PUC Science Class 11 - Karnataka Board PUC Question Bank Solutions for Mathematics

For any two sets A and B, prove that \[A - \left( A - B \right) = A \cap B\]

For any two sets A and B, prove that

\[A \cup \left( B - A \right) = A \cup B\]

For any two sets A and B, prove that \[\left( A - B \right) \cup \left( A \cap B \right) = A\]

If A and B are two sets such that \[n \left( A \cup B \right) = 50, n \left( A \right) = 28 \text{ and } n \left( B \right) = 32\]\[n \left( A \cap B \right)\]

If P and Q are two sets such that P has 40 elements, \[P \cup Q\]has 60 elements and\[P \cap Q\]has 10 elements, how many elements does Q have? 

In a school there are 20 teachers who teach athematics or physics. Of these, 12 teach mathematics and 4 teach physics and mathematics. How many teach physics? 

In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many like both coffee and tea? 

Let A and B be two sets such that :\[n \left( A \right) = 20, n \left( A \cup B \right) = 42 \text{ and } n \left( A \cap B \right) = 4\] Find\[n\left( B \right)\]

In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find: 

how many can speak both Hindi and English: 

In a group of 50 persons, 14 drink tea but not coffee and 30 drink tea. Find:
(i) how may drink tea and coffee both;
(ii) how many drink coffee but not tea. 

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 numbers of people who read at least one of the newspapers.
(ii) the number of people who read exactly one newspaper. 

Of the members of three athletic teams in a certain school, 21 are in the basketball team, 26 in hockey team and 29 in the football team, 14 play hockey and basket ball 15 play hockey and football, 12 play football and basketball and 8 play all the three games. How many members are there in all? 

In a group of 1000 people, there are 750 who can speak Hindi and 400 who can speak Bengali. How many can speak Hindi only? How many can speak Bengali? How many can speak both Hindi and Bengali? 

\[\cap\]A survey of 500 television viewers produced the following information; 285 watch football, 195 watch hockey, 115 watch basketball, 45 watch football and basketball, 70 watch football and hockey, 50 watch hockey and basketball, 50 do not watch any of the three games. How many watch all the three games? How many watch exactly one of the three games?

In a survey of 100 persons it was found that 28 read magazine A, 30 read magazine B, 42 read magazine C, 8 read magazines A and B, 10 read magazines A and C, 5 read magazines B and C and 3 read all the three magazines. Find:
(i) How many read none of three magazines?
(ii) How many read magazine C only? 

In a survey of 100 students, the number of students studying the various languages were found to be : English only 18, English but not Hindi 23, English and Sanskrit 8, English 26, Sanskrit 48, Sanskrit and Hindi 8, no language 24. Find:
(i) How many students were studying Hindi?
(ii) How many students were studying English and Hindi? 

If n is a positive integer, prove that \[3^{3n} - 26n - 1\]  is divisible by 676.