JEE Advanced Maths: Sets questions with solutions

Q1. In a chemistry class, there are 20 students, while a physics class has 30 students. If 10 students are enrolled in both classes, and the two classes are held at separate times, determine the value of k/8, where k represents the total number of students attending either class.

1. 5
2. 6
3. 7
4. 8

Answer: 5

Students in either class = 20 + 30 - 10 = 40 by inclusion-exclusion, so k = 40 and k/8 = 5. The stored answer (6) is wrong; correct value is 5.

Q2. Given that A represents the divisors of 15, B contains prime numbers less than 10, and C includes even numbers less than 9, how many elements are there in the intersection of (A ∪ C) and B?

1. 1
2. 2
3. 3
4. 4

Answer: 3

A (divisors of 15) = {1,3,5,15}, B (primes <10) = {2,3,5,7}, C (even <9) = {2,4,6,8}. A U C = {1,2,3,4,5,6,8,15}; intersecting with B gives {2,3,5}, which has 3 elements (index 2). The stored answer (1) is wrong.

Q3. Consider the set S = {a + b√2: a, b ∈ Z}, and the sets T₁ = {⌊(−1 + √2)ⁿ⌋: n ∈ N} and T₂ = {⌊(1 + √2)ⁿ⌋: n ∈ N}. Which of the following statements is/are correct?

1. The union of Z, T₁, and T₂ is a subset of S.
2. The intersection of T₁ with {0, 1/2024} is the empty set.
3. The set T₂ has elements greater than 2024.
4. For any integers a and b, the expression cos(π(a + b√2)) + i sin(π(a + b√2)) belongs to the integers if and only if b equals 0, where i is the imaginary unit.

Answer: The union of Z, T₁, and T₂ is a subset of S.

The union of Z, T₁, and T₂ is a subset of S because the elements of T₁ and T₂ are of the form ⌊(−1 + √2)ⁿ⌋ and ⌊(1 + √2)ⁿ⌋, which can be expressed as a + b√2, where a and b are integers.

Q4. If set A contains 3 elements and set B contains 6 elements, what is the minimum number of elements in A union B?

1. 3
2. 6
3. 9
4. 18

Answer: 6

By inclusion-exclusion, |A union B| = |A| + |B| - |A intersect B|. This is minimized when the intersection is largest. Since |A|=3, the maximum intersection is 3 (A subset of B), giving |A union B| = 3 + 6 - 3 = 6.

Q5. Let A = {1, 2, 3, 4, 5} and B = {2, 3, 6, 7}. Find the number of elements in (A x B) ∩ (B x A).

1. 18
2. 6
3. 4
4. 0

Answer: 4

A ∩ B = {2, 3}. The set (A x B) ∩ (B x A) consists of all pairs (x, y) where x and y both belong to A ∩ B = {2, 3}. The number of such pairs = |A ∩ B|² = 2² = 4.

Q6. In a school, 21 students are in the cricket team, 26 in the hockey team, and 29 in the football team. Among them, 14 play both hockey and cricket, 15 play both hockey and football, and 12 play both football and cricket. Eight students play all three games. If k is the total number of members across the three athletic teams, find the sum of digits of k.

1. 6
2. 7
3. 8
4. 9

Answer: 7

Using the principle of inclusion-exclusion: k = |C| + |H| + |F| - |C∩H| - |H∩F| - |C∩F| + |C∩H∩F| = 21 + 26 + 29 - 14 - 15 - 12 + 8 = 76 - 41 + 8 = 43. Sum of digits of 43 = 4 + 3 = 7.

Q7. Given n(U) = 600, n(A) = 100, n(B) = 200, n(A intersection B) = 50, where U is the universal set. Find n(A-complement intersection B-complement).

1. 300
2. 350
3. 250
4. 200

Answer: 350

De Morgan's law: (A intersection B)' = A' union B', and (A union B)' = A' intersection B'. Here A' intersection B' = (A union B)'. n(A union B) = 100 + 200 - 50 = 250. n(A' intersection B') = 600 - 250 = 350.

Q8. Let A be the set of all real solutions of x(x² + 3|x| + 5|x - 1| + |6x - 2|) = 0, and B be the set of all real solutions of x² - |x| - 12 = 0. How many subsets does the set A x B have?

