Sets Test

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Sets Test - 22

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Weekly Quiz Competition

Question 1
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 2
1 / -0

Find the equivalent set for $$A - B $$.

A
$$A \cup (A \cap B)$$

B
B - A

C
$$A - (A \cap B)$$

D
$$A \cap B$$

Solution

Hence By this graph we see that $$A-B = A - (A \cap B)$$
Question 3
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 4
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 5
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.
Question 6
1 / -0

If $$A - B = \phi$$ and $$B - A = \phi$$ then A and B are

A
Overlapping

B
Equivalent

C
Disjoint

D
Equal

Solution

if $$A-B = \phi$$ and $$B-A = \phi$$
hence this relation contains $$A-B$$ and $$B-A$$ doesn't contains any element,
so, the elements in $$A$$ & $$B$$ are same.
Question 7
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 8
1 / -0

In a group of 15 women, 7 have nose studs, 8 have ear rings and 3 have neither. How many of these have both nose studs and ear rings?

A
$$0$$

B
$$2$$

C
$$3$$

D
$$7$$

Solution

Since 3 women have neither nose studs nor earrings

$$n(N\cup E) = 15-3 = 12$$

By set theory

$$n(N\cap E) = n(N) + n(E)-n(N\cup E)$$

$$=7+8-12 = 3$$
Question 9
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 10
1 / -0

If $$A - B = \emptyset $$, then relation between A and B is :

A
$$A \neq B$$

B
$$B \subset A$$

C
$$A \subset B$$

D
$$A = B$$

Solution

If A and B are disjoint it would mean A is a null set. Otherwise A and B must be equal to $$A\cap B$$ AT LEAST.

Hence option C is correct.

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Re-Attempt Weekly Quiz Competition

Sets Test - 22

Result

Sets Test - 22

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SHARING IS CARING

Question 1

(0, 1)

$$(-\infty, 1)$$

(-1,1)

$$(1,\infty)$$

$$(0,\infty)$$

Solution

Question 2

$$A \cup (A \cap B)$$

B - A

$$A - (A \cap B)$$

$$A \cap B$$

Solution

Question 3

$$300$$

$$400$$

$$500$$

$$600$$

Solution

Question 4

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

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

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

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

Solution

Question 5

0%

20%

25%

50%

Solution

Question 6

Overlapping

Equivalent

Disjoint

Equal

Solution

Question 7

$$60$$

$$100$$

$$77$$

$$35$$

Solution

Question 8

$$0$$

$$2$$

$$3$$

$$7$$

Solution

Question 9

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

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

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

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

Solution

Question 10

$$A \neq B$$

$$B \subset A$$

$$A \subset B$$

$$A = B$$

Solution

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