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

ncert cbse 11th mathematics chapter 1 sets miscellaneous exercise

15. 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

 

H = set of people who read H

T  = set of people who read T

 I = set of people who read I

 

n(H)  = 25

n(T) = 26

n(I) =26

n( H intersection I ) = 9

n( H intersection T ) = 11

 n( T intersection I ) = 8

 n( H intersection T intersection I ) = 3

 

i)

at least one of the newspapers means ( H union T union I )

 

 n( H union T union I ) = n(H) + n(T)+n(I) - n( H intersection I ) -n( H intersection T ) -n( T intersection I ) +  n( H intersection T intersection I )

 

n( H union T union I ) =  25+26+26-9-11-8+3 =52

therefore 52 people read at least one of the newspapers.

 

(ii)

 

let n(H and I only and not T) = a

n(H and T only and not I) = b

n(T and I only and not H) = c

 

using n( H intersection T intersection I ) = 3

and given values of n( H intersection I ), n( H intersection T ),( T intersection I )

 

n( H intersection I ) =>   a+3 =   9

n( H intersection T ) =>  b+3 = 11

 n( T intersection I ) =>   c+3 =  8

 ----------------------------------------------------------------

adding

                          a + b + c + 9 =28

 

so a+b+c = 28-9

a+b+c = 19 

19 people read exactly two of the newspapers

 

now we add the people who read exactly three of the newspapers namely

adding n( H intersection T intersection I ) = 3 on both sides

a+b+c +n( H intersection T intersection I ) =19+3 = 22

22 people read more than one newspaper

remove these 22 people to get

 

n( H union T union I ) - [a+b+c +n( H intersection T intersection I )]

= 52 - 22 = 30

the number of people who read exactly one newspaper = 30

 

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 chapter 1 miscellaneous sets ncert cbse

15. 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

solution

 

14. In a group of students, 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group? 

solution

 

 

13. In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many
students were taking neither tea nor coffee?

solution

 

 

11th cbse ncert chapter 2 relations and functions miscellaneous exercise

 

12. Let A = {9,10,11,12,13} and let f : A → N be defined by f (n) = the highest prime factor of n. Find the range of f.

solution

 

11. Let f be the subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a
function from Z to  Z? Justify your answer.

solution

 

10. Let A ={1,2,3,4}, B = {1,5,9,11,15,16} and f = {(1,5), (2,9), (3,1), (4,5), (2,11)}
Are the following true?
(i) f is a relation from A to B
(ii) f is a function from A to B.
Justify your answer in each case.

solution 

 

9. Let R be a relation from N to N defined by

 R = {(a, b) : a, b ∈ N and a = (b^2) }. 

Are the following true?
(i) (a,a) ∈ R, for all a ∈ N
(ii) (a,b) ∈ R, implies (b,a) ∈ R
(iii) (a,b) ∈ R, (b,c) ∈ R implies (a,c) ∈ R.

 solution 

 

 8. Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined by
f(x) = ax + b, for some integers a, b. Determine a, b.

solution  

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