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CBSE Questions for Class 11 Commerce Applied Mathematics Set Theory Quiz 10 - MCQExams.com
CBSE
Class 11 Commerce Applied Mathematics
Set Theory
Quiz 10
In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only.
Report Question
0%
10
0%
11
0%
12
0%
13
Explanation
Let
A
,
B
, and
C
be the set of people who like
p
r
o
d
u
c
t
A
,
p
r
o
d
u
c
t
B
,
a
n
d
p
r
o
d
u
c
t
C
respectively.
Accordingly,
n
(
A
)
=
21
,
n
(
B
)
=
26
,
n
(
C
)
=
29
,
n
(
A
∩
B
)
=
14
,
n
(
C
∩
A
)
=
12
,
n
(
B
∩
C
)
=
14
,
n
(
A
∩
B
∩
C
)
=
8
The Venn diagram for the given problem can be drawn as above.
It can be seen that number of people who like product C only is
=
29
–
(
4
+
8
+
6
)
=
11
In a battle
70
%
of the combatants lost one eye,
80
%
an ear,
75
%
an arm,
85
%
a leg and
x
%
lost all the four limbs the minimum value of
x
is
Report Question
0%
10
0%
12
0%
15
0%
n
o
n
e
o
f
t
h
e
s
e
The number of subsets
R
of
P
=
(
1
,
2
,
3
,
.
.
.
.
,
9
)
which satisfies the property "There exit integers a<b<c with a
∈
R, b
∈
R,c
∈
R" is
Report Question
0%
512
0%
466
0%
467
0%
None of these
Explanation
Every subset of P containing at least 3 elements from P will always satisfy the required property since each number is distinct in the set P.
So, Total no. of ways =
9
C
3
+
9
C
4
+
.
.
.
.
+
9
C
9
=
(
9
C
0
+
9
C
1
+
.
.
.
.
.
+
9
C
9
)
−
9
C
0
−
9
C
1
−
9
C
2
=
2
9
−
46
=
512
−
46
=
466
If
A
=
{
3
,
{
4
,
5
}
,
6
}
, then the statement is true or false
{
3
}
⊆
A
Report Question
0%
True
0%
False
Explanation
A
=
{
3
,
{
4
,
5
}
,
6
}
We have to find out if
{
3
}
is subset of
A
or not.
Since,
3
is present in elements of
A
, then
{
3
}
⊆
{
A
}
∴
Given statement is true.
In a universal set x,
n
(
x
)
=
50
,
n
(
A
)
=
35
,
n
(
B
)
=
20
,
n
(
A
′
∩
B
′
)
=
5
,then
n
(
A
∪
B
)
,
n
(
A
∩
B
)
are repsectively
Report Question
0%
45
,
10
0%
10
,
45
0%
25
,
30
0%
15
,
25
Explanation
Given
n
(
X
)
=
50
,
n
(
A
)
=
35
,
n
(
B
)
=
20
n
(
A
′
∩
B
′
)
=
5
(
i
)
n
(
A
∪
B
)
n
(
A
′
∩
B
′
)
=
n
(
X
)
−
n
(
A
∪
B
)
n
(
A
∪
B
)
=
n
(
X
)
−
n
(
A
′
∩
B
′
)
n
(
A
∪
B
)
=
50
−
5
=
45
(
i
i
)
n
(
A
∩
B
)
n
(
A
∪
B
)
=
n
(
A
)
+
n
(
B
)
−
n
(
A
∩
B
)
n
(
A
∩
B
)
=
n
(
A
)
+
n
(
B
)
−
n
(
A
∪
B
)
n
(
A
∩
B
)
=
35
+
20
−
45
=
10
If
p
and
q
are two proposition, then
∼
(
p
↔
q
)
is
Report Question
0%
∼
p
∧
∼
q
0%
∼
p
∨
∼
q
0%
(
p
∧
∼
q
)
∨
(
∼
p
∧
q
)
0%
n
o
n
e
o
f
t
h
e
s
e
If
A
and
B
are two non empty sets then
(
A
∪
B
)
C
=
?
