The chapter “Relations and Functions” is crucial for both board exams and competitive tests because it builds the foundation for understanding how mappings behave under composition, inverses, injectivity/surjectivity, and equivalence classes. Concepts like inverse functions, functional equations, and classification of relations directly appear in higher-difficulty problems across CBSE and entrance exams.
25
Minutes
20
Questions
1 / -0
Marking
Q1. Let be defined by . Which of the following is true?
is not injective.
is bijective and .
is injective but not surjective.
is surjective but not injective.
Q2. Let be defined by . The range of is
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Q3. Let be given by and let . Then
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Q4. Let be non-empty sets and , be functions. Consider the statements:
P: "If is bijective then is bijective."
Q: "If is bijective then is bijective." Which option correctly describes P and Q?
Both P and Q are true.
P is true but Q is false.
Both P and Q are false.
P is false but Q is true.
Q5. Let be defined by , where . For which real values of is one-to-one?
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Q6. Let be defined by . Find .
Q7. Let be given by . For let denote the -fold composition of with itself. Then, for (),
Q8. Let be defined by . Which of the following is true?
is bijective on
is injective but not surjective
is surjective but not injective
is neither injective nor surjective
Q9. Assertion (A): For functions and , if is bijective then is bijective.
Reason (R): If is bijective then has a left inverse (so is injective) and has a right inverse (so is surjective).
Both A and R are true and R is the correct explanation of A.
A is true but R is false.
A is false but R is true.
Both A and R are false.
Q10. Let be defined by . Then the inverse function is
Q11. Let . A function satisfies for all . How many such functions exist?
Q12. Let be defined by . Then equals:
Q13. Define a relation on by . How many distinct equivalence classes does have?
Q14. Let and be functions.
Statement (A): If both and are bijections then is a bijection.
Statement (R): If is a bijection then both and must be bijections.
A is true and R is false.
Both A and R are true, but R is not a correct explanation of A.
Both A and R are true and R is a correct explanation of A.
A is false and R is true.
Q15. Let be a set with . A function is idempotent if for all . How many idempotent functions exist from to ?
...and 5 more challenging questions available in the interactive simulator.