JEE Mathematics Sets MCQs Set 01

Answer: (b) {2, 3}

Explanation:
1. x² − 5x + 6 = (x − 2)(x − 3).
2. The roots are x = 2 and x = 3.

Teacher's Note:
1. Solve the condition in the set-builder rule first.
2. Check each root in the equation.

Answer: (a) 5

Explanation:
1. The integers satisfying −2 < x ≤ 3 are −1, 0, 1, 2, 3.
2. That is 5 elements.

Teacher's Note:
1. −2 is excluded (strict inequality) but 3 is included.
2. Do not forget 0.

Answer: (d) 16

Explanation:
1. A set with n elements has 2ⁿ subsets.
2. 2⁴ = 16 (this includes the empty set and the set itself).

Teacher's Note:
1. Each element is either in or out of a subset.
2. Proper subsets number 2ⁿ − 1.

Answer: (c) 7

Explanation:
1. Total subsets = 2³ = 8.
2. A proper subset excludes the set itself, so 8 − 1 = 7.

Teacher's Note:
1. Proper subsets exclude only the set itself; the empty set is counted.
2. General formula 2ⁿ − 1.

Answer: (b) {3, 4}

Explanation:
1. Intersection means the elements that belong to both sets.
2. Only 3 and 4 appear in both.

Teacher's Note:
1. A ∩ B keeps common elements only.
2. {1, 2, 3, 4, 5, 6} is the union.

Answer: (a) 6

Explanation:
1. A ∪ B = {1, 2, 3, 4, 5, 6}.
2. Or n(A ∪ B) = 4 + 4 − 2 = 6.

Teacher's Note:
1. Common elements are counted once in a union.
2. Use n(A) + n(B) − n(A ∩ B).

Answer: (d) 27

Explanation:
1. n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
2. = 20 + 15 − 8 = 27.

Teacher's Note:
1. Subtract the overlap so common elements are not counted twice.
2. The union can never exceed n(A) + n(B).

Answer: (c) B

Explanation:
1. Every element of A already lies in B.
2. So joining A to B adds nothing new and A ∪ B = B.

Teacher's Note:
1. Draw A inside B in a Venn diagram.
2. In the same situation A ∩ B = A.

Answer: (b) 4

Explanation:
1. x² = 9 gives x = 3 or x = −3, so the set is {−3, 3}.
2. A set with 2 elements has 2² = 4 subsets.

Teacher's Note:
1. Do not forget the negative root.
2. Subsets: ∅, {−3}, {3}, {−3, 3}.

Answer: (a) ∅

Explanation:
1. The empty set has no element, so nothing can be common to A and ∅.
2. Hence A ∩ ∅ = ∅.

Teacher's Note:
1. {0} is a set with one element, not the empty set.
2. A ∪ ∅ = A, but A ∩ ∅ = ∅.

Answer: (d) The set of even integers and the set of odd integers

Explanation:
1. No integer is both even and odd, so their intersection is empty.
2. Primes and odd numbers share 3, 5, 7, …; multiples of 2 and 3 share 6, 12, …; N is inside Z.

Teacher's Note:
1. Disjoint sets have no common element.
2. Find one common element to show two sets are not disjoint.

Answer: (c) 11

Explanation:
1. The common element is 10 only.
2. n(A ∪ B) = 4 + 8 − 1 = 11.

Teacher's Note:
1. List the divisors of 70: 1, 2, 5, 7, 10, 14, 35, 70.
2. Count the common element once.

Answer: (b) {2, 5, 7}

Explanation:
1. C = {2, 3, 5, 7, 11, 13}.
2. Elements of B that are also prime and below 17: 2, 5, 7.
3. 1 is not prime; 3, 11, 13 do not divide 70.

Teacher's Note:
1. Check each element of B for primality and size.
2. 1 is not a prime number.

Answer: (a) The set of months in a year

Explanation:
1. There are exactly 12 months.
2. The other three sets have no last element.

Teacher's Note:
1. A finite set has a countable end.
2. Points on a line are infinite.

Answer: (d) 8

Explanation:
1. Letters: M, A, T, H, E, M, A, T, I, C, S.
2. Distinct letters are M, A, T, H, E, I, C, S, which is 8.

Teacher's Note:
1. A set does not repeat elements.
2. M, A and T each appear twice but count once.

Answer: (c) {1, 2} ⊂ A

Explanation:
1. Every element of {1, 2} is in A and A has the extra element 3, so {1, 2} is a proper subset of A.
2. {1, 2} is not an element of A, and 4 ∉ A.

Teacher's Note:
1. Use ∈ between an element and a set, and ⊂ between sets.
2. {1, 2} ∈ A would need {1, 2} to be listed inside A.

Answer: (b) 5

Explanation:
1. A = {2, 4, 6, 8, 10}.
2. A′ = {1, 3, 5, 7, 9}, which has 5 elements.

Teacher's Note:
1. The complement is taken within U.
2. n(A′) = n(U) − n(A) = 10 − 5.

Answer: (a) 32

Explanation:
1. The power set has 2ⁿ elements.
2. 2⁵ = 32.

Teacher's Note:
1. Do not confuse 2ⁿ with n².
2. Each element has two choices: in or out.

Answer: (d) 6

Explanation:
1. 2ⁿ = 64 = 2⁶.
2. So n = 6.

Teacher's Note:
1. Write 64 as a power of 2.
2. Check: 2⁶ = 64.

Answer: (c) 5

Explanation:
1. x = 1, 2, 3, 4, 5 give squares 1, 4, 9, 16, 25, all below 30.
2. 6² = 36 is not below 30.

Teacher's Note:
1. Natural numbers start at 1.
2. Test x = 5 and x = 6.