NCERT Solution Class XI Mathematics Sets Question 15 (Ex Text Sol)

Question 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.

Answer

Let A be the set of people who read newspaper H. Let B be the set of people who read newspaper T. Let C be the set of people who read newspaper I.

Accordingly, n(A) = 25, n(B) = 26, and n(C) = 26 n(A ∩ C) = 9, n(A ∩ B) = 11, and n(B ∩ C) = 8

n(A ∩ B ∩ C) = 3

Let U be the set of people who took part in the survey.

(i) Accordingly, n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C) = 25 + 26 + 26 −11 − 8 − 9 + 3 = 52

Hence, 52 people read at least one of the newspapers.

(ii) Let a be the number of people who read newspapers H and T only.

Let b denote the number of people who read newspapers I and H only. Let c denote the number of people who read newspapers T and I only. Let d denote the number of people who read all three newspapers.

Accordingly, d = n(A ∩ B ∩ C) = 3

Now, n(A ∩ B) = a + d

n(B ∩ C) = c + d

n(C ∩ A) = b + d

∴ a + d + c + d + b + d = 11 + 8 + 9 = 28

⇒ a + b + c + d = 28 − 2d = 28 6 = 22

Hence, (52 − 22) = 30 people read exactly one newspaper.

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