Set Theory Test 30

Result

Set Theory Test 30

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Weekly Quiz Competition

Question 1
1 / -0

S = {1, 2, 3, 5, 8, 13, 21, 34 }. Find $$\displaystyle \sum $$ max (A), where the sum is taken over all 28 elements subsets A to S.

A
844

B
480

C
484

D
488

Solution
Question 2
1 / -0

There are 6 boxes numbered 1, 2, ... 6. Each box is to be filled up either with a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is:

A
5

B
21

C
33

D
60

Solution
Question 3
1 / -0

The dual of $$-p\wedge (q\vee \sim r)$$ is

A
$$p\vee (\sim q \wedge r)$$

B
$$\sim p\vee (q\wedge r)$$

C
$$p\wedge (q\wedge r)$$

D
$$\sim q\vee(q\wedge \sim r)$$
Question 4
1 / -0

If $$S$$ represents the set of all real numbers $$x$$ such that $$1\le x \le 3$$ and $$T$$ represents the set of all real numbers $$x$$ such that $$2 \le x \le 5$$, the set represented by $$S \cap T$$ is

A
$$2 \le x \le 3$$

B
$$1 \le x \le 5$$

C
$$x \le 5 $$

D
$$x\ge 5$$

E
none of these

Solution

$$S\cap T$$(common part of S and T) = $$2\leq x\leq 3$$
Question 5
1 / -0

In a town of $$10,000$$ families it was found that $$40\%$$ families buy newspaper $$A$$, $$20\%$$ families buy newspaper $$B$$ and $$10\%$$ families buy newspaper $$C$$. $$5\%$$ families buy $$A$$ and $$B$$, $$3\%$$ buy $$B$$ and $$C$$ and $$4\%$$ buy $$A$$ and $$C$$. If $$2\%$$ families buy all the three newspaper, find the number of families which buy (i) $$A$$ only (ii) $$B$$ only (iii) none of $$A, B$$ and $$C$$

A
(i) $$3000$$ (ii) $$1800$$ (iii) $$4600$$

B
(i) $$3300$$ (ii) $$1400$$ (iii) $$4000$$

C
(i) $$3500$$ (ii) $$1600$$ (iii) $$3800$$

D
none

Solution
Question 6
1 / -0

Suman is given an aptitude test containing 80 problems, each carrying I mark to be tackled in 60 minutes. The problems are of 2 types; the easy ones and the difficult ones. Suman can solve the easy problems in half a minute each and the difficult ones in 2 minutes each. (The two type of problems alternate in the test). Before solving a problem, Suman must spend one-fourth of a minute for reading it. What is the maximum score that Suman can get if he solves all the problems that he attempts?

A
30

B
40

C
53

D
60

Solution

$$Assuming\quad that\quad all\quad the\quad problems\quad suman\quad solves\quad are\quad right,\\ marks\quad earned\quad per\quad minute\quad solving\quad easy\quad problem=\dfrac { 1 }{ 3/4 } =\dfrac { 4 }{ 3 } \\ marks\quad earned\quad per\quad minute\quad solving\quad difficult\quad problem=\frac { 1 }{ 9/4 } =\frac { 4 }{ 9 } \\ since\quad we\quad have\quad to\quad maximize\quad the\quad marks,\quad we\quad will\quad solve\quad all\quad the\quad easy\quad problems\quad first.\\ marks\quad earned\quad by\quad solving\quad easy\quad problems=40,\\ time\quad spent=\frac { 3 }{ 4 } *40=30\quad minutes\\ time\quad left\quad for\quad difficult\quad problems=30\quad minutes,\\ problems\quad solved=\frac { 30 }{ 9/4 } =13,\\ marks\quad earned=13,\\ total\quad marks=40+13=53$$
Question 7
1 / -0

Which one of the following is correct?

A
$$A \times (B - C) = (A - B) \times (A - C)$$

B
$$A \times (B - C) = (A \times B) - (A \times C)$$

C
$$\displaystyle A\cap \left ( B\cup C \right )=\left ( A\cap B \right )\cup C$$

D
$$\displaystyle A\cup \left ( B\cap C \right )=\left ( A\cup B \right )\cap C$$

Solution

$$(A) A\times(B-C) = (A\times B) - (A-C)$$
$$ A\times(B-C) \ne (A\times B) - (A-C)$$

$$(B) A\times(B-C) = (A\times B) - (A\times C)$$

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

$$(D) A\cup (B\cap C) = (A\cup B) \cap (A\cup C)$$
$$ (A\cup B)\cap C \ne (A\cup B) \cap (A\cup C)$$

So , B is correct.
Question 8
1 / -0

Let $$A = [\theta : sin (\theta) = tan (\theta)]$$ and $$B = [\theta : cos (\theta) = 1]$$ be two sets. Then,

A
$$A = B$$

B
$$A \nsubseteq B$$

C
$$B \nsubseteq A$$

D
$$A \subset B$$ and $$B - A \neq \phi$$

Solution
Question 9
1 / -0

If $$20$$% of three subsets (i.e., subsets containing exactly three elements) of the set $$A = \left \{a_{1}, a_{2}, ...., a_{n}\right \}$$ contain $$a_{2}$$, then the value of $$n$$ is

A
$$15$$

B
$$16$$

C
$$17$$

D
$$18$$

Solution
Question 10
1 / -0

Let $$A_{1}, A_{2}, ........., A_{m}$$ be m sets such that $$O(A_{i}) = p \forall i = 1, 2, ......... m$$ and $$B_{1}, B_{2}, .........., B_{n}$$ be n sets such that $$O(B_{j}) = q \forall j = 1, 2, ........., n$$. If $$\bigcup_{i=1}^{m} A_{i}$$ = $$\bigcup_{j=1}^{n} B_{j} = S$$ and each element of S belongs to exactly $$\alpha$$ number of $$A_{i}'s$$ and $$\beta$$ number of $$B_{j}'s$$, then

A
pm = nq

B
$$\alpha pm = \beta nq$$

C
$$\beta pm = \alpha nq$$

D
$$(pm)^{\beta} = (nq)^{\alpha}$$