1. 2
2. 4
3. 8
4. 16

Answer: 8

For set A: x(x² + 3|x| + 5|x-1| + |6x-2|) = 0. Either x = 0, or the bracket = 0. For x > 0: bracket = x² + 3x + 5(x-1) + (6x-2) = x² + 14x - 7 for x > 1/3. Discriminant = 196 + 28 = 224 > 0 but both roots are irrational and checking signs shows no positive real root makes bracket zero... Actually let me recheck. For x > 1: bracket = x² + 3x + 5(x-1) + (6x-2) = x² + 14x - 7. Setting = 0: x = (-14 + sqrt(224))/2 which is negative, so no positive root > 1. For 0 < x < 1/3: bracket = x² + 3x + 5(1-x) + (2-6x) = x² - 8x + 7. At x approaching 0+: bracket approaches 7 > 0. For 1/3 < x < 1: bracket = x² + 3x + 5(1-x) + (6x-2) = x² + 4x + 3 > 0. So bracket > 0 for all x > 0. By symmetry or direct check, bracket > 0 for x < 0 too. Hence A = {0}. For B: |x|² - |x| - 12 = 0, so t² - t - 12 = 0, (t-4)(t+3) = 0, t = 4 (since t >= 0), so |x| = 4, B = {-4, 4}. A x B = {(0, -4), (0, 4)}, which has 2 elements. Number of subsets = 2² = 4.

Q9. Two sets A and B satisfy n(A) = 150, n(B) = 250, and n(A union B) = 300. Find (n(B - A) - n(A - B)) / 100.

1. 0
2. 1
3. 2
4. 3

Answer: 1

n(B-A) - n(A-B) = [n(B) - n(A intersect B)] - [n(A) - n(A intersect B)] = n(B) - n(A) = 250 - 150 = 100. Therefore (n(B-A) - n(A-B))/100 = 100/100 = 1.

Q10. Members of three school athletic teams: 21 in cricket, 26 in hockey, 29 in football. Of these, 14 play both hockey and cricket, 15 play both hockey and football, 12 play both football and cricket, and 8 play all three sports. Find the total number of members across all three teams.

1. 43
2. 40
3. 45
4. 43

Answer: 43

The inclusion-exclusion principle for three sets gives the total count of distinct members across all three teams.

Q11. A zoo has 6 Bengal white tigers and 6 Bengal royal tigers (12 total). Of these 12 tigers, 5 are males and 10 are either Bengal royal tigers or males (or both). Find the number of female Bengal white tigers.

1. a
2. b
3. c
4. d

Answer: a

Using set theory: the count of tigers that are royal OR male is 10. With 6 royal and 5 male, inclusion-exclusion gives 1 royal male tiger. The remaining 4 males are white tigers. White tigers total 6, so 6-4 = 2 are female white tigers.

Q12. If A = {1, 2, 4}, B = {2, 4, 5}, C = {2, 5}, find (A - C) x (B - C).

1. {(1, 4)}
2. {(1, 4), (4, 4)}
3. {(4, 1), (4, 4)}
4. {(1, 2), (2, 5)}

Answer: {(1, 4), (4, 4)}

A - C = {1, 2, 4} {2, 5} = {1, 4}. B - C = {2, 4, 5} {2, 5} = {4}. (A-C) x (B-C) = {1,4} x {4} = {(1,4), (4,4)}.

Q13. A survey shows that 73% of people in an office like coffee and 65% like tea. If x denotes the percentage who like both coffee and tea, which of the following values of x is NOT possible?

1. 63
2. 38
3. 54
4. 86

Answer: 86

From the inclusion-exclusion principle: P(C union T) = 73 + 65 - x <= 100, so x >= 38. Also x cannot exceed either set: x <= min(73,65) = 65. Therefore 38 <= x <= 65. Checking options: 63 in [38,65] — valid; 38 in [38,65] — valid (boundary); 54 in [38,65] — valid; 86 > 65 — NOT valid.

Q14. Let A and B be two sets with n(A) = 70, n(B) = 60, and n(A union B) = 110. Find n(A intersection B).

1. 240
2. 20
3. 100
4. 120

Answer: 20

By inclusion-exclusion: n(A intersection B) = n(A) + n(B) - n(A union B) = 70 + 60 - 110 = 20.

Q15. Given A = {2, 3, 4, 8, 10}, B = {3, 4, 5, 10, 12}, C = {4, 5, 6, 12, 14}, find the set (A ∩ B) ∪ (A ∩ C).

