Exercise 1.2 — Sets

1. 1.2.12 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   Which of the following are examples of the null set? (i) odd natural numbers divisible by 2, (ii) even prime numbers, (iii) {x : x is a natural number, x < 5 and x > 7}, (iv) {y : y is a point common to any two parallel lines}.

   Hint. A set is null only when NO object at all can satisfy its defining condition -- check each one for a genuine contradiction.

   (i) no odd number is divisible by 2, so this set is empty. (ii) 2 is an even prime, so this set is {2}, not empty. (iii) no natural number is simultaneously less than 5 and greater than 7, so this set is empty. (iv) parallel lines never meet, so this set is empty.

   ✦ Working through each part gives: null sets: (i), (iii), (iv). Not null: (ii), since {2} has one element.

2. 1.2.22 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   State whether each is finite or infinite: (i) months of a year, (ii) {1,2,3,...}, (iii) {1,2,3,...,99,100}, (iv) positive integers greater than 100, (v) prime numbers less than 99.

   Hint. A set is finite exactly when counting its elements eventually stops.

   (i) exactly 12 months. (ii) the naturals continue without end. (iii) stops at 100. (iv) continues without end past 100. (v) a fixed, countable list of primes below 99.

   ✦ Working through each part gives: finite: (i),(iii),(v). Infinite: (ii),(iv).

3. 1.2.32 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   State whether each is finite or infinite: (i) lines parallel to the x-axis, (ii) letters in the English alphabet, (iii) multiples of 5, (iv) animals living on the earth, (v) circles passing through the origin.

   Hint. For geometric sets, ask whether infinitely many configurations satisfy the condition.

   (i) infinitely many horizontal lines exist at every height. (ii) exactly 26 letters. (iii) multiples of 5 continue without end. (iv) a large but definite, countable number at any moment. (v) infinitely many circles of every radius pass through the origin.

   ✦ Working through each part gives: finite: (ii),(iv). Infinite: (i),(iii),(v).

4. 1.2.43 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   In each case, state whether A = B: (i) A={a,b,c,d}, B={d,c,b,a}, (ii) A={4,8,12,16}, B={8,4,16,18}, (iii) A={2,4,6,8,10}, B={x : x is a positive even integer and x<=10}, (iv) A={x : x is a multiple of 10}, B={10,15,20,25,30,...}.

   Hint. Two sets are equal only if they contain exactly the same elements, regardless of order.

   (i) both list the same four letters, only reordered. (ii) A has 12 but not 18; B has 18 but not 12 -- they differ. (iii) B written out is exactly {2,4,6,8,10}, matching A. (iv) B contains 15, which is not a multiple of 10, so B is not the same set as A.

   ✦ Working through each part gives: a=B in (i) and (iii). A is not equal to B in (ii) and (iv).

5. 1.2.52 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   Are the following pairs of sets equal? Give reasons. (i) A={2,3}, B={x : x is a solution of x^2+5x+6=0}. (ii) A=letters in FOLLOW, B=letters in WOLF.

   Hint. Solve the equation in (i) before comparing; list distinct letters only in (ii).

   (i) x^2+5x+6=0 factors as (x+2)(x+3)=0, giving x=-2,-3, so B={-2,-3}, which does not match A={2,3}. (ii) the distinct letters of FOLLOW are F,O,L,W, and the distinct letters of WOLF are the same four letters.

   ✦ Working through each part gives: (i) Not equal. (ii) Equal.

6. 1.2.62 marksNCERT Class 11 Mathematics, Sets, Reprint 2026-27

   From A={2,4,8,12}, B={1,2,3,4}, C={4,8,12,14}, D={3,1,4,2}, E={-1,1}, F={0,a}, G={1,-1}, H={0,1}, select the equal sets.

   Hint. Order and repetition never matter for set equality -- rewrite each set as a plain list and compare.

   B={1,2,3,4} and D={3,1,4,2} contain exactly the same four numbers, just listed in a different order, so B=D. E={-1,1} and G={1,-1} contain exactly the same two numbers, so E=G. No other pair matches.

   ✦ Working through each part gives: b=D and E=G.

Solutions written by the tuition.in editorial team and checked against the NCERT Class 11 Mathematics textbook, Reprint 2026-27 (kemh101.pdf) — five numbered exercises (1.1-1.5, 39 questions total) plus the chapter's Miscellaneous Exercise (10 questions); every question solved directly from the given sets rather than from memory, including the multi-part true/false and subset-vs-membership questions in Exercise 1.3 Q3 and the Miscellaneous Exercise Q2, which require care to avoid conflating 'element of' with 'subset of'. Questions are referenced from the NCERT textbook for identification.