Question 1

Let A be the set {(n, 2n): n ∈ N} and let B be the set {(2n, 3n): n ∈ N}. What is the intersection A ∩ B?

Accepted Answer

∅. The sets A and B consist of pairs where the first elements are defined differently, leading to no common pairs. Specifically, the first element of A is n while in B it is 2n, meaning there are no values of n that satisfy both conditions simultaneously.

Question 2

Let set A contain 3 elements and set B contain 6 elements. Then the cardinality of their union must satisfy

Accepted Answer

6 ≤ n(A ∪ B) ≤ 9. The union of sets A and B can have a minimum of 6 elements if all elements of A are also in B, and a maximum of 9 elements if all elements are distinct. Therefore, the cardinality of the union must fall within the range of 6 to 9.

Question 3

Given the sets A = {1, 2, 5} and B = {3, 4, 5, 9}, what is A ∩ B?

Accepted Answer

None of these. The intersection contains only elements common to both sets. A = {1,2,5} and B = {3,4,5,9} share only 5, so A intersect B = {5}. Since {5} is not among the options, the answer is 'None of these'.

Question 4

At a conference with 100 attendees, 29 are Indian women and 23 are Indian men. Among the Indian attendees, 4 are doctors, and 24 are either men or doctors. If there are no foreign doctors, how many foreigners and how many women doctors are present at the conference?

Accepted Answer

48, 1. Indians = 29 + 23 = 52, so foreigners = 100 - 52 = 48 (all doctors are Indian). For Indians, |men or doctors| = 24 = 23 + 4 - (male doctors), giving 3 male doctors, so women doctors = 4 - 3 = 1. Answer: 48 foreigners and 1 woman doctor.

Question 5

Let X and Y be two non-empty sets, and let A be a non-empty set such that X ∩ A = Y ∩ A = A and X ∪ A = Y ∪ A. Which of the following must be true?

Accepted Answer

X and Y are equal. X cap A = A means A is a subset of X, so X cup A = X. Similarly Y cup A = Y. The given X cup A = Y cup A then becomes X = Y, so the two sets must be equal.

Question 6

If A and B are non-empty sets with A containing B, then which of the following is true?

Accepted Answer

B' − A' = A − B. With B contained in A: B' - A' = B' intersect A = A intersect B' = A - B. Since B is a subset of A, B - A is empty, so option 'B'-A' = B-A' is false. The correct identity is B' - A' = A - B.

Question 7

In a city with 10,000 households, it is observed that 40% of the households subscribe to newspaper A, 20% subscribe to newspaper B, and 10% subscribe to newspaper C. Also, 5% subscribe to both A and B, 3% subscribe to both B and C, and 4% subscribe to both A and C. If 2% of the households

Accepted Answer

All of the above statements are true. With N=10000: A-only = A-AB-AC+ABC = 4000-500-400+200 = 3300; B-only = 2000-500-300+200 = 1400; union = 4000+2000+1000-500-300-400+200 = 6000, so none = 10000-6000 = 4000. All three statements hold, so the correct choice is 'All of the above are true'.

Question 8

In a conflict, 70% of the fighters lost one eye, 80% lost an ear, 75% lost an arm, and 85% lost a leg. If x% of them lost all four of these parts, what is the least possible value of x?

Accepted Answer

10. The minimum percentage losing all four = 100 - [(100-70)+(100-80)+(100-75)+(100-85)] = 100 - (30+20+25+15) = 100 - 90 = 10. So the least value of x is 10.

Question 9

Given a universal set U with n(U) = 700, n(A) = 200, n(B) = 300, and n(A ∩ B) = 100, what is the value of n(A' ∩ B')?

Accepted Answer

300. By De Morgan, A' n B' = (A u B)'. n(A u B) = 200 + 300 - 100 = 400, so n(A' n B') = 700 - 400 = 300.

Question 10

Consider the following statements:
Statement-1: If B = U A, then n(B) = n(U) − n(A), where U denotes the universal set.
Statement-2: For any three sets A, B and C, if C = A − B, then n(C) = n(A) − n(B).
Which of the following is correct?

Accepted Answer

Statement-1 is true, while Statement-2 is false.. If B is the complement of A then n(B) = n(U) - n(A), so Statement-1 is true. But for C = A - B, n(A-B) = n(A) - n(A intersect B), which equals n(A) - n(B) only when B is a subset of A; in general it is false. Hence Statement-1 true, Statement-2 false.

