Set Theory Test 15

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Set Theory Test 15

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Weekly Quiz Competition

Question 1
1 / -0

Look at the set of numbers below.
Set : $$\left \{6, 12, 30, 48\right\}$$
Which statement about all the numbers in this set is NOT true?

A
They are all multiples of $$3$$

B
They are all even numbers

C
They are all factors of $$48$$

D
They are all divisible by $$2$$

Solution

Here above statement C is not true because 30 is not factors of 48.
So, Answer (C) They are all factors of 48
Question 2
1 / -0

In a class of 55 students, the number of students studying different subjects are 23 in Mathematics and 24 in Physics, 19 in Chemistry, 12 in Mathematics and Physics, 9 in Mathematics and Chemistry, 7 in Physics and Chemistry and 4 in all the three subjects, The number of students who have taken exactly one subject is

A
20

B
24

C
23

D
22

Solution

Total no. of student=55
Let n(M)=student who studying mathematics
n(C)=student who studying chemistry
n(P)=student who studying Physics
$$\therefore n(M)=23,n(P)=24,n(C)=19,n(M\cap P)=12,n(M\cap C)=9,n(P\cap C)=7,n(M\cap P\cap C)=4$$
Number of student who studying mathematics but not physic and chemistry
$$\Rightarrow n(M)-[(n(M\cap C)+n(M\cap P)]+n(M\cap P\cap C)$$
$$\Rightarrow 23-[9+12]+4$$
$$\Rightarrow 23-21+4=6$$

Number of student who studying chemistry but not physic and matehematics
$$\Rightarrow n(C)-[(n(M\cap C)+n(P\cap C)]+n(M\cap P\cap C)$$
$$\Rightarrow 19-[9+7]+4$$
$$\Rightarrow 19-16+4=7$$

Number of student who studying physics but not mathematics and chemistry
$$\Rightarrow n(P)-[(n(M\cap P)+n(P\cap C)]+n(M\cap P\cap C)$$
$$\Rightarrow 24-[12+7]+4$$
$$\Rightarrow 24-19+4=9$$

$$\therefore$$no. of student studying exactly one subject$$=6+7+9=22$$
Question 3
1 / -0

Let $$A$$ and $$B$$ be two sets such that $$n(A)=16$$, $$n(B)=12$$, and $$n(A\cap B)=8$$. Then $$n(A\cup B)$$ equals

A
$$28$$

B
$$20$$

C
$$36$$

D
$$12$$

Solution

$$n(A\cup B)=n(A)+n(B)-n(A\cap B)=16+12-8=20$$
Question 4
1 / -0

Let $$A=\{1,2,3,4,5,6\}$$. How many subsets of $$A$$ can be formed with just two elements, one even and one odd?

A
$$6$$

B
$$8$$

C
$$9$$

D
$$10$$

Solution

$$\{1,2\}$$, $$\{1,4\}$$, $$\{1,6\}$$, $$\{2,3\}$$, $$\{2,5\}$$, $$\{3,4\}$$, $$\{3,6\}$$, $$\{4,5\}$$, $$\{5,6\}$$,
Question 5
1 / -0

In a class 60% of the students were boys and 30% of them had I class. If 50%of the students in the class had I class, find the fraction of the girls in the class who did not have a I class.

A
$$1/5$$

B
$$4/5$$

C
$$1/4$$

D
$$1/3$$

Solution

$$Let\quad the\quad number\quad of\quad students\quad be\quad x,\\ boys\quad with\quad first\quad class=0.3*0.6x=0.18x,\\ number\quad of\quad girls=0.4x,\\ students\quad with\quad first\quad class=0.5x,\\ girls\quad with\quad first\quad class=0.5x-0.18x=0.32x,\\ fraction\quad of\quad girls\quad without\quad first\quad class=\dfrac { 0.4-0.32x }{ 0.4x } =\dfrac { 1 }{ 5 } $$
Question 6
1 / -0

If $$A=\{\dots,-6,-4,-2,0,2,4,6,\dots\}$$, then

A
$$10\notin A$$

B
$$-10\notin A$$

C
$$5\in A$$

D
$$50\in A$$

Solution

50 is even.
Question 7
1 / -0

The sets $$\displaystyle S_{x}$$ are defined to be $$(x, x + 1, x + 2, x + 3, x + 4)$$ where $$x=1, 2, 3,.....80$$. How many of these sets contain $$6$$ or its multiple?

