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Sets — JEE MCQs with answers

NCERT Class 11 · Mathematics · Ch 110 free questions530 in the full bank

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Question 1 · 1.10 Complement of a Set · Medium
If $U = \{a, b, c, d, e, f, g, h\}$ and $C = \{a, c, e, g\}$, then $C'$ is
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Answer: C — Option C is correct: removing $a, c, e, g$ from $U$ leaves $b, d, f, h$. Option A is wrong — this is $C$ itself. Option B is wrong — this omits $h$, which lies in $U$ and not in $C$. Option D is wrong — $c \in C$, so $c$ cannot appear in $C'$.

Class 11 · Mathematics · Ch 1 · §1.10 · p.20 · Exercise 1.5 Q2 ("U = {a, b, c, d, e, f, g, h}")

Question 2 · 1.2 Sets and their Representations · Hard
Write $\{1, 4, 9, \ldots, 100\}$ in set-builder form (Exercise 1.1 Question 4).
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Answer: B — B is correct: the members are the squares $1^2, 2^2, 3^2, \ldots$, and the listing stops at $100 = 10^2$, so $n$ runs from 1 to 10. A is Example 3's unbounded set of all squares, which does not stop at 100. C would run as far as $100^2 = 10000$. D describes the even numbers $\{2, 4, \ldots, 20\}$.

Class 11 · Mathematics · Ch 1 · §1.2 · p.5 · Exercise 1.1 Q4(v)

Question 3 · 1.9 Operations on Sets · Medium
Let $B = \{x : x \text{ is an even natural number}\}$ and $D = \{x : x \text{ is a prime number}\}$. Then $B \cap D$ is
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Answer: C — $2$ is the only even prime, so the only element common to $B$ and $D$ is $2$: $B \cap D = \{2\}$; option (C) is correct. Option (A) is wrong: $2$ is both even and prime, so the intersection is not empty. Option (B) is wrong: odd primes such as $3$ and $5$ are not even, so they are not in $B$. Option (D) is wrong: these even numbers are composite, so none of them is in $D$.

Class 11 · Mathematics · Ch 1 · §1.9.2 · p.15 · ("Definition 6 ... A ∩ B = {x : x ∈ A and x ∈ B}")

Question 4 · 1.6 Subsets · Medium
Here the first set is an **element** of the second: $A = \{5\}$, $B = \{\{5\}, 7\}$, $C = \{\{5\}, 7, 9\}$. Given that $B$ is contained in $C$, which conclusion about $A$ is justified?
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Answer: A — $A = \{5\}$ is listed as an element of $B$, so $A \in B$; every element of $B$ (namely $\{5\}$ and $7$) lies in $C$, so $B \subset C$. But $5 \in A$ while $5 \notin C$ (the elements of $C$ are $\{5\}, 7, 9$), so $A \not\subset C$. “$A \in B$, $B \subset C$ and $A \subset C$” fails at the last claim, for the reason just given — this is exactly the trap Example 11 of §1.6 warns about. “$A \subset B$” is wrong, since $5 \notin B$. “$B \in C$” is wrong: $B$ itself is not listed as an element of $C$.

Class 11 · Mathematics · Ch 1 · §1.6 · p.10 ("Example 11 Let A, B and C be three sets. If A ∈ B and B ⊂ C, is it true that A ⊂ C?")

Question 5 · 1.5 Equal Sets · Medium
In Example 7, $E = \{x : x \text{ is an integral positive root of the equation } x^2 - 2x - 15 = 0\}$. In roster form $E$ is:
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Answer: A — A is correct: $x^2 - 2x - 15 = 0$ gives $(x - 5)(x + 3) = 0$, so the roots are $5$ and $-3$; only $5$ is a positive integral root, hence $E = \{5\}$, as stated in the solution. B is wrong because $-3$ is negative and the definition of $E$ admits only positive roots. C is wrong for the same reason and additionally omits the qualifying root $5$. D is wrong because $E$ is not empty — the solution explicitly uses $E = \{5\}$ to conclude $C = E$.

