If $A=(x: x=4 n, n \in Z)$ and $B=\{x: x=6 n, n \in Z\}$, then $A \cap B=$

$\left\{ {x:x = 10n,n \in Z} \right\}$

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JEE • MATHEMATICS • SET THEORY

Set theory practice · Mixed difficulty

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 1-30 of 273 questions

1 Medium

1 of 273

Analyzing +4 | -1

Let

S

be the set of all those 2007-place decimal integers

2 a_1 a_2 a_3 \ldots . . a_{2006}

which contain odd number of digit 9 in each sequence

a_1, a_2, a_3, \ldots ., a_{2006}

. The cardinal number of

S

View answer and explanation

Correct option: B

\frac{1}{2}\left(10^{2006}-8^{2006}\right)

Option A (

\frac{1}{2}(10^{2006}+8^{2006})

) is incorrect. The correct answer is

\frac{1}{2}(10^{2006}-8^{2006})

because it represents the number of 2007-place decimal integers with an odd number of 9s in each sequence.

2 Medium

2 of 273

Analyzing +4 | -1

a N=\{a n: n \in N: b N=\{n b: n \in N\}

c N=\{c N: n \in N\}

and

b N \cap c N=d N

, where

a, b, c \in N

and

b, c

are coprime, then

View answer and explanation

Correct option: C

d=b c

Option A (0) is correct because the equation

x^3 - 3x - k = 0

cannot have two distinct real roots in the interval (0, 1) based on Rolle's theorem.

3 Easy

3 of 273

Understanding +4 | -1

Which of the following cannot be the number of elements in the power set of any finite set?

View answer and explanation

Correct option: A

Option A (26) is correct because the number of elements in a power set must be a power of 2, and 26 is not a power of 2.

4 Medium

4 of 273

Understanding +4 | -1

A=\{x: x

is a multiple of 3

\}

and

B=\{x: x

is a multiple of 5}, then

A-B

is equal to

View answer and explanation

Correct option: B

A \cap \bar{B}

Option B (

A \cap \bar{B}

) is correct because A - B represents elements in A but not in B, which is the intersection of A and the complement of B.

5 Medium

5 of 273

Application +4 | -1

F(n)

denotes the set of all divisors of

n

except 1 , what is the least value of y satisfying

[\mathrm{F}(20) \cap \mathrm{F}(16)] \subseteq \mathrm{F}(\mathrm{y})

View answer and explanation

Correct option: B

Option B (2) is correct because it's the smallest number whose divisors (excluding 1) contain the intersection of the divisors of 20 and 16 (excluding 1).

6 Easy

6 of 273

Application +4 | -1

In a town of 10000 families, it was found that

40 \%

families buy newspaper

A, 20 \%

families buy newspaper B and

10 \%

families buy newspaper

C

, 5% families buy newspapers

A

and B, 3% buy newspapers B and

C

and

4 \%

buy newspapers

A

and

C

. If

2 \%

families buy all the three newspapers, then number of families which buy

A

only is

View answer and explanation

Correct option: B

3300

Option B is correct because by using the principle of inclusion-exclusion, the number of families buying newspaper A only is calculated to be 3300.

7 Easy

7 of 273

Application +4 | -1

A

and

B

are two sets such that

A

has 12 elements,

B

has 17 elements and

A \cup B

has 21 elements, then number of elements in

{A} \cap {B}

are

View answer and explanation

Correct option: C

Option C (8) is correct because it correctly uses the inclusion-exclusion principle to find the number of elements in the intersection of sets A and B.

8 Medium

8 of 273

Understanding +4 | -1

A

and

B

are subsets of a set

X

, then what is

\{A \cap(X-B)\} \cup B

equal to?

View answer and explanation

Correct option: A

A \cup B

Option A (

A \cup B

) is correct because the expression simplifies to the union of sets A and B using set identities.

9 Easy

9 of 273

Analyzing +4 | -1

Let

A = \left\{ {x \in R:|x + 1| < 2} \right\}{\text{ }}\,\,{\text{and }}\,\,\,B = \left\{ {x \in R:|x - 1| \geqslant 2} \right\}.

Then which one of the following statements is NOT true?

View answer and explanation

Correct option: B

B - A = R - ( - 3,1)

The statement

B - A = R - ( - 3,1)

is NOT true because

B - A

should be

R - (-3, 3)

. Therefore, option B is the correct answer.