A
$$65$$

B
$$66$$

C
$$59$$

D
$$60$$

Solution

There are $$14$$ multiples of $$6$$ till $$84$$.

Since $$5$$ consecutive no.s are chosen only one set in $$6$$ consecutive sets will not have a multiple of $$6$$. So till $$78$$ sets there are

$$78-\dfrac{78}6=78-13=65$$ sets containing 6 or multiples of 6.

$$S_{79}$$ does not contain any multiple of 6

Hence $$S_{80} $$ must contain a multiple of 6.

Answer $$=66$$ sets
Question 8
1 / -0

If $$\displaystyle Q=\left((x|x=\frac{1}{y}\:\ \text{wher} \ e\:y\in N\right)$$, then

A
$$0\notin Q$$

B
$$1\in Q$$

C
$$2\in Q$$

D
$$\displaystyle\frac{2}{3}\in Q$$

Solution

When $$y=1$$, $$\displaystyle x=\frac{1}{1}=1$$.
Question 9
1 / -0

If A={a,b,c,d,e}, B={a,c,e,g} and C={b,d,e,g} then which of the following is true?

A
$$\displaystyle C\subset \left ( A\cup B \right )$$

B
$$\displaystyle C\subset \left ( A\cap B \right )$$

C
$$\displaystyle A\cup B=A\cup C$$

D
Both(1) and (3)

Solution

For the given sets,
$$ (A \cup B ) = $$ { $$ a,b,c,d,e,g $$} (Combination of all elements of both sets)

Clearly, elements of C are a part of $$ A \cup B $$ as well,
So, $$ C\subset (A\cup B) $$

And, $$ (A \cap B ) = $$ { $$ a,c,e $$} (Common elements of both sets )
As the elements of C are not completely a part of $$ A \cap B $$ , Option B is not True.

Also,
$$ (A \cup C ) = $$ { $$ a,b,c,d,e,g $$}

Clearly, $$ (A \cup B ) = (A \cup C ) $$

Hence, both first option and third options are True.
Question 10
1 / -0

If the universal set is U = $$ \displaystyle \left \{ 1^{2},2^{2},3^{2},4^{2},5^{2},6^{2} \right \} $$ What is the complement of the intersection of set A = $$ \displaystyle \left \{ 2^{2},4^{2},6^{2} \right \} $$ and set B=$$ \displaystyle \left \{ 2^{2},3^{2},4^{2} \right \} $$ ?

A
$$ \displaystyle \left \{ 2^{2},4^{2} \right \} $$

B
$$ \displaystyle \left \{ 1^{2},5^{2} \right \} $$

C
$$ \displaystyle \left \{ 1^{2},5^{2},6 ^{2} \right \} $$

D
$$ \displaystyle \left \{ 1^{2},3^{2},5^{2},6^{2} \right \} $$

E
Answer required

Solution

$$A\cap B=\{2^2,4^2\}$$
$$\bar{A\cap B}=U-(A\cap B)=\{1^2,3^2,5^2,6^2\}$$
Option D is correct.

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Set Theory Test 15

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Set Theory Test 15

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SHARING IS CARING

Question 1

They are all multiples of $$3$$

They are all even numbers

They are all factors of $$48$$

They are all divisible by $$2$$

Solution

Question 2

20

24

23

22

Solution

Question 3

$$28$$

$$20$$

$$36$$

$$12$$

Solution

Question 4

$$6$$

$$8$$

$$9$$

$$10$$

Solution

Question 5

$$1/5$$

$$4/5$$

$$1/4$$

$$1/3$$

Solution

Question 6

$$10\notin A$$

$$-10\notin A$$

$$5\in A$$

$$50\in A$$

Solution

Question 7

$$65$$

$$66$$

$$59$$

$$60$$

Solution

Question 8

$$0\notin Q$$

$$1\in Q$$

$$2\in Q$$

$$\displaystyle\frac{2}{3}\in Q$$

Solution

Question 9

$$\displaystyle C\subset \left ( A\cup B \right )$$

$$\displaystyle C\subset \left ( A\cap B \right )$$

$$\displaystyle A\cup B=A\cup C$$

Both(1) and (3)

Solution

Question 10

$$ \displaystyle \left \{ 2^{2},4^{2} \right \} $$

$$ \displaystyle \left \{ 1^{2},5^{2} \right \} $$

$$ \displaystyle \left \{ 1^{2},5^{2},6 ^{2} \right \} $$

$$ \displaystyle \left \{ 1^{2},3^{2},5^{2},6^{2} \right \} $$

Answer required

Solution

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