Set Theory Test 9

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Set Theory Test 9

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Weekly Quiz Competition

Question 1
1 / -0

All the students of a batch opted Psychology, Business, or both. 73% of the students opted Psychology and 62% opted Business. If there are 220 students, how many of them opted for both Psychology and business?

A
$$60$$

B
$$100$$

C
$$77$$

D
$$35$$

Solution

By set theory

$$n(P\cap B) = n(P) + n(B)-n(P\cup B)$$

$$=0.73+0.62-1.00=0.35$$

$$35\%$$ of $$220 = 77$$
Question 2
1 / -0

In an examination $$80\%$$ passed in English, $$85\%$$ in Maths, $$75\%$$ in both and $$40$$ students failed in both subjects. Then the number of students appeared are

A
$$300$$

B
$$400$$

C
$$500$$

D
$$600$$

Solution

$$n(E)=80$$
$$n(M)=85$$
$$n(E \cap M)=75$$
$$n(E \cup M)=n(E)+n(M)-n(E \cap M)=80+85-75=90$$
$$n(E \cup M)'=10$$
Let n be the total number of students appeared
$$\dfrac{10}{100} \times n=40$$
$$\therefore n=400$$
Question 3
1 / -0

$$p\cap (q\cup r)=?$$

A
$$p\cup (q\cap r)$$

B
$$(p\cap q)\cap (q\cap p)$$

C
$$(p\cap q)\cup r$$

D
$$(p\cap q)\cup (p\cap r)$$

Solution
Question 4
1 / -0

There are 19 hockey players in a club. On a particular day 14 were wearing the prescribed hockey shirts, while 11 were wearing the prescribed hockey pants. None of then was without hockey pant or hockey shirt. How many of them were in complete hockey uniform?

A
8

B
6

C
9

D
7

Solution

We can look at it in 2 ways

First by set theory

$$n(A\cap B) = n(A) + n(B)-n(A\cup B)$$

$$=14+11-19=6$$

Qualitatively, we know that 14 people are wearing prescribed hockey shirts,which leaves us with 5 players who must be wearing hockey pants. So out of 11 players who are wearing hockey pants, 5 are not wearing hockey shirts while the other 6 are in complete uniform.
Question 5
1 / -0

In an examination, $$34\%$$ of the candidates fail in Arithmetic and $$42\%$$ in English. If $$20\%$$ fail in Arithmetic and English, the percentage of those passing in both subjects is :

A
$$44$$

B
$$45$$

C
$$46$$

D
$$47$$

Solution

Let $$A$$ denote students fail in Arithmetic, $$B$$ denote students fail in English
$$n(A)=34$$
$$n(B)=42$$
$$n(A \cap B)=20$$
$$n(A \cup B) = n(A) + n(B) - n(A \cap B) = 34+42-20=56$$
$$n(A \cup B)' = 100 - n(A \cup B) = 100-56=44$$
Question 6
1 / -0

If $$A$$ and $$B$$ have some elements in common, then $$n(A \cup B)$$ is:

A
$$= n(A) + n (B)$$

B
$$ > n (A) + n(B)$$

C
$$< n(A) + n(B)$$

D
$$\geq n (A)+ n (B)$$

Solution

We know that $$n(A\cup B)+n(A\cap B) = n(A)+n(B)$$

But $$n(A\cap B) >0$$ always
$$\Rightarrow n(A)+n(B)=n(A\cup B)+n(A\cap B)>n(A\cup B)$$
Hence, $$n(A\cup B)<n(A)+n(B)$$.
Question 7
1 / -0

The smallest set $$A$$ such that $$A\cup \left\{ 1,2 \right\} =\left\{ 1,2,3,5,9 \right\} $$ is

A
$$\left\{ 2,3,5 \right\} $$

B
$$\left\{ 3,5,9 \right\} $$

C
$$\left\{ 1,2,5,9 \right\} $$

D
$$\left\{ 1,2 \right\} $$

Solution

$$A\cup \{1,2\}=\{1,2,3,5,9\}$$
Thus
$$A=\{1,2,3,5,9\}-\{1,2\}$$

Hence
$$A=\{3,5,9\}$$
Question 8
1 / -0

If $$A \cap B = \phi$$, then $$A \Delta B$$ =

A
$$A \cap B$$

B
$$A \cup B$$

C
$$A - B$$

D
$$B - A$$

Solution

We know that $$A\Delta B=A\cup B-A\cap B $$

Given $$A\cap B=\phi $$

$$\therefore A\Delta B=A\cup B-\phi=A\cup B $$

Hence, the answer is $$A\cup B$$.
Question 9
1 / -0

The equation x + cosx $$=$$ a has exactly one positive root. Complete set of values of 'a' is

A
(0, 1)

B
$$(-\infty, 1)$$

C
(-1,1)

D
$$(1,\infty)$$

E
$$(0,\infty)$$

Solution

Let f(x) = x + cosx a
$$\Rightarrow f'(x)=1-sinx\geq 0\forall x \epsilon R.$$
Thus f(x) is increasing in $$\left ( -\infty ,\infty \right )$$, as zero of f'(x) don't for an interval. f(0) = 1 a

For a positive root, $$1-a< 0$$

$$\Rightarrow a> 1$$
Question 10
1 / -0

In a science talent examination, 50% of the candidates fail in Mathematics and 50% fail in Physics. If 20% fail in both these subjects, then the percentage who pass in both Mathematics and Physics is:

A
0%

B
20%

C
25%

D
50%

Solution

By set theory

$$n(M\cup P) = n(M) + n(P)-n(M\cap P)$$ where M and P are sets of students failing in respective subjects.

$$=0.5 + 0.5 - 0.2 = 0.8$$

This indicates $$80\%$$ of the class fails in at least one of the given subjects while $$20\%$$ pass in both.

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Re-Attempt Weekly Quiz Competition

Set Theory Test 9

Result

Set Theory Test 9

Score

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Rank

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SHARING IS CARING

Question 1

$$60$$

$$100$$

$$77$$

$$35$$

Solution

Question 2

$$300$$

$$400$$

$$500$$

$$600$$

Solution

Question 3

$$p\cup (q\cap r)$$

$$(p\cap q)\cap (q\cap p)$$

$$(p\cap q)\cup r$$

$$(p\cap q)\cup (p\cap r)$$

Solution

Question 4

8

6

9

7

Solution

Question 5

$$44$$

$$45$$

$$46$$

$$47$$

Solution

Question 6

$$= n(A) + n (B)$$

$$ > n (A) + n(B)$$

$$< n(A) + n(B)$$

$$\geq n (A)+ n (B)$$

Solution

Question 7

$$\left\{ 2,3,5 \right\} $$

$$\left\{ 3,5,9 \right\} $$

$$\left\{ 1,2,5,9 \right\} $$

$$\left\{ 1,2 \right\} $$

Solution

Question 8

$$A \cap B$$

$$A \cup B$$

$$A - B$$

$$B - A$$

Solution

Question 9

(0, 1)

$$(-\infty, 1)$$

(-1,1)

$$(1,\infty)$$

$$(0,\infty)$$

Solution

Question 10

0%

20%

25%

50%

Solution

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