Question 11

In a class containing 80 students, labeled 1 to 80, every student with an odd number chooses Cricket, every student whose number is a multiple of 5 chooses Football, and every student whose number is a multiple of 7 chooses Hockey. How many students choose none of these three games?

Accepted Answer

28. Odd numbers (Cricket), multiples of 5 (Football) and multiples of 7 (Hockey) are chosen. A student picks none only if the number is even and not a multiple of 5 or 7. Counting 1-80 gives 28 such students.

Question 12

A group of 60 students contains 23 who play hockey, 15 who play basketball, and 20 who play cricket. Also, 7 students play both hockey and basketball, 5 play both cricket and basketball, 4 play both hockey and cricket, and 15 students do not play any of these three games. Which statement i

Accepted Answer

1 student plays hockey and cricket but not basketball. Union = 60 - 15 = 45. By inclusion-exclusion 45 = 23+15+20 -7-5-4 + n(all), giving n(all)=3. Then hockey-and-cricket-only = 4 - 3 = 1, so exactly 1 student plays hockey and cricket but not basketball.

Question 13

Let A denote the set of all divisors of 15, B the set of prime numbers less than 10, and C the set of even numbers less than 9. Then the set (A ∪ C) ∩ B is

Accepted Answer

{2, 3, 5}. The correct option includes the elements that are both in the union of the divisors of 15 and the even numbers less than 9, and also in the set of prime numbers less than 10. The divisors of 15 are {1, 3, 5, 15}, and the even numbers less than 9 are {0, 2, 4, 6, 8}. The union of these sets is {1, 2, 3, 4, 5, 6, 8, 15}, and the intersection with the prime numbers less than 10, which are {2, 3, 5, 7}, results in {2, 3, 5}.

Question 14

Two finite sets contain m and n elements, respectively. The first set has 112 more subsets than the second set. What are the values of m and n?

Accepted Answer

7, 4. The number of subsets of a set with m elements is given by 2^m, and for a set with n elements, it is 2ⁿ. The equation 2^m - 2ⁿ = 112 holds true for m = 7 and n = 4, as it results in 128 - 16 = 112.

Question 15

In a school, 30 students study both Mathematics and Chemistry. This common group is 10% of all Mathematics enrolments and 12% of all Chemistry enrolments. What is the total number of students who study at least one of these two subjects?

Accepted Answer

520. The total number of Mathematics students can be calculated as 30 is 10% of the total, leading to 300 Mathematics students. Similarly, for Chemistry, 30 is 12% of the total, resulting in 250 Chemistry students. Using the principle of inclusion-exclusion, the total number of students studying at least one subject is 300 + 250 - 30, which equals 520.

Question 16

Given two sets A and B with n(A) = 1000 and n(B) = 500, and with n(A ∩ B) at least 1, let n(A ∪ B) = p. Which range must p satisfy?

Accepted Answer

1000 ≤ p ≤ 1499. n(AuB) = 1000 + 500 - n(AnB). Since n(AnB) can be from 1 (max union) up to 500 (min union, B subset of A), p ranges from 1000 to 1499, i.e. 1000 <= p <= 1499.

Question 17

How many ordered pairs {a, b} with integers a and b satisfy 2a² + 3b² = 35, where Z denotes the set of all integers?

Accepted Answer

8. Testing b^2 in {0,1,4,9}: 3b^2 = 3 leaves 2a^2 = 32 -> a = +/-4, and 3b^2 = 27 leaves 2a^2 = 8 -> a = +/-2. So (a,b) = (+/-4,+/-1) and (+/-2,+/-3), giving 8 ordered integer pairs.

Question 18

Let A, B and C be finite sets with n(A)=10, n(B)=15, n(C)=20, n(A∩B)=8 and n(B∩C)=9. Which of the following can be the value of n(A∪B∪C)?

Accepted Answer

Any one of 26, 27, or 28 may occur. n(AUBUC) = 45 - 8 - 9 - n(A∩C) + n(A∩B∩C). Since n(A∩C) and n(A∩B∩C) are not fixed, testing feasible (nonnegative-region) values yields totals of 26, 27 and 28. Hence any one of 26, 27 or 28 may occur.

