CBSE Class 12 Mathematics Relations and Functions Assignment Set 21

This assignment has practice questions on Chapter 1 Relations and Functions for Class 12 CBSE Mathematics. Each question is followed by its answer from the answer key so students can check their work.

Question 1. Let L denote the set of all straight lines in a plane and a relation R in L defined by l R m if l is perpendicular to m for all l, m ∈ L. Check whether R is reflexive, symmetric or transitive.
Answer:
Symmetric

Question 2. Let R be the relation in the set A = {1, 2, 3, 4} defined by R = {(1, 2), (1, 1), (2, 3), (2, 2), (3, 1), (3, 3), (2, 1), (4, 4), (1, 3), (3, 2)}. Check whether R is equivalence relation or not.
Answer:
Equivalence

Question 3. State whether the function f : {–2, –1, 1, 2} → {4, 5, 6, 7, 8} defined as f(x) = x² + 4 is one-one onto.
Answer:
Many one into

Question 4. Show that the relation R defined on set A = {1, 2, 3} such that R = {(1, 1), (2, 2), (3, 3)} is equivalence relation.
Answer:
R is reflexive, symmetric and transitive, hence an equivalence relation.

Question 5. State whether the function f : N → N, given by f(x) = x – 1, ∀ x > 2 and f(1) = 1 = f(2) is bijective or not.
Answer:
Not bijective

Question 6. If f(x) = \( \frac{x - |x|}{|x|} \), then find f(–1).
Answer:
–2

Question 7. If R be the relation on N given by R = {(x, y) : x + 2y = 8}, then find domain of R.
Answer:
{2, 4, 6}

Question 8. Let A = {1, 2, 3, 4, 5, 6, 7, 8} and R = {(x, y) : y is half of x; x, y ∈ A} is a relation in A, then write R in roster form.
Answer:
R = {(2, 1), (4, 2), (6, 3), (8, 4)}

Question 9. Let R = {(a, a³) : a is prime number less than 5} be a relation. Then find range of R.
Answer:
{8, 27}

Question 10. Let R be an equivalence relation on set A = {x ∈ Z : 0 ≤ x ≤ 12} given by R = {(a, b) : |a – b| is a multiple of 4}, then find equivalence class of element 2 ∈ A.
Answer:
{2, 6, 10}

Question 11. If f(x) = \( \frac{x}{1-x} = \frac{1}{y} \), then find f(y) in terms of x.
Answer:
\( \frac{1 - x}{2x - 1} \)

Question 12. Show that the function f : R → R given by f(x) = ax + b, where a, b ∈ R, a ≠ 0 is bijective.
Answer:
f is one-one and onto, hence bijective.

Question 13. State whether the function f : R → R, f(x) = e^x, is bijective or not.
Answer:
Not bijective

Question 14. If f(x) = \( \log\left(\frac{1+x}{1-x}\right) \), then find the value of \( f\left(\frac{2x}{1+x^2}\right) \).
Answer:
2f(x)

Question 15. Let A = {1, 2, 3, ......., 9} and R be an equivalence relation on A × A defined by (a, b) R (c, d) ⇔ a + d = b + c, for all (a, b), (c, d) ∈ A × A, then find the equivalence class [(2, 5)].
Answer:
{(1, 4), (2, 5), (3, 6), (4, 7), (5, 8), (6, 9)}