Practice Set theory - quantitative aptitude Online Quiz (set-1) For All Competitive Exams
Q-1) What is {[(A ∪ B)' ∩ A]} - (A - B) equal to?
(a)
(b)
(c)
(d)
{(A ∪ B)' ∩ A} - (A - B)
= {(U – (A ∪ B)) ∩ A} – (A – B)
= {(U ∩ A) – {(A ∪ B) ∩ A}} – (A – B)
= {A – A} – (A – B)
= Φ – (A – B) = Φ
Q-2) Consider the following for the next 04 (four) items that follow :In an examination of Class XII, 55% students passed in Biology, 62% passed in Physics, 60% passed in Chemistry, 25% passed in Physics and Biology, 30% passed in Physics and Chemistry, 28% passed in Biology and Chemistry. Only 2% failed in all the subjects.What percentage of students passed in all the three subjects?
(a)
(b)
(c)
(d)
Total passed student = 98%
7 + x + 2 + x + 2+ x + 30 – x + 25 – x + 28 – x + x = 98
94 + x = 98 ; x = 4%
Q-3) If A is a non-empty subset of a set E, then what is E ∪ (A ∩ Φ) – (A – Φ) equal to?
(a)
(b)
(c)
(d)
E ∪ (A ∩ Φ) – (A – Φ)
= E ∪ Φ – A = E – A = A'
Q-4) In the quadratic equation $x^2$ + ax + b = 0, a and b can take any value from the set {1, 2, 3, 4}. How many pairs of values of a and b are possible in order that the quadratic equation has real roots?
(a)
(b)
(c)
(d)
| $x^2$ + ax + b = 0 | value 0f (a, b) |
| $a^2$ - ab ≥ 0 | [1, 2, 3, 4] |
| for real roots | |
| $1^2$ - 4 × 1 ≥ 0 | (1, 1) |
not possible
(2, 1) → $2^2$ – 4 × 1 ≥ 0 possible
(3, 1) → possible
(3, 2), (4, 1), (4, 2) (4, 3) (4, 4) → possible
So → possible values can be possible
Q-5) If A = {1, 2, 3, 4}, then what is the number of subsets of A with atleast three elements?
(a)
(b)
(c)
(d)
Given, A = {1, 2, 3, 4}
So, the required subsets are {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}
and {1, 2, 3, 4}
Q-6) Consider the following for the next 04 (four) items that follow : In an examination of Class XII, 55% students passed in Biology, 62% passed in Physics, 60% passed in Chemistry, 25% passed in Physics and Biology, 30% passed in Physics and Chemistry, 28% passed in Biology and Chemistry. Only 2% failed in all the subjects.What percentage of students passed in exactly one subject?
(a)
(b)
(c)
(d)
Percentage of students passed in exactly one subject
= 7 + x + 2 + x + 2 + x = 11 + 3x = 11 + 12 = 23%
Q-7) In an examination, 52% candidates failed in English and 42% failed in Mathematics. If 17% failed in both the subjects, then what percent passed in both the subjects ?
(a)
(b)
(c)
(d)
venn diagram of no. of failed students
No. of students failed in English only = 52 – 17 = 35
No. of students failed in maths only = 42 – 17 = 25
Total no. of failed students in either of the subjects
= 35 + 17 + 25 = 77
No. of passed student in both subjects = 100 – 77
= 23
Q-8) In a class of 110 students, x students take both Mathematics and Statistics, 2x + 20 students take Mathematics and 2x + 30 students take Statistics. There are no students who take neither Mathematics nor Statistics. What is x equal to?
(a)
(b)
(c)
(d)
n(M) = 2x + 20
n(S) = 2x + 30
n(M ∩ S) = x
n(M ∪ S) = 110
We know that,
n(M ∪ S) = n(M) + n(S) – n(M ∩ S)
⇒ 110 = 2x + 20 + 2x + 30 – x
⇒ 110 = 3x + 50
⇒ 3x = 60
∴ x = 20
Q-9) If two sets A and B have 2n and 4n elements, respectively. When n is a natural number. What can be the minimum number of elements in A ∪ B?
(a)
(b)
(c)
(d)
Here, n(A ∩ B) = 2n
∴ n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
= 2n + 4n – 2n = 4n
Hence, minimum number of elements of A ∪ B is 4n.
Q-10) If a set A contains 60 elements and another set B contains 70 elements and there are 50 elements in common, then how many elements does A ∪ B contain?
(a)
(b)
(c)
(d)
Here, n(A) = 60, n(B) = 70, n(A ∩ B) = 50 and n(A ∪ B) = ?
We know that:
n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
= 60 + 70 – 50 = 130 – 50 = 80
Q-11) A is a set of positive integers such that when divided by 2, 3, 4, 5 and 6 leaves the remainder 1, 2, 3, 4 and 5 respectively. How many integers between 0 and 100 belong to the set A ?
(a)
(b)
(c)
(d)
LCM of 2, 3, 4, 5, 6 = 60
Number when divided by 2, 3, 4, 5, 6 gives
remainder 1, 2, 3, 4, 5 respectively here 2 – 1 = 1
3 – 2 = 1, 4 – 3, = 1, 5 – 4 = 1, 6 – 5 = 1
So required no. = 60 – 1 = 59
other no. 59 × 2 = 118
i.e. there is one no. below 100.