Question 19

In a city with 10,000 households, it is observed that 40% of the households purchase newspaper A, 20% purchase newspaper B, and 10% purchase newspaper C. Also, 5% purchase both A and B, 3% purchase both B and C, and 4% purchase both A and C. If 2% of the households subscribe to all three n

Accepted Answer

3300. To find the number of households that buy only newspaper A, we subtract those who buy A and B, A and C, and all three newspapers from the total purchasing A. This calculation shows that 3300 households exclusively purchase newspaper A.

Question 20

Consider the following assertions:

Assertion 1: If A ∪ B = A ∪ C and A ∩ B = A ∩ C, then B = C.
Assertion 2: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).

Choose the correct statement about these assertions.

Accepted Answer

Assertion 1 is true and Assertion 2 is true; Assertion 2 does not correctly explain Assertion 1.. Assertion 1 is a valid theorem: if A∪B=A∪C and A∩B=A∩C then B=C. Assertion 2 is the distributive law A∪(B∩C)=(A∪B)∩(A∪C), also true. Both are true, but the distributive law does not directly explain the cancellation result, so 'both true, A2 does not explain A1' is correct.

Question 21

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

Accepted Answer

4. (A×B)∩(B×A) consists of pairs (x,y) with x,y in both A and B, i.e. in A∩B = {2,3}. So the count is |A∩B|^2 = 2^2 = 4.

Question 22

Given that the number of elements in set A is 4 and in set B is 3, and the Cartesian product A × B × C has 24 elements, what is the number of elements in set C?

Accepted Answer

2. |A x B x C| = 4 * 3 * |C| = 24, so |C| = 24/12 = 2.

Question 23

Let S = {1, 2, 3, 4, 5}. If R = {(x, y): x + y < 6}, then how many ordered pairs are in R?

Accepted Answer

10. For x+y<6 with x,y in {1..5}: x=1 gives y=1..4 (4), x=2 gives y=1..3 (3), x=3 gives y=1,2 (2), x=4 gives y=1 (1), x=5 gives none. Total = 4+3+2+1 = 10.

Question 24

If the ordered pairs (1, 3), (2, 5), and (3, 3) belong to A × B, and A × B contains exactly 6 elements in all, which of the following are the other elements of A × B?

Accepted Answer

(1, 5), (2, 3), (3, 5). The correct option includes pairs that combine the elements of sets A and B in a way that maintains the total of 6 unique ordered pairs, while also ensuring that the first elements (from set A) and the second elements (from set B) are consistent with the given pairs.

Question 25

Determine the collection of real numbers x for which both |x − 1| ≤ 3 and |x − 1| ≥ 1 hold simultaneously.

Accepted Answer

[−2, 0] ∪ [2, 4]. |x-1|<=3 gives -2<=x<=4. |x-1|>=1 gives x<=0 or x>=2. Intersection is [-2,0] U [2,4].

Question 26

For real x, the solution set of the inequality |x - |x - 1| - 1| ≤ 1 is

Accepted Answer

None of these. For x>=1, x-|x-1|-1 = x-(x-1)-1 = 0, so |0|<=1 holds for all x>=1. For x<1, the expression is |2x-2|<=1 giving 1/2<=x<1. Combining, the solution set is [1/2, infinity), which is none of the listed finite intervals.

Question 27

For which real values of x does the inequality log base 0.2 of ((x + 2)/x) ≤ 1 hold true?

Accepted Answer

(-∞, -5/2] ∪ (0, ∞). The inequality log base 0.2 of ((x + 2)/x) ≤ 1 holds true for values of x where the expression inside the logarithm is positive and the logarithm itself is less than or equal to 1. This occurs for x in the intervals (-∞, -5/2] and (0, ∞), as these intervals ensure the argument of the logarithm is valid and satisfies the inequality.

Question 28

How many squares in total can be counted on a standard chessboard?

Accepted Answer

204. Number of axis-aligned squares on an 8x8 board = 1^2+2^2+...+8^2 = 8*9*17/6 = 204.

Question 29

A set has (2n + 1) members. If the total number of its subsets having no more than n elements is 4096, then what is the value of n?

Accepted Answer

6. For a set of 2n+1 elements, by symmetry C(2n+1,k)=C(2n+1,2n+1-k), the number of subsets with at most n elements is exactly half of 2^(2n+1), i.e. 2^(2n) = 4^n. Setting 4^n = 4096 = 4^6 gives n = 6.