Report Question
0%
A
C
∪
B
C
0%
A
C
∩
B
C
0%
A
∪
B
C
0%
A
C
∩
B
Explanation
If A and B are two no empty sets then
(
A
∪
B
)
C
=
A
C
∩
B
C
Let
S
=
1
,
2
,
3
,
.
.
.
.
.10
and
P
=
1
,
2
,
3
,
4
,
5
The number of subsets
′
Q
′
of
S
such that
p
∪
Q
=
S
, are.....
Report Question
0%
128
0%
256
0%
32
0%
64
If 'p' is true and 'q' is false, then which of the following statement is not true?
Report Question
0%
p
∨
q
0%
p
⇒
q
0%
p
∧
(
∼
q
)
0%
q
⇒
p
If the number of
5
elements subsets of the set
A
{
a
1
,
a
2
.
.
.
.
.
a
20
}
of
20
distinct elements is
k
times the number of
5
elements subsets containing
a
4
, then
k
is
Report Question
0%
5
0%
20
7
0%
4
0%
10
3
Explanation
No of
5
elements subset
=
20
C
5
=
20
!
15
!
15
!
=
20
×
19
×
18
×
17
×
16
×
15
!
15
!
5
×
4
×
3
×
2
×
1
=
15504
No of
5
elements subset containing
a
4
=
19
C
4
=
19
!
4
!
15
!
⇒
20
!
15
!
15
!
=
k
19
!
4
!
15
!
⇒
20
×
19
!
5
×
4
!
=
k
19
!
4
!
⇒
k
=
4
Let
A
and
B
be two sets.
A
∩
B
=
{
1
,
2
}
,
A
∪
B
=
{
1
,
2
,
3
,
4
,
5
}
and if
n
1
is the maximum number of function from
A
to
B
and
n
2
is the minimum number of function form
A
to
B
then
n
1
−
n
2
equals
Report Question
0%
32
0%
56
0%
64
0%
81
The value of set
(
A
∪
B
∪
C
)
∩
(
A
∩
B
1
∩
C
1
)
1
∩
C
1
is equal to
Report Question
0%
B
∩
C
1
0%
A
∩
C
0%
B
∩
C
1
0%
A
∩
C
1
If two sets
P
and
Q
,
n
(
P
)
=
5
,
n
(
Q
)
=
4
then
n
(
P
×
Q
)
=
Report Question
0%
20
0%
9
0%
25
0%
5
/
4
Explanation
n
(
P
×
Q
)
=
n
(
P
)
×
n
(
Q
)
=
5
×
4
=
20
∴
option
A
is correct.
The statement that is true among the following is
Report Question
0%
p
⟹
q
is equivalent to
p
∧
−
q
0%
p
∨
q
and
p
∧
q
have the same truth value
0%
The converse of
t
a
n
x
=
0
⟹
x
=
0
is
x
≠
0
⟹
t
a
n
x
=
0
0%
The contrapositive of
3
x
+
2
=
8
⟹
x
=
2
is
x
≠
2
⟹
3
x
+
2
≠
8
Let
Q
be a non empty subset of
N
and
q
is a
statement as given below :
q
:
There exists an even number
a
∈
Q
Negation of the statement
q
will be :
Report Question
0%
There is no even number in the set
Q
0%
Every
a
∈
Q
is an odd number.
0%
(
a
)
and
(
b
)
both
0%
None of these
Explanation
Let
Q
be a non empty subset of
N
and
q
.
There exists even number
a
∈
Q
negation of statement
q
.
There does not exist a even number in set Q.
Hence, the answer is there is no even number in the set
Q
.
