Number system|solution set 1|number basics,recurring decimals

Find the value of .1777..+ .131313...?

         (1)5/15 (2) 17/55 (3) 31/99 (4) 147/990

Ans: option (2). Simply convert both into fractions and add. 

 .177.. =(17-1)/90 and .1313.. = 13/99

Which among the following is a rational number?

         (1)  2^∛8  (2) .122222…(3)  .101001000…(4) both 1 and 2

Ans: option (4). In option one ∛8 is nothing but 2 and option two is a recurring decimal , but option three has no repeating pattern.

What is the fraction form of 123.34565656....?

Ans: Apply the formula given in the theory. It is nothing but (1233456-12334)/9900

If N = .mnmnmn... and m and n are not necessarily distinct, how many values can the denominator take in N?

(1)5 (2) 1 (3) 3 (4) 4

Ans: option (5)

It is a very important type.

case1: Both m and n are distinct then denominator is 99. However what if the number mn has a common factor with 99? the factors of 99 are 1,99,3,9,11,33. So the denominator can get reduced to 33 or 11

case2:when m and n are same denominator is 9. However if m or n has a common factor with 9 then the denominator can get reduced to 3.

For eg 

m=0, n=9 Denominator=11

m=0,n=3 Denominator=33

m=2,n=3 Denominator=99

m=n=3 Denominator =3

m=n=5 Denominator=9

So total values is nothing but the factors of 99 greater than 1.

Which of the following statement(s) is or are correct?

           (1)0 is an integer (2) the sum of two irrational numbers is always irrational (3) For two real   numbers a and b if a (4) product of two irrational numbers is always irrational (5) the sum and product of one rational and one irrational number is always irrational.

Ans: (1) is correct,(2) is correct,(3) not correct in case of negative numbers like -3 is less than -2 but 9 is greater than 4 , (4) once again not always as √2×√2  is rational however √2×√3 is irrational ,(5) always correct