10 Easy

10 of 273

Understanding +4 | -1

If the set

{A}

has 3 elements and the set

{B}=\{3,4\}

, then the number of elements in

{A} \times {B}

View answer and explanation

Correct option: A

6 is the correct answer because the number of elements in the Cartesian product A x B is the product of the number of elements in A and B, which is 3 * 2 = 6.

11 Easy

11 of 273

Application +4 | -1

{X}

and

{Y}

are two sets such that

{X} \cup {Y}

has 18 elements,

{X}

has 8 elements and

{Y}

has 15 elements; then number of elements in

{X} \cap {Y}

are

View answer and explanation

Correct option: A

Option A (5) is correct because it accurately applies the inclusion-exclusion principle to determine the number of elements in the intersection of sets X and Y.

12 Easy

12 of 273

Application +4 | -1

If the sets

A

and

B

are defined as

A=\{(x, y): y

\left.=\frac{1}{x}, 0 \neq x \in R\right\} B=\{(x, y): y=-x, x \in R\}

, then:

View answer and explanation

Correct option: C

A \cap B=\phi

Option C (

A \cap B=\phi

) is correct because the equations

y=\frac{1}{x}

and

y=-x

have no real solutions, meaning the sets have no common elements.

13 Medium

13 of 273

Understanding +4 | -1

A=\{1,2,3\}, B=\left\{x \in R: x^2-2 x+1=0, C=\{1,2,3\}\right.

and

D=\left\{x \in R: x^3-6 x^2+11 x-6=0 ;\right.

then the equal sets are

View answer and explanation

Correct option: D

A, C

and

D

Option D (

A, C

and

D

) is correct because sets A, C, and D all contain the elements {1, 2, 3}, making them equal.

14 Easy

14 of 273

Understanding +4 | -1

A=(x: x=4 n, n \in Z)

and

B=\{x: x=6 n, n \in Z\}

, then

A \cap B=

View answer and explanation

Correct option: C

\{x: x=12 n, n \in Z\}

Option C (

\{x: x=12 n, n \in Z\}

) is correct because the intersection of the two sets contains elements that are multiples of both 4 and 6, thus multiples of 12.

15 Medium

15 of 273

Application +4 | -1

In a class of 30 pupils, 12 take needle work, 16 take physics and 18 take history. If all the 30 students take at least one subject and no one takes all three then the number of pupils taking 2 subjects is

View answer and explanation

Correct option: A

Option A (16) is correct because it represents the number of pupils taking exactly two subjects, calculated using the principle of inclusion-exclusion and the given conditions.

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16 Easy

16 of 273

Understanding +4 | -1

A=\{x: x=2 n+1, n \in Z

and

B=\{x: x=2 n, n \in Z\}

, then

A \cup B

View answer and explanation

Correct option: C

set of integers

Option C (set of integers) is correct because the union of the set of odd integers and the set of even integers is the set of all integers.

17 Medium

17 of 273

Application +4 | -1

In a group of 50 people, two tests were conducted, one for diabetes and one for blood pressure. 30 people were diagnosed with diabetes and 40 people were diagnosed with high blood pressure. what is the minimum number of people who were having diabetes and high blood pressure?

View answer and explanation

Correct option: C

Option C (20) is correct because it is calculated using the formula for the number of elements in the union of two sets: n(D ∪ H) = n(D) + n(H) - n(D ∩ H), and considering the total number of people.

18 Medium

18 of 273

Application +4 | -1

n(A \cup B \cup C)+n(A \cup B \cup C)^{\prime}=50, n(A)=10, n(B)=15, n(C)=20, n(A \cap B)=5, n(B \cap C)=6, n(C \cap A)=10, n(A \cap B \cap C)=5,

then the value of

n(A \cup B \cup C)^{\prime}

View answer and explanation

Correct option: B

Option B (21) is correct because using the principle of inclusion-exclusion to find

n(A \cup B \cup C)

and subtracting it from the total number of elements gives the correct value for

n(A \cup B \cup C)'

19 Easy

19 of 273

Understanding +4 | -1

B

is the set whose elements are obtained by adding 1 to each of the even numbers, then the set builder notation of

B

View answer and explanation

Correct option: C

B=\{x: x

is odd and

x \in Z\}

Option C (

B=\{x: x

is odd and

x \in Z\}

) is correct because adding 1 to each even number results in odd numbers, and the set B contains all odd integers.

20 Easy

20 of 273

Understanding +4 | -1

Which of the following relations is/are correct?