Question 30

A class contains 125 students. Out of these, 70 cleared Mathematics, 55 cleared Statistics, and 30 cleared both subjects. If one student is chosen at random from the class, what is the probability that the student has cleared exactly one of the two subjects?

Accepted Answer

13/25. Only Maths = 70-30 = 40; only Statistics = 55-30 = 25; exactly one = 65. Probability = 65/125 = 13/25.

Question 31

On the set {0, 1, 2, 3, 4, 5}, a binary operation * is defined by a * b = a + b when a + b < 6, and a * b = a + b - 6 when a + b ≥ 6. Which element acts as the identity for this operation?

Accepted Answer

0. For the identity e we need a*e=a. Since a+0=a<6 for every a in {0..5}, a*0=a. Hence 0 is the identity element.

Question 32

Find the area enclosed by the graphs of y = |x - 1| and y = 3 - |x|.

Accepted Answer

4 square units. The curves y=|x-1| and y=3-|x| enclose a region; integrating (3-|x|)-|x-1| over where it is positive (from x=-1 to x=2) gives total area 4 square units.

Question 33

If A, B and C are three sets such that A ∩ B = A ∩ C and A ∪ B = A ∪ C, then

Accepted Answer

B = C. The conditions A ∩ B = A ∩ C and A ∪ B = A ∪ C imply that the elements of B and C must be identical when considering their relationship with A. Since both intersections and unions yield the same results, it follows that B and C must contain the same elements, leading to the conclusion that B = C.

Question 34

Let X = {1,2,3,4,5}. The number of different ordered pairs (Y,Z) that can formed such that Y ⊆ X, Z ⊆ X and Y ∩ Z is empty is:

Accepted Answer

3⁵. Each element of set X can either belong to set Y, set Z, or neither, but not both. Since there are 5 elements in X, and each element has 3 independent choices (in Y, in Z, or in neither), the total number of combinations is 3 raised to the power of 5.

Question 35

Let S = {x ∈ R: x ≥ 0 and 2|√x − 3| + √x(√x − 6) + 6 = 0}. Then S:

Accepted Answer

contains exactly two elements.. The equation simplifies to a quadratic form that can be analyzed for its roots. Upon solving, it reveals two valid solutions for x that satisfy the conditions of the set S, confirming that it contains exactly two elements.

Question 36

If a ∈ R and the equation −3(x − [x])² + 2(x − [x]) + a² = 0, (where [x] denotes the greatest integer ≤ x) has no integral solution, then all possible values of a lie in the interval:

Accepted Answer

(−1, 0) ∪ (0, 1). The equation involves the term (x - [x]), which represents the fractional part of x. For the equation to have no integral solutions, the values of a must be such that the quadratic expression does not yield any real roots, which occurs when a is in the interval (-1, 0) ∪ (0, 1), ensuring that the discriminant is negative.

Question 37

If a, b, c are distinct +ve real numbers and a² + b² + c² = 1 then ab + bc + ca is

Accepted Answer

less than 1. The sum of the squares of distinct positive real numbers a, b, and c being equal to 1 implies that their pairwise products ab, bc, and ca must collectively be less than 1, as the maximum value of the products cannot exceed the total sum of their squares.

Question 38

If set A has 4 elements and set B has 2 elements, then how many subsets of the Cartesian product A × B contain at least 3 elements?

Accepted Answer

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

Question 39

Let X = {4ⁿ - 3n - 1: n ∈ N} and Y = {9(n - 1): n ∈ N}, where N denotes the set of natural numbers. Then X ∪ Y is:

Accepted Answer

Y. The set Y consists of multiples of 9, which are all natural numbers of the form 9(n - 1) for n in the natural numbers. The elements of set X, while defined, do not cover all natural numbers and thus do not include all elements of Y. Therefore, the union X ∪ Y is equal to Y.

Question 40

Two sets A and B are as under:
A = {(a, b) ∈ R × R: |a − 5| < 1 and |b − 5| < 1};
B = {(a, b) ∈ R × R: 4(a − 6)² + 9(b − 5)² ≤ 36}.
Then:

Accepted Answer

A ⊂ B. A is the open square 4<a<6, 4<b<6. On it, 4(a-6)^2 + 9(b-5)^2 is largest at a corner, value 4(4)+9(1)=25, which is < 36, so every point of A satisfies B. Hence A is a subset of B.