If A= {1, 2, 5} and B= {3, 4, 5, 9}, then
A
⋃
B
is equal to :
Report Question
0%
{
1
,
2
,
5
,
9
}
0%
{
1
,
2
,
3
,
4
,
9
}
0%
{
1
,
2
,
3
,
4
,
5
,
9
}
0%
None of these
Explanation
A
=
{
1
,
2
,
5
}
B
=
{
3
,
4
,
5
,
9
}
A
∪
B
=
{
1
,
2
,
3
,
4
,
5
,
9
}
Option C.
In a battle
70
% of the combatants lost one eye,
80
% an ear,
75
% an arm,
85
% a leg,
x
% lost all the four limbs the minimum value of
x
is
Report Question
0%
10
0%
12
0%
15
0%
5
Explanation
70
%
of the combatants lost one eye,
80
%
an ear,
75
%
an arm and
85
%
a leg.
Now,
⇒
The combatants who lost one eye and one ear
=
(
70
+
80
−
100
)
%
=
50
%
⇒
The combatants who lost one eye, one ear and one erm
=
(
50
+
75
−
100
)
%
=
25
%
⇒
The combatants who lost one eye, one ear one arm and one leg
=
(
25
−
85
−
100
)
%
=
10
%
∴
Combatant who lost all the four limbs
=
10
%
∴
x
=
10
%
Two sets A and B are defined as follows
A
=
{
(
x
,
y
)
:
y
=
e
2
x
,
x
∈
R
}
and
B
=
{
(
x
,
y
)
:
y
=
x
2
,
x
∈
R
}
, then
Report Question
0%
A
⊂
B
0%
B
⊂
A
0%
A
⋃
B
0%
A
∩
B
=
ϕ
Explanation
A
=
{
(
x
,
y
)
:
y
=
e
2
x
,
x
∈
R
}
a
n
d
B
=
{
(
x
,
y
)
:
y
=
x
2
,
x
∈
R
}
t
h
e
n
,
A
=
(
e
∘
,
e
1
,
e
2
,
.
.
.
.
,
e
∞
)
=
(
1
,
e
1
,
e
2
,
.
.
.
.
.
.
.
)
a
n
d
,
B
=
(
0
,
1
,
4
,
.
.
.
.
.
)
∴
A
⊂
B
H
e
n
c
e
,
t
h
e
o
p
t
i
o
n
A
i
s
t
h
e
c
o
r
r
e
c
t
a
n
s
w
e
r
.
If
A
=
{
1
,
2
,
3
,
4
}
;
B
=
{
2
,
4
,
6
,
8
}
;
C
=
{
3
,
4
,
5
,
8
}
then
A
∩
B
∩
C
=
Report Question
0%
ϕ
0%
4
0%
μ
0%
2
,
4
Explanation
A
=
{
1
,
2
,
3
,
4
}
B
=
{
2
,
4
,
6
,
8
}
C
=
{
3
,
4
,
5
,
8
}
A
∩
B
∩
C
=
{
4
}
.
Examine whether the following statements are true or false:
{
b
,
c
}
⊂
{
a
,
{
b
,
c
}
}
Report Question
0%
True
0%
False
In a group of 800 people, 550 can speak Hindi and 450 can speak English. How many can speak both Hindi and English ?
Report Question
0%
100
0%
150
0%
200
0%
None of these
Explanation
Let H denote the set of people speaking Hindi and E denote the set of people speaking English. We are given :
n
(
H
)
=
550
,
n
(
E
)
=
450
,
n
(
H
∪
E
)
=
800
Now,
n
(
H
∪
E
)
=
n
(
H
)
+
n
(
E
)
−
n
(
H
∩
E
)
n
(
H
∩
E
)
=
550
+
450
−
800
=
200
Hence, 200 persons can speak both Hindi and English.