(a)

A-B=A-(A \cap B)

(b)

A=(A \cap B) \cup(A-B)

(c)

A-(B \cup C)=(A-B) \cup(A-C)

View answer and explanation

Correct option: D

(a) and (b)

Option D ((a) and (b)) is correct because the set identities A - B = A - (A ∩ B) and A = (A ∩ B) ∪ (A - B) are both correct.

21 Easy

21 of 273

Understanding +4 | -1

{A}

has 3 elements and

{B}

has 6 elements, then the minimum number of elements in the set

A \cup B

View answer and explanation

Correct option: A

Option A (6) is correct because the minimum number of elements in the union of two sets occurs when one set is a subset of the other.

22 Easy

22 of 273

Application +4 | -1

Suppose

A_1, A_2, \ldots A_{30}

are thirty sets, each with five elements and

{B}_1, {~B}_2, \ldots, {B}_n

are

n

sets each with three elements Let

\bigcup_{i=1}^{30} {~A}_i=\bigcup_{j=1}^n {~B}_j={S}

If each elements of

{S}

belongs to exactly ten of the

{A}_i

's and exactly nine of the

{B}_j

's then

n=

View answer and explanation

Correct option: A

Option A (45) is correct because it equates the total number of elements counted in two different ways using the given information about the sets.

23 Easy

23 of 273

Understanding +4 | -1

A, B, C

are any three sets, then

A \times(B \cup C)

is equal to

View answer and explanation

Correct option: A

(A \times B) \cup(A \times C)

Option A (

(A \times B) \cup (A \times C)

) is correct based on the distributive property of the Cartesian product over union.

24 Easy

24 of 273

Application +4 | -1

Let

A = \left\{ {1,\;2,\;3,\;4} \right\},\;B = \left\{ {2,\;4,\;6} \right\}

Then, the number of sets

C

such that

A \cap B \subseteq C \subseteq A\; \cup B

View answer and explanation

Correct option: C

Option C (8) is correct because the number of sets C is determined by the number of possible combinations of elements from A ∪ B that include A ∩ B.

25 Medium

25 of 273

Understanding +4 | -1

Let

A

and

B

have 3 and 6 elements, respectively. What can be minimum number of elements in

A \cup B

View answer and explanation

Correct option: B

Option B (6) is correct because the minimum number of elements in the union of two sets occurs when one set is a subset of the other.

26 Easy

26 of 273

Understanding +4 | -1

{P}, {Q}

and

{R}

are subsets of a set

{A}

, then

{R} \times\left({P}^c \cup {Q}^c\right)^c

equals

View answer and explanation

Correct option: A

({R} \times {P}) \cap({R} \times {Q})

Option A ((

R \times P) \cap(R \times Q)

) is correct because it follows from De Morgan's law and the properties of Cartesian products.

27 Easy

27 of 273

Understanding +4 | -1

A=\{a, b\}, B=\{c, d\}, C=\{d, e\},

then

\{(a, c),(a, d),(a, e),(b, c),(b, d),(b, e)\}

is equal to

View answer and explanation

Correct option: C

A \times(B \cup C)

Option C (

A \times (B \cup C)

) is correct because it represents the Cartesian product of A with the union of B and C, resulting in the given set of ordered pairs.

28 Easy

28 of 273

Application +4 | -1

In a college of 300 students, every student reads 5 newspapers and every newspaper is read by 60 students. The number newspapers is

View answer and explanation

Correct option: C

exactly 25

Option C (exactly 25) is correct because the total number of readings is the number of students times the number of newspapers each student reads, which equals the number of newspapers times the number of students reading each newspaper.

29 Easy

29 of 273

Understanding +4 | -1

A=\{1,2,5\}

and

B=\{3,4,5,9\}

, then number of elements in

A \Delta B

is equal to

View answer and explanation

Correct option: A

Option A (5) is correct because A Δ B contains elements in A or B but not in both, resulting in {1, 2, 3, 4, 9}, which has 5 elements.

30 Medium

30 of 273

Understanding +4 | -1

The Cartesian product of two sets

\mathrm{P}

and

\mathrm{Q}

, i.e.,

\mathrm{P} \times \mathrm{Q}=\phi

, if

View answer and explanation

Correct option: A

either

\mathrm{P}

\mathrm{Q}

is the null set

Option A is correct because the Cartesian product of two sets is empty if at least one of the sets is empty.

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Key Takeaways

✓ This set contains 273 single-correct-answer MCQs on SET THEORY.
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