Exercise 2.2 — Relations and Functions

1. 2.2.12 marksNCERT Class 11 Mathematics, Relations and Functions, Reprint 2026-27

   Let A={1,2,...,14}. Define R from A to A by R={(x,y): 3x-y=0, x,y in A}. Write down its domain, codomain and range.

   Hint. 3x-y=0 means y=3x -- find every x in A for which 3x also lands inside A.

   y=3x, and y must also be in A (so y<=14), which restricts x to 1,2,3,4 (since 3x5=15 is already too large). This gives the pairs (1,3),(2,6),(3,9),(4,12).

   ✦ Working through each part gives: r={(1,3),(2,6),(3,9),(4,12)}. Domain={1,2,3,4}. Codomain=A={1,...,14}. Range={3,6,9,12}.

2. 2.2.22 marksNCERT Class 11 Mathematics, Relations and Functions, Reprint 2026-27

   Define R on N by R={(x,y): y=x+5, x is a natural number less than 4; x,y in N}. Depict in roster form. Write the domain and range.

   Hint. x less than 4 restricts x to exactly three natural numbers -- work out y=x+5 for each.

   x can be 1, 2, or 3, giving y=6, 7, 8 respectively.

   ✦ Working through each part gives: r={(1,6),(2,7),(3,8)}. Domain={1,2,3}. Range={6,7,8}.

3. 2.2.33 marksNCERT Class 11 Mathematics, Relations and Functions, Reprint 2026-27

   A={1,2,3,5}, B={4,6,9}. Define R from A to B by R={(x,y): x-y is odd}. Write R in roster form.

   Hint. Check every one of the 4x3=12 possible pairs for whether x-y comes out odd.

   Testing each pair systematically: x=1 gives odd differences with 4 and 6 but not 9; x=2 gives an odd difference only with 9; x=3 gives odd differences with 4 and 6 but not 9; x=5 gives odd differences with 4 and 6 but not 9.

   ✦ Working through each part gives: r = {(1,4),(1,6),(2,9),(3,4),(3,6),(5,4),(5,6)}.

4. 2.2.42 marksNCERT Class 11 Mathematics, Relations and Functions, Reprint 2026-27

   Fig 2.7 shows a relationship between P={5,6,7} and Q={3,4,5}, with arrows 5 to 3, 6 to 4, 7 to 5. Write this relation (i) in set-builder form (ii) in roster form. What is its domain and range?

   Hint. Compare each arrow's start and end value to spot the arithmetic pattern connecting them.

   Every arrow subtracts 2: 5 to 3, 6 to 4, 7 to 5, so the rule is y = x - 2.

   ✦ Working through each part gives: (i) R={(x,y): y=x-2, x in P, y in Q}. (ii) R={(5,3),(6,4),(7,5)}. Domain={5,6,7}. Range={3,4,5}.

5. 2.2.53 marksNCERT Class 11 Mathematics, Relations and Functions, Reprint 2026-27

   Let A={1,2,3,4,6}. R is defined on A by {(a,b): a,b in A, b is exactly divisible by a}. (i) Write R in roster form (ii) find the domain (iii) find the range.

   Hint. For each a in A, list every b in A that a divides exactly, including b=a itself.

   a=1 divides every element of A; a=2 divides 2,4,6; a=3 divides 3,6; a=4 divides only 4; a=6 divides only 6.

   ✦ Working through each part gives: r={(1,1),(1,2),(1,3),(1,4),(1,6),(2,2),(2,4),(2,6),(3,3),(3,6),(4,4),(6,6)}. Domain=A={1,2,3,4,6}. Range=A={1,2,3,4,6}.

6. 2.2.62 marksNCERT Class 11 Mathematics, Relations and Functions, Reprint 2026-27

   Determine the domain and range of R defined by R={(x, x+5): x in {0,1,2,3,4,5}}.

   Hint. Add 5 to each of the six given values of x to get the matching second element.

   x runs over 0 through 5, and each maps to x+5.

   ✦ Working through each part gives: r={(0,5),(1,6),(2,7),(3,8),(4,9),(5,10)}. Domain={0,1,2,3,4,5}. Range={5,6,7,8,9,10}.

7. 2.2.72 marksNCERT Class 11 Mathematics, Relations and Functions, Reprint 2026-27

   Write the relation R = {(x, x^3): x is a prime number less than 10} in roster form.

   Hint. List the primes below 10 first, then cube each one.

   The primes less than 10 are 2, 3, 5, 7; cubing each gives 8, 27, 125, 343 respectively.

   ✦ Working through each part gives: r = {(2,8),(3,27),(5,125),(7,343)}.

8. 2.2.82 marksNCERT Class 11 Mathematics, Relations and Functions, Reprint 2026-27

   Let A={x,y,z} and B={1,2}. Find the number of relations from A to B.

   Hint. A relation is any subset of AxB, so the count is 2 raised to the size of AxB.

   n(A)=3, n(B)=2, so n(AxB)=6, and the number of subsets of a 6-element set is 2^6.

   ✦ Working through each part gives: 2^6 = 64 relations.

9. 2.2.92 marksNCERT Class 11 Mathematics, Relations and Functions, Reprint 2026-27

   Let R be the relation on Z defined by R={(a,b): a,b in Z, a-b is an integer}. Find the domain and range of R.

   Hint. Since a and b are already required to be integers, check whether the stated condition on a-b restricts anything at all.

   The difference of any two integers is always itself an integer, so the condition a-b is an integer is automatically true for every pair of integers -- it excludes nothing.

   ✦ Working through each part gives: r is all of Z x Z; domain = Z and range = Z, since every integer pairs with every other integer.

Solutions written by the tuition.in editorial team and checked against the NCERT Class 11 Mathematics textbook, Reprint 2026-27 (kemh102.pdf) — three numbered exercises (2.1-2.3, 24 questions) plus the chapter's Miscellaneous Exercise (12 questions); Exercise 2.2 Q4's arrow-diagram figure (Fig 2.7) was rendered directly from the PDF at 250dpi and read visually to confirm the y=x-2 mapping, and the piecewise function boundaries in Miscellaneous Exercise Q1 were re-rendered at high resolution after the raw text extraction garbled the fraction/subscript layout, confirming f's pieces agree at x=3 (function) while g's pieces disagree at x=2 (not a function). Questions are referenced from the NCERT textbook for identification.