CBSE Class 12 Mathematics 2026 Main Paper 65/4/1 · Q263 Marksinjectivity_surjectivity
CBSE Class 12 Mathematics 2026 Main Paper 65/4/1 · Question 26 (Relations And Functions)
Is f:ℝ∖{3}→ℝ∖{1}, f(x)=(x−2)/(x−3), one-one and onto? OR For a fixed positive integer n, determine whether xRy if n divides x−y defines an equivalence relation on ℤ.
Alternative 1 3 Marks
Is f:ℝ∖{3}→ℝ∖{1}, f(x)=(x−2)/(x−3), one-one and onto?
Answer: Both.
- For any y≠1 solve y=(x−2)/(x−3) to obtain x=(3y−2)/(y−1).
- This value differs from 3 and substitution returns y.
- Each target has exactly one preimage, so f is bijective.
Alternative 2 3 Marks
For a fixed positive integer n, determine whether xRy if n divides x−y defines an equivalence relation on ℤ.
Answer: It is an equivalence relation.
- Reflexivity follows since n divides x−x=0.
- Symmetry follows by negating the divisible difference.
- Transitivity follows by adding x−y and y−z to get x−z.