Class 11 · Mathematics · Ch 1 · §1.5 · p.8 · Example 7 ("E = {x : x is an integral positive root of the equation x2 – 2x –15 = 0}" ; solution: "Since E = {5}, C = E.")

Question 6 · 1.4 Finite and Infinite Sets · Hard
If $P$ is the set of prime numbers less than 99, then $n(P)$ is:
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Answer: C — C is correct: the primes below 99 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 — exactly 25 numbers, so $n(P) = 25$. A is wrong: 24 omits one prime (a common slip is dropping 2, the only even prime). B is wrong: 26 counts one number too many — 91 $= 7 \times 13$ is composite, and 99 is in any case ruled out by the bound “less than 99”. D is wrong: the bound 'less than 99' makes the set finite, as Exercise 1.2 Q2(v) requires.

Class 11 · Mathematics · Ch 1 · Exercise 1.2 · p.8 · Q2(v) ("The set of prime numbers less than 99")

Question 7 · 1.8 Venn Diagrams · Easy
In Venn diagrams, where are the elements of the sets written?
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Answer: C — The text says "In Venn diagrams, the elements of the sets are written in their respective circles", so C is correct. A is wrong because listing elements outside the rectangle would place them outside the universal set. B is wrong because the elements are not written on the boundary but inside the region. D is wrong because the convention places elements inside the diagram, not in an accompanying table.

Class 11 · Mathematics · Ch 1 · §1.8 · p.13 · ¶2 ("In Venn diagrams, the elements of the sets are written in their respective circles (Figs 1.2 and 1.3)")

Question 8 · 1.1 Introduction · Easy
According to the Historical Note of the chapter, Cantor's papers on set theory appeared sometime during the period
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Answer: C — C is correct: the Historical Note states 'His papers on set theory appeared sometimes during 1874 to 1897.' A is wrong: 1845-1918 are Cantor's life dates, not the period of his papers. B is wrong: 1879 is the year from which, according to the Note, he published several papers showing various properties of abstract sets, and 1902 is the year of Russell's result; the pair is not the period given for his papers. D is wrong: 1908 and 1940 are the years of Zermelo's first axiomatisation and Gödel's monograph respectively, both long after Cantor's papers.

Class 11 · Mathematics · Ch 1 · Historical Note · p.22 · ¶1 ("His papers on set theory appeared sometimes during 1874 to 1897.")

Question 9 · 1.7 Universal Set · Easy
While studying the system of numbers, the text says we are interested in the set of natural numbers and its subsets such as
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Answer: C — C is correct: the text says 'we are interested in the set of natural numbers and its subsets such as the set of all prime numbers, the set of all even numbers, and so forth.' A is wrong: the irrationals and the reals are not subsets of the natural numbers, and they are not the examples given here. B is wrong: intervals and singleton sets belong to earlier sections and are not named in this sentence. D is wrong: the rationals and the integers are not subsets of $\mathbb{N}$ — in fact $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q}$ runs the other way.

Class 11 · Mathematics · Ch 1 · §1.7 Universal Set · p.12 · ¶1 ("…we are interested in the set of natural numbers and its subsets such as the set of all prime numbers, the set of all even numbers, and so forth.")

Question 10 · 1.3 The Empty Set · Easy
Which of the following sets contains no element at all?
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Answer: B — B is correct: the text observes that a student cannot study simultaneously in both Classes X and XI, so this set contains no element at all. A is wrong because the Class XI students of a school can be counted, and the text states this set is not an empty set. C is wrong because $\{1, 2, 3, 4, 5\}$ plainly has five elements. D is wrong because $\{a, b, c, d, e, g\}$ plainly has six elements.

Class 11 · Mathematics · Ch 1 · §1.3 · p.5–6 ("Thus, the set B contains no element at all... B is an empty set while A is not an empty set.")

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