1. {3, 4, 10}
2. {2, 8, 10}
3. {4, 5, 6}
4. {3, 5, 14}

Answer: {3, 4, 10}

By the distributive law, (A ∩ B) ∪ (A ∩ C) = A ∩ (B ∪ C). B ∪ C = {3,4,5,6,10,12,14}. A ∩ (B ∪ C) = {3,4,10}.

Q16. For any two sets A and B, the complement (A union B)' is equal to which of the following?

1. A' intersect B'
2. U minus (A union B)
3. ((A minus B) union (B minus A))'
4. A' intersect (U minus B)

Answer: A' intersect B'

By De Morgan's Law, the complement of a union equals the intersection of the complements: (A union B)' = A' intersect B'.

Q17. Let A = {a, e, i, o, u} and B = {i, o}. Which of the following statements is true?

1. A is a subset of B
2. B is a subset of A
3. A equals B
4. A is equivalent to B

Answer: B is a subset of A

Every element of B (namely i and o) is also an element of A, so B is a proper subset of A; the reverse does not hold since A has elements not in B.

Q18. Let A = {a, b, c, d, e}. Find the total number of proper subsets of A.

1. 16
2. 31
3. 32
4. 63

Answer: 31

A set with n elements has 2ⁿ subsets in total (including the empty set and the set itself). A proper subset is any subset that is not equal to the set itself. For A = {a, b, c, d, e} with |A| = 5: total subsets = 2⁵ = 32. Proper subsets = 32 - 1 = 31.

Q19. In a universal set U with n(U) = 600, two subsets A and B satisfy: n(A) = 200, n(A∩B) = 100, and n(A^c ∩ B^c) = 200. Find n(A^c ∩ B).

1. 100
2. 150
3. 200
4. 300

Answer: 200

By De Morgan: n(A^c ∩ B^c) = n(U) - n(A∪B) -> n(A∪B) = 600 - 200 = 400. From inclusion-exclusion: n(B) = n(A∪B) - n(A) + n(A∩B) = 400 - 200 + 100 = 300. Finally, n(A^c ∩ B) = n(B) - n(A∩B) = 300 - 100 = 200.

Q20. In a group of 40 students, 26 drink tea, 18 drink coffee, and 8 drink neither. How many students drink only tea or only coffee (but not both)?

1. 5
2. 12
3. 20
4. 32

Answer: 20

32 students drink tea or coffee. Those who drink both = 26+18-32 = 12. Only tea = 26-12 = 14; only coffee = 18-12 = 6; total only one = 14+6 = 20.

Q21. Which of the following is a null (empty) set?

1. A = {x: x > 1 and x < 1}
2. B = {x: x + 3 = 3}
3. C = {phi}
4. D = {x: x >= 1 and x <= 1}

Answer: A = {x: x > 1 and x < 1}

Set A requires x to be simultaneously greater than 1 and less than 1, which is impossible for any real number, making A the empty set.

Q22. Write the roster (tabular) form of the set A = {x: x is a natural number and x² < 30}.

1. {1, 2, 3, 4, 5}
2. {0, 1, 2, 3, 4, 5}
3. {-5, -4, -3,.... 4, 5}
4. None of these

Answer: {1, 2, 3, 4, 5}

Natural numbers satisfying x² < 30 are 1, 2, 3, 4, 5 (since 5² = 25 < 30 and 6² = 36 > 30). Zero is not a natural number in the standard Indian curriculum definition.

Q23. If n(A) = 10, n(B) = 15, and n(A union B) = 23, what is n(A intersection B)?

1. 1
2. 2
3. 3
4. 4

Answer: 2

By inclusion-exclusion: n(A intersection B) = 10 + 15 - 23 = 2.

Q24. Let A and B be two sets. Simplify the expression (A union B)' union (A' intersection B).

1. A'
2. A
3. B'
4. None of these

Answer: A'

Using De Morgan's law and set algebra: (A union B)' union (A' intersection B) = (A' intersection B') union (A' intersection B) = A' intersection (B' union B) = A' intersection U = A'.

Q25. Let A and B be two sets with |A| = 4 and |B| = 2. Find the number of subsets of A x B that each contain at least three elements.

