R}\). Since…

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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List-I	List-II
(A) \(P(x=2)\)	(I) \(\frac{4}{15}\)
(B) \(P(x=3)\)	(II) \(\frac{1}{15}\)
(C) \(P(x=4)\)	(III) \(\frac{2}{15}\)
(D) \(P(x=5)\)	(IV) \(\frac{1}{5}\)