Question 41

The Boolean Expression (p∧~q)∨q∨(~p∧q) is equivalent to:

Accepted Answer

p∨q. (p AND ~q) OR q OR (~p AND q): the last term absorbs into q, leaving (p AND ~q) OR q = (p OR q) AND (~q OR q) = p OR q.

Question 42

The Boolean expression ~(p∨q)∨(~p∧q) is equivalent to:

Accepted Answer

~p. The expression can be simplified using De Morgan's laws and distribution, ultimately showing that it simplifies to ~p, indicating that the negation of p is true when the entire expression holds.

Question 43

Find the area enclosed by the graph of y = |x - 2|, the vertical lines x = 1 and x = 3, and the x-axis.

Accepted Answer

1. The area is calculated by integrating the function |x - 2| between the limits x = 1 and x = 3. The graph forms a triangle with a base of 2 units and a height of 1 unit, resulting in an area of 1 square unit.

Question 44

If X = {4ⁿ − 3n − 1: n ∈ N} and Y = {9(n − 1): n ∈ N}, where N is the set of natural numbers, then X∪Y is equal to:

Accepted Answer

Y. The set Y consists of all numbers of the form 9(n - 1), which generates multiples of 9 starting from 0. The elements of set X do not produce any multiples of 9 for natural numbers n, meaning all elements of X are already included in Y, making Y the union of both sets.

Question 45

Let S = {1, 2,.... 20}. A subset B of S is said to be "nice" if the sum of the elements of B is 203. Then the probability that a randomly chosen subset of S is "nice" is:

Accepted Answer

5/2²⁰. The total sum of the elements in set S is 210, and for a subset B to have a sum of 203, it must exclude elements that sum to 7. There are 5 subsets of S that can achieve this by excluding combinations of elements that total 7, leading to a probability of 5 out of the total 2²⁰ possible subsets.

Question 46

In a class of 60 students, 40 opted for NCC, 30 opted for NSS and 20 opted for both NCC and NSS. If one of these students is selected at random, then the probability that the students selected has opted neither for NCC nor for NSS is:

Accepted Answer

1/6. n(NCC or NSS) = 40 + 30 - 20 = 50, so neither = 60 - 50 = 10. P = 10/60 = 1/6.

Question 47

Let [t] denote the greatest integer ≤ t. Then the equation in x, [x]² + 2[x + 2] - 7 = 0 has:
(1) exactly two solutions.
(2) exactly four integral solutions.
(3) infinitely many solutions.
(4) no integral solution.

Accepted Answer

infinitely many solutions.. Let n=[x]. n^2 + 2(n+2) - 7 = 0 -> n^2 + 2n - 3 = 0 -> n = 1 or n = -3. Then x in [1,2) or x in [-3,-2), each containing infinitely many x, so there are infinitely many solutions.

Question 48

A survey shows that 63% of the people in a city read newspaper A whereas 76% read newspaper B. If x% of the people read both the newspapers, then a possible value of x can be:
(1) 55
(2) 37
(3) 29
(4) 65

Accepted Answer

55. The value of x, representing the percentage of people who read both newspapers, must be less than or equal to the smaller percentage of readers for either newspaper. Since 63% read newspaper A, the maximum overlap with newspaper B (76%) can be 63%, but the minimum overlap must also satisfy the inclusion-exclusion principle, leading to a possible value of 55%.

Question 49

A survey shows that 73% of the persons working in an office like coffee, whereas 65% like tea. If x denotes the percentage of them, who like both coffee and tea, then x cannot be:

Accepted Answer

36. The percentage of people who like both coffee and tea cannot exceed the total percentage of those who like either beverage. Since 73% like coffee and 65% like tea, the maximum overlap (both) can be calculated using the principle of inclusion-exclusion, which shows that x must be at least 5% (73 + 65 - 100) and cannot exceed 65%. Therefore, 36% is not a valid percentage for those who like both, as it would imply an impossible overlap.

Question 50

Let A, B, C and D be four non-empty sets. The contrapositive statement of "If A ⊆ B and B ⊆ D, then A ⊆ C" is:

Accepted Answer

(3) If A ⊄ C, then A ⊄ B or B ⊄ D. The contrapositive of a conditional statement reverses and negates both the hypothesis and conclusion. In this case, if A is not a subset of C, it logically follows that either A cannot be a subset of B or B cannot be a subset of D, which aligns with option (3).

JEE Main Maths: Sets questions with solutions

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