TEST PAPER (SET THEORY AND RELATION (OR FUNCTION)

1) For three sets A, B and C, if A∩C = B∩ C= ∅ and AU C = B UC, BC then prove that A= B.

2) For any three sets A, B and C, If A∩B= A∩ C and AUB = AUC then show that B= C.

3) Two finite sets A and B consist of m and n elements respectively. The number of subsets in A exceeds that of B by 112. Find the value of m and n.

4) A survey shows that 75% of the student of a school like Mathematics and 65% like Physics. If x% of the students like both Mathematics and Physics, find the maximum and minimum values of x.

5) In a survey of 35 students of a class it was found that 17 students like Mathematics and 10 like Mathematics but not Biology. Find the number of students who like 

a) Biology 

b) Biology but not Mathematics ,

It being given that each student takes at least one of the two subjects.

6) An enquiry into 100 candidates of failed in English of HS Examination revealed the following data:

failed in aggregate-66, failed in paper I-37, failed in paper II -59, failed in aggregate and paper I- 17, failed in aggregate and paper II -43 and failed in both papers -13.

 Find the number of candidates who failed in

a) aggregate of paper II but not paper I.

b) aggregate but not in paper I and paper II.

7) In a survey it was found that 76 men read magazine A, 30 read magazine B, 40 read magazine C and and 6 men read all the three magazines . If the total number of men who read magazines be 116, find how many men read exactly two of the three magazines.    

8) For two sets A and B, the 3 elements of A x B are (a,x),(b,y),(c,x); find B x A.    

9) If (a,b) and (b,c) are elements of A x A, find the set A and other elements of A x A.

10) Find the domain of definition of each of the following functions:

a) f(x)= 1/log₁₀(1- x)  + √(x +2).

b) f(x)=log{(3+ x)/(3- x)}.

11) If f(x+y)= f(x)+ f(y) for all real number x and y, show that f(x)=xf(1).

12) f(x + 1/x)= x²+ 1/x², find f(x).

13) If 2f(1/x)+ f(x)=3x, find f(x - 1/x).

14) If f(x)= tan(x - π/4), find f(x). f(-x).

15) A cubic function f(x) satisfies the relation f(x)+ f(1/x)= f(x). f(1/x), show that f(x)= 1+ x³ or 1- x³. Further, if f(2)= 9, show that f(4)= 65.