1. 275
2. 510
3. 219
4. 256

Answer: 219

A x B has 4*2 = 8 elements. Subsets with at least 3 elements = 2⁸ - C(8,0) - C(8,1) - C(8,2) = 256 - 1 - 8 - 28 = 219.

Q26. Set A contains m elements. B and C are two subsets of A such that n(B) = n(C), B union C = A, and n(B intersection C) = 1. Identify the correct statement(s).

1. m must be odd
2. m must be even
3. Number of ways in which subset B and C are to be formed is m * C((m-1), (m-1)/2)
4. Number of ways in which subsets B and C are to be formed is 1/2 * C((m-1), (m-1)/2)

Answer: m must be odd

From n(B union C)=m and n(B)=n(C)=k, n(B cap C)=1: m=2k-1, so m must be odd. The number of (ordered) ways to form B and C: choose the 1 element in B cap C (m ways), then split remaining m-1 elements into two equal groups of (m-1)/2 each: C(m-1,(m-1)/2) ways. Total = m * C(m-1,(m-1)/2). If B and C are unordered (B,C same as C,B), divide by 2 but since B and C are labeled subsets, the ordered count applies.

Q27. Two finite sets P and Q have m and n elements respectively. The total number of subsets of P exceeds the total number of subsets of Q by 56. Find the value of (m + n).

1. 13
2. 9
3. 10
4. 15

Answer: 9

The number of subsets of a set with k elements is 2^k. Setting 2^m - 2ⁿ = 56 and factoring gives 2ⁿ(2^(m-n) - 1) = 8*7, so n=3 and m=6, hence m+n=9.

Q28. Given P(A) = 3/4, P(A intersection B intersection C complement) = 1/3, and P(A intersection B complement intersection C complement) = 1/3. Find P(A intersection C).

1. 1/6
2. 1/15
3. 1/12
4. 1/9

Answer: 1/12

P(A) = P(A*B*C) + P(A*B*C') + P(A*B'*C) + P(A*B'*C') = 3/4. Subtracting the two given terms: P(A*B*C) + P(A*B'*C) = 3/4 - 1/3 - 1/3 = 1/12. This sum equals P(A*C).

Q29. If A and B are two subsets of the universal set U, then (A - B) union (B - A) union (A complement intersection B complement) equals:

1. A complement union B complement
2. (A union B) complement
3. empty set
4. U (where U is universal set and A and B are subsets of U)

Answer: A complement union B complement

(A-B)u(B-A)u(A'*B') = {only in A} u {only in B} u {neither in A nor B} = U minus {in both A and B} = (A intersection B)' = A' union B' by De Morgan's law.

Q30. Set A has m elements. B and C are subsets of A with n(B) = n(C), B union C = A, and n(B intersect C) = 1. Identify all correct statements.

1. m must be odd
2. m must be even
3. Number of ordered pairs (B,C) is m * ^(m-1)C_((m-1)/2)
4. Number of ordered pairs (B,C) is (1/2) * m * ^(m-1)C_((m-1)/2)

Answer: m must be odd

From n(B)+n(C)-n(B intersect C)=m with n(B)=n(C)=k and n(B intersect C)=1: 2k-1=m, so m is odd. Counting: choose 1 element for the intersection (m ways), then split the remaining m-1 elements equally into B-only and C-only sets: C(m-1,(m-1)/2) ways. Total = m*C(m-1,(m-1)/2) ordered pairs.

Q31. For two sets A and B (subsets of universal set U), the expression (A - B) union (B - A) union complement(A intersection B) is equal to:

1. complement(A) union complement(B)
2. complement(A union B)
3. empty set (phi)
4. U (universal set)

Answer: complement(A) union complement(B)

We have (A-B) union (B-A) = (A union B) - (A intersection B) which lies entirely within (A intersection B)^c. So their union equals (A intersection B)^c = A^c union B^c by De Morgan's law.

Q32. Two finite sets A and B have m and n elements respectively. The total number of subsets of A exceeds the total number of subsets of B by 56. Find the value of m + n.

1. 13
2. 9
3. 10
4. 15

Answer: 9

2^m - 2ⁿ = 56. Factor out 2ⁿ: 2ⁿ*(2^(m-n)-1) = 56 = 8*7. Since 2^(m-n)-1 must be odd (= 7) and 2ⁿ = 8, we get n=3 and m-n=3 => m=6. So m+n = 9.