Q-12) In an examination, 52% candidates failed in English and 42% failed in Mathematics. If 17% candidates failed in both English and Mathematics, what percentage of candidates passed in both the subjects?
(a)
(b)
(c)
(d)
Total number of candidates = 100%
Percentage of candidates passed in both the subjects
= {100 – (25 + 17 + 35)}% = 23%
Q-13) In a gathering of 100 people, 70 of them can speak Hindi, 60 can speak English and 30 can speak French Further, 30 of them can speak both Hindi and English. 20 can speak both Hindi and French. If x is the number of people who can speak both English and French, then which one of the following is correct? (Assume that everyone can speak at least one of the three languages)
(a)
(b)
(c)
(d)
Let n(A) be no. of people who speak Hindi
⇒ n(A) = 70
Let n(B) be no. of people who speak English
⇒ n(B) = 60
Let n(C) be no. of people who speak French
⇒ n(C) = 30
Given n (A ∪ B ∪ C) = 100
n(A ∩ B) = 30, n(A ∪ C) = 20
n (B ∩ C) = x, n(A ∩ B ∩ C) = 1
We know that
n(A ∪ B ∪ C) = n(A) + n(B) + n (C) - n(A ∩ B)
- n (B ∩ C) - n(C ∩ A) - n(A ∩ B ∩ C)
100 = 70 + 60 + 30 – 30 – x – 20 – 1
100 = 109 – x
⇒ x = 109 – 100
⇒ x = 9
∴ Option (a) is correct.
Q-14) Consider the following for the next 04 (four) items that follow : In an examination of Class XII, 55% students passed in Biology, 62% passed in Physics, 60% passed in Chemistry, 25% passed in Physics and Biology, 30% passed in Physics and Chemistry, 28% passed in Biology and Chemistry. Only 2% failed in all the subjects.What is the ratio of number of students who passed in both Physics and Chemistry to number of students who passed in both Biology and Physics but not Chemistry?
(a)
(b)
(c)
(d)
Ratio = ${30}/{2 - x} = {30}/{25 - 4} = {30}/{21} ={10}/7$
Q-15) If A = {(22n – 3n – 1)|n ∈ N} and B = {9(n – 1)|n ∈ N}, then which one of the following is correct?
(a)
(b)
(c)
(d)
Given:
A = {($2^{2n}$ – 3n – 1) | n ε N}
= {0, 9, 54, 243, ...}
and B = {9(n – 1) | n ε N}
= {0, 9, 18, 27, ...}
From the above, it is clear that A ε B.
Q-16) Let :
P = Set of all integral multiples of 3
Q = Set of all integral multiples of 4
R = Set of all integral multiples of 6
Consider the following relations:
I. P ∪ Q = R
II. P ⊂ R
III. R ⊂ (P ∪ Q)
Which of the relations given above is/are correct?
(a)
(b)
(c)
(d)
Here, P = {..., –6, –3, 0, 3, 6, ...}
Q = {..., –8, – 4, 0, 4, 8, ...}
and R = {..., –36, –6, 0, 6, 36, ...}
I. P ∪ Q = {..., –8, –6, – 4, –3, 0 3, 4, 6, 8, ...} ≠ R
II. Here, P ⊄ R
III. Here, R ⊂ (P ∪ Q) is true.
Q-17) Consider the following in respect of the sets A and B.
I. (A ∩ B) ⊆ A
II. (A ∩ B) ⊆ B
III. A ⊆ (A ∪ B)
Which of the above are correct?
(a)
(b)
(c)
(d)
From figure,
(A ∩ B) ⊆ A (true)
(A ∩ B) ⊆ B (true)
and A ⊆ (A ∪ B) also (true)
Thus, all three statements are correct.
Shaded region = (A ∪ B).
Q-18) If A = {x : x is an odd integer} and B = {x : $x^2$ – 8x + 15 = 0}. Then, which one of the following is correct?
(a)
(b)
(c)
(d)
Given that,
A = {x : x is an odd integer}
and B = {x : $x^2$ – 8x + 15 = 0}
= (x : $x^2$ – 5x – 3x + 15 = 0)
= {x : x (x – 5) – 3(x – 5) = 0}
= {x : (x – 5) (x – 3) = 0} = {3, 5}
Since, B has the odd elements,
∴ B ⊆ A
Q-19) Which one of the following is a null set?
(a)
(b)
(c)
(d)
From option (a),
A = {x is a real number : x > 1 and x < 1}.
So, there is no element which is greater or less than 1.
So, A is a null set.
From option (b),
B = {x : x + 3 = 3} = {0}
= Singleton set
From option (c):
C = {Φ} = Singleton set
From option (d),
D = {x is a real number : x ≥ 1 and x ≤ 1}
= {1} = Singleton set
Q-20) The set S = {x ε N : x + 3 = 3} is a
(a)
(b)
(c)
(d)
Given:
S = x ε N : {x + 3 = 3}
S = { }
Thus, S is a null set.