.In a certain school, 40 students were asked about whether they like tennis or football or both. It was found that the number of students who like both tennis and football was three times the number of students who like tennis only. Furthermore, the number of students who like football only was 6 more than twice the number of students who like tennis only. However, 4 students like neither tennis nor football.

(a)Represent this information in a Venn diagram, letting x be the number of students who like tennis only.

(b)Use the results obtained in part (a) to determine number of students who likes;

(i)Football only.

(ii)Both football and tennis.

In a class of 35 students, 21 study commercial subjects, 15 study both commercial and science subjects and 4 students study science subjects only. Use a Venn diagram to find the number of students who study:

(i)either science or commercial subjects.

(ii) neither science nor commercial subjects.

(iii) commercial subjects only. (iv) science subjects.

In a certain village, 300 people were interviewed about their food preference. It was found that, 200 people like banana, 120 people like rice and 60 people like both banana and rice. By using formula, find the number of people who like neither banana nor rice.

In a certain school, 40 students were asked about whether they like tennis or football or both. It was found that the number of students who like both tennis and football was three times the number of students who like tennis only. Furthermore, the number of students who like football only was 6 more than twice the number of students who like tennis only. However, 4 students like neither tennis nor football.

(a) Represent this information in a Venn diagram, letting x be the number of students who like tennis only. (b) Use the results obtained in part (a) to determine the probability that a student selected at random likes;

(i)Football only.

(ii)Both football and tennis.

 3.(a)Use the following diagram to answer the question that follows

(i) Find the number of subsets of set B (ii) Find the elements of set AnB.

(iii) If an element is picked at random from the universal set (U), find the probability that it is not the element of set B.

 Use the following diagram to answer the question that follows

(i) Find the number of subsets of set B (ii) Find the elements of set AnB.

(iii) If an element is picked at random from the universal set (U), find the probability that it is not the element of set B.

 The Ministry of Business and Industries has planned to employ 54 people who  will work in the business sector, 36 people who will work in industries sector only, 12  people who will work in both sectors and 21 people who will neither work in business sector nor in industries sector. How many people will be employed by the Ministry? (Use a Venn diagram).

SECTION B (40 Marks)

21. Given that the universal set ξ = {all counting numbers less than 29}. M and N are the subsets of set ξ where M = {Numbers which are multiples of 4} and N = {Numbers which are perfect

squares}. Find (M∩N).

  In a school of 60 teachers, some drink Fanta and some drink Coca-Cola. If 46 drink Fanta, 18 drink Coca-Cola and 14 drink both Coca-Cola and Fanta. How many teachers drink neither Fanta nor Coca-Cola? (Use Venn diagram)

22. In a group of 40 people, 10 are healthy and every person of the remaining 30 has either high blood pressure, high level of sugar content in the body or both. If 15 people have high blood pressure and 25 people have high level of sugar content in their bodies, how many people have high blood pressure and high level of sugar content?

  At Moiva’s graduation ceremony 45 people drank Pepsi-Cola, 80 drank Coca-Cola and 35 drank both Pepsi-Cola and Coca-Cola. By using a Venn diagram, found out how many people were at the ceremony if each person drank Pepsi-Cola or Coca-Cola.

24.Draw a Venn diagram to represent the relationship between the sets 

={1,2,5,6,7,9,10} and ={1,3,4,5,6,8,10}  

20.In a class of 75 form two students, 50 like Physics and 10 like Physics and Chemistry. Apply the general formula to find the number of students who like Chemistry.  

Let U be a universal set and A and B be the subsets of U where:

U = {1,2,3,4,5,6,7,8,9,10}, A = {odd numbers} and B = {prime numbers} (i) Represent this information in a venn diagram.

(ii)  Find A n B? and (A U B) ?

24.In a class of 45 students, some study physics or chemistry or both. If 23 students study physics, 33 study chemistry and 10 students do not study neither physics nor chemistry, find the number of students who study both physics and chemistry using the formula.

The Venn diagram below shows the universal set U and its two subsets A and B.

Write down the elements of:

(i)  A'

(ii)  B',

(iii)  A UB ,

(iv)  A'U B'.

Verify that n(A U B) = n(A) + n(B) - n(A n B) where A and B are the sets given in part 3(b).

Let U be a universal set and A and B be the subsets of U where,  and.

1. Find the number of subsets of set A’
2. Find

In a class of 30 students, 17 participate in English debate and 12 participate in both English debate and Mathematics club. If every student is required to participate in at least one of these two events, find the number of students who participate in;

i.English debate only.

ii.Mathematics club only

In a class of 15 students who take either Mathematics or Biology, 12 students take Mathematics, 8 students take Biology. If each student takes either subjects find by using formula the number of students who take Biology but not Mathematics.

In a class of 45 students, 30 study Chemistry, 20 study Physics and 5 study neither of the two subjects.

(a) Represent this information in a well labeled Venn diagram.

(b) Using part (a) find the number of students studying both subjects.

The Venn diagram below shows the number of elements in sets P and Q.

If n(P U Q) = 95, calculate:

(i) The value of x,

(ii) n(P n Q)?.

There are 48 men at a meeting of whom 24 are teachers, 36 are parents and 16 are both teachers and parents. By using a Venn diagram, find the number of men who are neither teachers nor parents.

A and B are subsets of the universal set U . Find n(A?B) given that n(A) = 39, n(A??B?) = 4, n(B?) = 24 and n(U) = 65 .

A survey of 240 houses showed that all of them kept a farm or a garden or both. If 180 kept gardens and 79 kept farms, how many houses kept both?

Given that A= {x : 0 ≤ x ≤ 8}

B = {x : 3 ≤ x ≤ 8}

where x is an integer, in the same form, represent in a Venn diagram

1. A u B
2. A n B

and hence find the elements in each set.

In a school of 75 pupils, 42% of the pupils take Biology but= not Chemistry, 32% take both subjects and 10% of them take Chemistry but not Biology. How many pupils do not take either Biology or Chemistry? (6 marks)

In a class of 50 students, 16 like watching television, 41 like reading story books and 7 do not like neither watching television nor reading story books. Find the number of students who like both watching television and reading story books using the formula.

If P ={ all multiples of 5 less than 35} and Q={all numbers between 14 and 30} find P

In a village of 50 farmers, 25 grow cashew nut and 16 grow both cashew nut and maize. If 10 farmers grow neither cashew nut nor maize, find the number of famers who grow maize only. Do not use Venn diagram.

In a class of 40 students, 24 students study Geography and 21 students study History. Use Venn diagrams to find the number of students who study both Geography and History.

In a primary school of 150 pupils 50 study Maths, 70 study Science and 40 study both subjects. By using the appropriate formula, calculate the number of pupils who study neither Maths nor Science.

The grades on a Mathematics test taken by 100 students arc as shown in the following distribution table: 

1. What is the size of each class interval of this distribution? 
2. Which class interval had the highest number of students? 
3. Find the class mark of the highest class interval. 
4. Find the number of students who passed if the pass mark was 70. 
5. Use the condition given in 10 (b) (iv), find the number of students who failed the test. 

In a village of 1500 villagers, 600 keep goats, 700 keep cows and 300 do not keep any of these animals. Use a Venn diagram to find the number of villagers who keep; 

1. both goats and cows. 
2. goats only. 
3. cows only. 
4. goats or cows.

In a class of 32 students, 18 play golf, 16 play piano and 7 play both golf and piano. Use a formula to find the number of students who play neither golf nor piano.