De Morgan's Law

Q.

Find the sum of all natural numbers from$1$ to $100.$

Q.

What is $32$ degrees Celsius in Fahrenheit?

Q.

Sets $A$ and $B$ have $3$ and $6$ elements respectively. What can be the minimum number of elements in $A\cup B$

1. $3$

2. $6$

3. $9$

4. $18$

Q.

Let U be the universal set containing 700 elements. If A, B are sub-sets of U such that n (A) = 200, n(B) =300 and $n(A\cap B)$= 100. Then , $n(A^{\prime }\cap B^{\prime })=$

1. 400

2. 600

3. 300

4. none of these

Q. In an examination, out of $100$ students, $75$ passed in English, $60$ passed in Mathematics and $45$ passed in both English and Mathematics. The number of students who passed in none of the two subjects, is (Assume all students gave both the exams)

1. $5$
2. $10$
3. $20$
4. $40$

Q. For any two non-empty sets $A$ and $B$, $(A\cup B{)}^C\cap (A^C\cap B)$ is equal to

1. $B^C$
2. $\varphi$
3. $A\cup B$
4. $A^C$

Q. In a class of $42$ students, $23$ are studying Mathematics, $24$ are studying Physics, $19$ are studying Chemistry. If $12$ are studying both Mathematics and Physics, $9$ are studying both Mathematics and Chemistry, $7$ are studying both Physics and Chemistry and $4$ are studying all the three subjects, then the number of students studying exactly one subject, is

1. $15$
2. $30$
3. $22$
4. $27$

Q. If $U$ is the universal set with 100 element; $A$ and $B$ are two set such that $n\left(A\right)=50,n\left(B\right)=60,n(A\cap B)=20$, then $n(A^{\prime }\cap B^{\prime })=$

1. $20$
2. $90$
3. $40$
4. $10$

Q.

If A and B are two given sets, then $A\cap (A\cap B{)}^c$ is equal to

1. $\varphi$

2. $A\cap B^c$

3. A

4. B

Q.

A survey shows that 63% of the Americans like cheese whereas 76% like apples. If x% of the Americans like both cheese and apples then find the value of x.

Q. Prove that:(2n)!/n! = { 1*3*5....(2n-1)} 2ⁿ

Q.

In a potato race, $20$ potatoes are placed in a line at intervals of $4\mathrm{meters}$ with the first potato$24\mathrm{meters}$ from the starting point. A contestant is required to bring the potatoes back to the starting place one at a time. How far would he run in bringing back all the potatoes?

1. $2340$

2. $2480$

3. $2540$

4. $2230$

Q.

If $S=\left\{1,2,3,4\right\}$, then the total number of unordered pairs of disjoint subsets of $S$ is

1. $25$

2. $34$

3. $42$

4. $41$

Q. For two sets $A$ and $B$, $(A\cup B)\cap (A^{\prime }\cup B^{\prime })=$

1. $A\cap B$
2. $A\cup B$
3. $A-B$
4. $A\Delta B$

Q.

If a committee of $3$ to be chosen from a group of $38$ people of which you are a member. What is the probability that you will be on the committee

1. ${}^{38}C_3$

2. ${}^{37}C_2$

3. $\frac{{}^{37}C_2}{{}^{38}C_3}$

4. $\frac{666}{8436}$

Q. 1) Let A = { 1, 2, 3, 4}. Let R be the equivalence relation on A x A defined by (a, b)R(c, d) iff a+d=b+c. Find the equivalence class [(1, 3)].

Q. In a class of $60$ students, $40$ opted for NCC, $30$ opted for NSS and $20$ opted for both NCC and NSS. If one of these students is selected at random, then the probability that the student selected has opted neither for NCC nor for NSS is :

1. $\frac{1}{6}$
2. $\frac{2}{3}$
3. $\frac{5}{6}$
4. $\frac{1}{3}$

Q. Number of ways in which $5$ different mobiles can be distributed among $3$ boys such that each gets atleast one mobile and none gets three is

1. $180$
2. $90$
3. $360$
4. $120$

Q. Four married couples are to be seated in a row having $8$ chairs. The number of ways so that spouses are seated next to each other is

1. $186$
2. $384$
3. $516$
4. $72$

Q.

Which of the following sets are finite or infinite

(i) The set of months of a year

(ii) {1, 2, 3 ...}

(iii) {1, 2, 3 ... 99, 100}

(iv) The set of positive integers greater than 100

(v) The set of prime numbers less than 99

Q. If $P\left(B\right)=\frac{3}{5},P\left(A|B\right)=\frac{1}{2}$ and $P(A\cup B)=\frac{4}{5}$, then find $P(A\cup B{)}^{\prime }+P(A^{\prime }\cap B)$.

Q.

Check the injectivity and surjectivity of the following functions:

(i) f: N → N given by f(x) = x²

(ii) f: Z → Z given by f(x) = x²

(iii) f: R → R given by f(x) = x²

(iv) f: N → N given by f(x) = x³

(v) f: Z → Z given by f(x) = x³

Q.

Assume that each child born is equaly likely to be boy or a girl. IF a family has two children, what is the conditional probability that both are girls given that
the youngest is a girl?

atleast one is a girl?

Q.

Write the following sets in roster form:

(i) A = {x: x is an integer and –3 < x < 7}.

(ii) B = {x: x is a natural number less than 6}.

(iii) C = {x: x is a two-digit natural number such that the sum of its digits is 8}

(iv) D = {x: x is a prime number which is divisor of 60}.

(v) E = The set of all letters in the word TRIGONOMETRY.

(vi) F = The set of all letters in the word BETTER.

Q.

The first jump in a unicycle high-jump contest is shown. The bar is raised $2$ centimeters after each jump. Solve the inequality $2n+10\ge 26$ to find the numbers of additional jumps needed to meet or exceed the goal of clearing a height of $26$ centimeters.

Q.

In a potato race, a bucket is placed at the starting point, which is $5m$ from the first potato and other potatoes are placed $3m$apart in a straight line.

There are ten potatoes in the line.

The total distance covered in $m$ is$?$

Q.

In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:

(i) the number of people who read at least one of the newspapers.

(ii) the number of people who read exactly one newspaper.

Q.

Cards are drawn one by one at random from a well shuffled full pack of $52$ cards until two aces are obtained for the first time. If $N$ is the number of cards required to be drawn, then $P_r\left\{N=n\right\}$, where $2\le n\le 50$, is

1. $\frac{\left[\right(n-1\left)\right(52-n\left)\right(51-n\left)\right]}{[50.49.17.13]}$

2. $\frac{\left[2\right(n-1\left)\right(52-n\left)\right(51-n\left)\right]}{[50.49.17.13]}$

3. $\frac{\left[3\right(n-1\left)\right(52-n\left)\right(51-n\left)\right]}{[50.49.17.13]}$

4. $\frac{\left[4\right(n-1\left)\right(52-n\left)\right(51-n\left)\right]}{[50.49.17.13]}$

Q.

Q. Let $A_{3\times 4},B_{4\times 5},C_{5\times 1}$ and $D_{1\times 4}$ be the matrices. Then which of the following is true

1. Order of $\left(ABCD\right)$ is $3\times 4$
2. Order of $\left(ABCD\right)$ is $4\times 3$
3. Order of $\left(ABCD\right)$ is $4\times 4$
4. Order of $\left(ABCD\right)$ can not be determined