Let A,B are two sets such that n(A)=4 and n(B)=Then the least possible number of elements in the power set of
(
A
∪
B
)
is
Report Question
0%
16
0%
64
0%
256
0%
1024
Explanation
Given,
n
(
A
)
=
4
,
n
(
B
)
=
6
Then the least number of possible elements in
n
(
A
∪
B
)
=
2
n
(
A
)
.2
n
(
B
)
=
2
4
.2
6
=
2
10
=
1024
Left A = {1, 2, 3, 4, 5, 6} and B= {1, 2, 3, 4} be two sets, then the number of functions that can be defined from A to B such that the element '2' in B has exactly 3 pre - images i A, is equal ot
Report Question
0%
540
0%
440
0%
810
0%
620
if universal set
∑
=
{
a
,
b
,
c
,
d
,
e
,
f
,
g
,
h
,
}
A
=
{
b
,
c
,
d
,
e
,
f
}
a
n
d
C
=
{
c
,
d
,
e
,
f
,
g
}
,
t
h
e
n
f
i
n
d
B
−
A
Report Question
0%
{
b
,
c
,
e
,
f
}
0%
{
a
,
b
,
f
,
h
}
0%
{
a
,
g
,
h
}
0%
{
a
,
c
,
e
,
g
}
Explanation
Given,
B
=
{
a
,
b
,
c
,
d
,
e
,
f
,
g
,
h
}
A
=
{
b
,
c
,
d
,
e
,
f
}
,
C
=
{
c
,
d
,
e
,
f
,
g
}
∴
B
−
A
=
{
a
,
b
,
c
,
d
,
e
,
f
,
g
,
h
}-{
b
,
c
,
d
,
e
,
f
}
⟹
B
−
A
=
{
a
,
g
,
h
}
If P(S) denotes the set of all subsets of a given set S, then the number of one to one function from the set s={1,2,3} to the set of P(S) is
Report Question
0%
336
0%
8
0%
36
0%
320
Explanation
G
i
v
e
n
P
(
s
)
=
{
1
,
2
,
3
}
P
(
s
)
=
a
l
l
s
u
b
s
e
t
o
f
5
=
{
ϕ
,
{
1
}
,
{
3
}
,
{
1
,
2
}
,
{
2
,
3
}
,
{
3
,
1
}
,
{
1
,
2
,
3
}
}
N
o
.
o
f
e
l
e
m
e
n
t
o
f
5
=
0
(
5
)
=
3
s
i
m
i
l
a
r
l
y
O
(
p
(
s
)
)
=
8
T
o
t
a
l
n
o
.
o
f
o
n
e
−
o
n
e
f
u
n
c
t
i
o
n
=
n
!
(
n
−
m
)
!
,
w
h
e
r
e
n
=
o
(
p
(
s
)
)
a
n
d
m
=
o
(
5
)
=
8
!
(
8
−
3
)
!
=
8
!
5
!
=
336
N
o
.
