Sets Test

Result

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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Sets Test - 22

Result

Sets Test - 22

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SHARING IS CARING

Question 1

(0, 1)

(−∞,1)(-\infty, 1)(−∞,1)

(-1,1)

(1,∞)(1,\infty)(1,∞)

(0,∞)(0,\infty)(0,∞)

Solution

Question 2

A∪(A∩B)A \cup (A \cap B)A∪(A∩B)

B - A

A−(A∩B)A - (A \cap B)A−(A∩B)

A∩BA \cap BA∩B

Solution

Question 3

300300300

400400400

500500500

600600600

Solution

Question 4

p∪(q∩r)p\cup (q\cap r)p∪(q∩r)

(p∩q)∩(q∩p)(p\cap q)\cap (q\cap p)(p∩q)∩(q∩p)

(p∩q)∪r(p\cap q)\cup r(p∩q)∪r

(p∩q)∪(p∩r)(p\cap q)\cup (p\cap r)(p∩q)∪(p∩r)

Solution

Question 5

0%

20%

25%

50%

Solution

Question 6

Overlapping

Equivalent

Disjoint

Equal

Solution

Question 7

606060

100100100

777777

353535

Solution

Question 8

000

222

333

777

Solution

Question 9

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

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

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

≥n(A)+n(B)\geq n (A)+ n (B)≥n(A)+n(B)

Solution

Question 10

A≠BA \neq BA=B

B⊂AB \subset AB⊂A

A⊂BA \subset BA⊂B

A=BA = BA=B

Solution

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$(-\infty, 1)$

$(1,\infty)$

$(0,\infty)$

$A \cup (A \cap B)$

$A - (A \cap B)$

$A \cap B$

$300$

$400$

$500$

$600$

$p\cup (q\cap r)$

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

$(p\cap q)\cup r$

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

$60$

$100$

$77$

$35$

$0$

$2$

$3$

$7$

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

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

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

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

$A \neq B$

$B \subset A$

$A \subset B$

$A = B$