CUET · MATHS · PYQ PAPER 2023

Let \(R\) be a relation on the set of integers \(Z\) such that \(R=\left\{(a, b), a=2^k b, a, b, k \in Z\right\}\), then \(R\) is :

A Reflexive but not Symmetric and Transitive
B Symmetric and Reflexive but not Transitive
C Equivalence relation
D Reflexive and Transitive but not Symmetric

Verified Solution

Answer & Solution

Correct Answer

(C) Equivalence relation

Step-by-step Solution

Detailed explanation

Reflexivity: For any \(a \in Z\), \(a = 2^0 a\) holds with \(0 \in Z\). Thus \((a,a) \in R\). R is reflexive. Symmetry: Let \((a, b) \in R\). Then \(a = 2^k b\) for some \(k \in Z\). If \(a=0\), then \(b=0\), so \((0,0)\in R\). If \(a \ne 0\), then…

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 R is the set of real numbers)

List-I	List-II
(A) \(sin\) \(x\) is continuous on :	(I) \(R -\{0\}\)
(B) \(tan\) \(x\) is continuous on :	(II) \(R\)
(C) \(\cot x\) is continuous on :	(III) \(R -\{n \pi: n \in Z \}\)
(D) \(x^{-n}, n \in N\) is continuous on:	(IV) \(R -\left\{(2 n+1) \frac{\pi}{2}: n \in Z \right\}\)

Choose the correct answer from the options given below :

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