o
f
o
n
e
−
o
n
e
f
u
n
c
t
i
o
n
i
s
336
H
e
n
c
e
,
o
p
t
i
o
n
A
i
s
c
o
r
r
e
c
t
,
The function
f
(
x
)
satisfies the condition
(
x
−
2
)
f
(
x
)
+
2
f
(
1
x
)
=
2
for all
x
≠
0
. Then the value of
f
(
2
)
is
Report Question
0%
1
2
0%
1
0%
7
4
0%
−
3
2
Explanation
Putting
x
=
2
in
(
x
−
2
)
f
(
x
)
+
2
f
(
1
x
)
=
2
(
2
−
2
)
f
(
2
)
+
2
f
(
1
2
)
=
2
2
f
(
1
2
)
=
2
f
(
1
2
)
=
1
If
A
=
{
1
,
2
,
3
,
4
}
, then the number of subsets of
A
that contain the element
2
but not
3
, is
Report Question
0%
16
0%
4
0%
8
0%
24
Explanation
The subsets are be
{
1
,
2
,
4
}
,
{
1
,
2
}
,
{
2
,
4
}
,
{
2
}
Number of subsets of
A
that contain the element
2
but not
3
is
4
Let
p
and
q
be two statements. amongst the following, the statement is equivalent to
p
→
q
is
Report Question
0%
p
∧
∼
q
0%
∼
p
∨
q
0%
∼
p
∧
q
0%
p
∨
∼
q
Explanation
Given,
P
⟶
q
ne trinow that,
→
- if then
⇒
if
P
then
q
p
q
p
→
q
1
1
0
1
0
1
0
1
1
0
0
1
⇒
Evaluating option
A
(
p
∧
∼
q
)
P
iq
p
∧
∼
q
111
1
0
1
0
0
0
0
1
0
0
Incorrect assumption
⇒
Eualuating
B
(
∼
p
∧
q
)
∼
p
q
∼
p
∧
q
0
0
1
1
0
Incorrect assumption
⇒
Evaluating
c
(
∼
p
∨
q
)
∼
p
q
∼
p
v
q
0
1
0
0
1
1
1
0
1
0
1
correct assumption
Let
S
=
{
a
∈
N
,
a
≤
100
}
if the equation
[
tan
2
x
]
−
tan
x
−
a
=
0
has real roots, then the number of elements
S
is (when
[
.
]
is greatest integer function)
Report Question
0%
10
0%
1
0%
9
0%
0
Explanation
Given equation is,
[
t
a
n
2
x
]
−
t
a
n
x
−
a
=
0
∴
D
=
b
2
−
4
a
c
=
(
−
1
)
2
−
4
(
1
)
(
−
a
)
=
1
+
4
a
So D must be a odd perfect square
⟹
√
1
+
4
a
=
2
λ
+
1
⟹
1
+
4
a
=
4
λ
2
+
1
+
4
λ
⟹
a
=
λ
(
λ
+
1
)
For different values og=f
λ
we get
a
=
0
,
2
,
6
,
12
,
20
,
30
,
42
,
56
,
72
,
90
Hence there are
10
values of
a
.
If
A
=
{
1
,
3
,
5
,
7
,
9
,
11
,
13
,
15
,
17
}
,
B
=
{
2
,
4
,
.
.
.
.
,
18
}
and N is the universal set, then
A
′
∪
(
(
A
∪
B
)
∩
B
′
)
Report Question
0%
A
0%
N
0%
B
0%
R
Explanation
A
=
{
1
,
3
,
5
,
7
,
9
,
11
,
13
,
15
,
17
}
A
′
=
N
−
A
=
N
−
{
1
,
3
,
5
,
7
,
9
,
11
,
13
,
15
,
17
}
=
{
2
,
4
,
6
,
8
,
10
,
12
,
14
,
16
,
18
}
A
∪
B
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
,
11
,
12
,
13
,
14
,
15
,
16
,
17
,
18
}
B
′
=
N
−
B
=
N
−
{
2
,
4
,
6
,
8
,
10
,
12
,
14
,
16
,
18
}
=
{
1
,
3
,
5
,
7
,
9
,
11
,
13
,
15
,
17
}
(
A
∪
B
)
∩
B
′
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
,
11
,
12
,
13
,
14
,
15
,
16
,
17
,
18
}
∩
{
1
,
3
,
5
,
7
,
9
,
11
,
13
,
15
,
17
}
=
{
1
,
3
,
5
,
7
,
9
,
11
,
13
,
15
,
17
}
A
′
∪
(
(
A
∪
B
)
∩
B
′
)
=
{
2
,
4
,
6
,
8
,
10
,
12
,
14
,
16
,
18
}
∪
{
1
,
3
,
5
,
7
,
9
,
11
,
13
,
15
,
17
}
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
,
11
,
12
,
13
,
14
,
15
,
16
,
17
,
18
}
=
N
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Practice Class 11 Commerce Applied Mathematics Quiz Questions and Answers
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