Sets Test

Result

Sets Test - 5

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Weekly Quiz Competition

Question 1
1 / -0

Directions: Study the following information and answer the question.

A newsagent sells the dailies Hindu, Express and Mail in equal number to 906 persons. 21 persons buy Express and Mail, 36 buy Hindu and Mail, 27 buy Hindu and Express, and buy 9 all the three dailies.

The percentage of people buying Hindu or Express but not Mail is

A
nearly 64%

B
less than 60%

C
more than 65%

D
65%

Solution

Since the dailies are sold in equal number, the total sale of each may be taken as x.
Hence, from the diagram, if we add up all the ''only'' regions, we get:
(x - 54) + (x - 48) + (x - 39) + 9 + 27 + 18 + 12 = 906
x = 327

Percentage of people buying Hindu or Express but not Mail =
Question 2
1 / -0

If S is the set of all the factors of 1024, then how many elements of S will have 6 as the last digit?

A
2

B
3

C
5

D
11

Solution

2⁰, 2¹, … 2¹⁰ are the factors of 1024.
S = {2⁰, 2¹, … 2¹⁰}
Thus, 2⁴and 2⁸ end with 6.
Question 3
1 / -0

If A B and B A, then

A
A ≠ B

B
A^c= B^c

C
A = B

D
A = B^c

Solution

A B and B A, then A = B
Question 4
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If A B, then for a non-empty set C,

A
A B × C

B
A × C B × C

C
C A × B

D
B A × C

Solution

If A B, then for non-empty set C:
A × C B × C
Question 5
1 / -0

Find the incorrect option.

For any two sets A and B, A = B is equivalent to

A
A - B = B - A

B
A ∪ B = A ∩ B

C
A ∪ C = B ∪ C and A ∩ C = B ∩ C for any set C

D
A ∩ B = φ

Solution

(1) A - B = B - A
A - A = A - A [ ∵ A = B ]

(2) Given: A = B
A ∪ B = A = B
A ∩ B = A = B
Hence, A ∪ B = A ∩ B

(3) A ∪ C = B ∪ C
A ∪ C = A ∪ C

(4) A ∩ B = φ
A ≠ φ
A ∩ A = A [ ∵ A = B]
So, option (4) is not correct.
Question 6
1 / -0

If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find A ∪ (B ∩ C).

A
{3}

B
{1, 2, 5}

C
{1, 2, 3, 4}

D
{1, 2, 3, 4, 5}

Solution

A = {1, 2, 3}, B = {3, 4}, C = {4, 5, 6}
B ∩ C = {4}
A ∪(B ∩C) = {1, 2, 3, 4}
Question 7
1 / -0

In a survey at Massachusetts Institute of Technology, 80 students were asked whether they used laptop or desktop or both. Of these, 50 students said they used desktop and 70 said they used laptop. What could be the respective minimum and maximum numbers of students who used both desktop and laptop?

A
40, 50

B
50, 60

C
30, 40

D
10, 20

Solution

n (A ∩ B) = n(A) + n(B) - n (A ∪ B)

For min n (A ∩ B), n (A ∪ B) should be maximum, which is 80.
Hence, n(A ∩ B) min = 50 + 70 - 80 = 40
Also, A ∩ B is maximum when all desktop users are also laptop users. Hence, 50.
Question 8
1 / -0

If A = {x: – 2 ≤ x ≤ 2, x ∈ integers} and B = {x: – 3 ≤ x ≤ 3, x ∈ integers}, find A ∪ B.

A
{– 2, – 1, 0, 1, 2, 3}

B
{0, 1, 2}

C
{– 3, – 2, – 1, 0, 1, 2, 3}

D
{0, 1, 2, 3}

Solution

A = {x: – 2 ≤ x ≤ 2, x ∈ integers} A = {-2, -1, 0, 1, 2}
B = {x: – 3 ≤ x ≤ 3, x ∈ integers} B = {-3, -2, 1, 0, 1, 2, 3}
A ∪ B = {-3, -2, -1, 0, 1, 2, 3}
Question 9
1 / -0

Which of the following sets is/are null set(s)?

A = {x | x = 7, 2x = 4}
B = {x | x = 3x, x ≠ 0}
C = {x | x - 5 = 4}

A
A, B and C

B
A and B only

C
B and C only

D
C only

Solution

In set A, as x = 7, 2x = 4 is not possible. So, the result is null.
In set B, if value of x is 0, then the result of x is 3x, but x ≠ 0. So, it is a null set.
Question 10
1 / -0

A survey of 90 children shows that 40 like Thumbs Up, 22 like Coke, 6 like both Thumbs Up and Pepsi, 13 like both Pepsi and Coke, 6 like both Thumbs Up and Coke, 2 like all three and 20 like none of these drinks.

How many like exactly two of these drinks?

A
12

B
15

C
17

D
19

Solution

Total number of children = 90
n(P T) = 6, n(T C) = 6, n(P C) = 13
n(P T C) = 2
Hence, total number of children who like exactly two drinks = n(P T) + n(T C) + n(P C) - 3 × n(P T C) = 6 + 6 + 13 - 6 = 19
Question 11
1 / -0

^{_{If set A has 5 elements and set B has 7 elements, then A}_{B has}}

(i) 5 elements
(ii) 7 elements
(iii) 12 elements
(iv) 8 elements
(v) 13 elements

A
Only i, ii, iii and iv

B
Only ii and iv

C
Only i and v

D
Only ii, iii and iv

Solution

n(A) = 5 n(B) = 7
Minimum of n(A B) = n(B) = 7
Maximum of n(A B) = n(A) + n(B) = 12
So, all values are correct, as these are in the given range 7 x 12.
So, option 4 is correct.
Question 12
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(A C) (B C) is best depicted by:

A

B

C

D

Solution

is the area common to both A and C.
is the area common to both B and C.
Union of common areas gives (3) as the required Venn diagram.
Question 13
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Which of the following is/are the example(s) of a singleton set?

i. {x : x ∈ N x ≤ 5, x ≥ 6}
ii. {x : x ∈ Z x > 5, x < 7}
iii. {x : x ∈ R x > 1}
iv. {x : x ∈ Z x > 3, x < 3}

A
i and ii

B
ii only

C
i, ii and iii

D
None of these

Solution

i = { } as no natural number can be less than
equal to 5 and greater than equal to 6.
ii = {6}
iii = {All real numbers greater than 1}
iv = { } No integer can be less than and greater than 3.
Question 14
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In a town, 48% people are educated, 51% people are young, 60% are servicemen, 24% are educated and young. There are 25% are young and servicemen, 27% are educated and servicemen and 5% fit in all the three categories. What is the ratio of educated or young, but not servicemen to young or servicemen but not educated?

A

B

C

D
None of these

Solution

Ratio =
Question 15
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In a group of 50 persons, everyone takes either tea or coffee. If 35 persons take tea and 25 persons take coffee, then the number of persons who take tea only (and not coffee) is

A
10

B
25

C
35

D
None of these

Solution

Total number of persons = 50
Number of persons who take tea = 35
Number of persons who take coffee = 25
Number of persons who take both tea and coffee = 35 + 25 - 50 = 10
Now, number of persons who take tea only = 35 - 10 = 25