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Q3 (3)   The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form, p/q what can you say about the prime factors of q?  

              43. \overline{123456789}

As the decimal part of the given number is non-terminating and repeating, the number is rational but its denominator will have factors other than 2 and 5.

Q3 (2)   The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form , p / q what can you say about the prime factors of q? 0.120120012000120000

Since the decimal part of the given number is non-terminating and non-repeating we can conclude that the given number is irrational and cannot be written in the form  where p and q are integers.

Q3 (1)   The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form , p / q what can you say about the prime factors of q? 43.123456789

The denominator is of the form 2a x 5b where a = 9 and b = 9. Therefore the given number is rational and has a terminating decimal expansion.

Q2   Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.

decimal expansions of  rational numbers are (i)  (ii)  (iv)  (vI)  (viii)  (ix) 

Q1 (10)   Without actually performing the long division, state whether the following rational
               numbers will have a terminating decimal expansion or a non-terminating repeating decimal
              expansion:  77 / 210 

The denominator is not of the form 2a x 5b. Therefore the given rational number will have a non-terminating repeating decimal expansion.

Q1 (9)   Without actually performing the long division, state whether the following rational
             numbers will have a terminating decimal expansion or a non-terminating repeating decimal
              expansion: 35 / 50 

The denominator is of the form 2a x 5b where a = 1 and b = 1. Therefore the given rational number will have a terminating decimal expansion.

Q1 (8)   Without actually performing the long division, state whether the following rational
              numbers will have a terminating decimal expansion or a non-terminating repeating decimal
             expansion: 6 / 15 

The denominator is of the form 2a x 5b where a = 0 and b = 1. Therefore the given rational number will have a terminating decimal expansion.

Q1 (7)    Without actually performing the long division, state whether the following rational
             numbers will have a terminating decimal expansion or a non-terminating repeating decimal
            expansion: \frac{129 }{2 ^ 2 7^5 5 ^ 7 }

The denominator is not of the form 2a x 5b. Therefore the given rational number will have a non-terminating repeating decimal expansion.

Q1 (6)   Without actually performing the long division, state whether the following rational
             numbers will have a terminating decimal expansion or a non-terminating repeating decimal
             expansion:  \frac{23}{2 ^3 5 ^ 2 }

The denominator is of the form 2a x 5b where a = 3 and b = 2. Therefore the given rational number will have a terminating decimal expansion.

Q 1 (5)   Without actually performing the long division, state whether the following rational
              numbers will have a terminating decimal expansion or a non-terminating repeating decimal
            expansion: 29 / 343

The denominator is not of the form 2a x 5b. Therefore the given rational number will have a non-terminating repeating decimal expansion.

Q 1 (4)    Without actually performing the long division, state whether the following rational
               numbers will have a terminating decimal expansion or a non-terminating repeating decimal
               expansion: 15 / 1600 

The denominator is of the form 2a x 5b where a = 6 and b = 1. Therefore the given rational number will have a terminating decimal expansion.

Q1 (3)   Without actually performing the long division, state whether the following rational
            numbers will have a terminating decimal expansion or a non-terminating repeating decimal
           expansion:  64 / 455 

The denominator is not of the form 2a x 5b. Therefore the given rational number will have a non-terminating repeating decimal expansion.

Q1 (2)   Without actually performing the long division, state whether the following rational
             numbers will have a terminating decimal expansion or a non-terminating repeating decimal
             expansion:   

              17 / 8 

The denominator is of the form 2a x 5b where a = 3 and b = 0. Therefore the given rational number will have a terminating decimal expansion.

Q3 (3)   Prove that the following are irrationals :  6 + \sqrt 2

Let us assume  is rational. This means  can be written in the form  where p and q are co-prime integers. As p and q are integers  would be rational, this contradicts the fact that  is irrational. This contradiction arises because our initial assumption that  is rational was wrong. Therefore  is irrational.

 Q3 (2)   Prove that the following are irrationals :

(ii) 7 \sqrt 5

Let us assume  is rational. This means  can be written in the form  where p and q are co-prime integers. As p and q are integers  would be rational, this contradicts the fact that  is irrational. This contradiction arises because our initial assumption that  is rational was wrong. Therefore  is irrational.

Q3  Prove that the following are irrationals : 

(i) 1/ \sqrt 2 

Let us assume  is rational. This means  can be written in the form  where p and q are co-prime integers. Since p and q are co-prime integers  will be rational, this contradicts the fact that   is irrational. This contradiction arises because our initial assumption that   is rational was wrong. Therefore  is irrational.

Q2  Prove that 3 + 2 \sqrt 5  is irrational.

Let us assume  is rational. This means  can be written in the form  where p and q are co-prime integers. As p and q are integers  would be rational, this contradicts the fact that  is irrational. This contradiction arises because our initial assumption that  is rational was wrong. Therefore  is irrational.

Q1  Prove that \sqrt 5  is irrational.

Let us assume  is rational. It means  can be written in the form  where p and q are co-primes and  Squaring both sides we obtain From the above equation, we can see that p2 is divisible by 5, Therefore p will also be divisible by 5 as 5 is a prime number.  Therefore p can be written as 5r p = 5r  p2 = (5r)2 5q2 = 25r2 q2 = 5r2 From the above equation, we can see that q2 is divisible by 5,...

Q7   There is a circular path around a sports field. Sonia takes 18 minutes to drive one round
        of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the
        same point and at the same time, and go in the same direction. After how many minutes
        will they meet again at the starting point?

The time after which they meet again at the starting point will be equal to the LCM of the times they individually take to complete one round. Time taken by Sonia = 18 = 2 x 32 Time taken by Ravi = 12 = 22 x 3  LCM(18,12) = 22 x 32 = 36 Therefore they would again meet at the starting point after 36 minutes.

Q6  Explain why 7 \times 11 \times 13 + 13 and 7 \times  6 \times  5 \times 4 \times 3 \times 2 \times 1 + 5 are composite numbers.

7 x 11 x 13 + 13 = (7 x 11 + 1) x 13 = 78 x 13 = 2 x 3 x 132 7 x 6 x 5 x 4 x 3 x 2 x 1 + 5 = (7 x 6 x 4 x 3 x 2 x 1 + 1) x 5 = 5 x 1008  After Solving we observed that both the number are even numbers and the number rule says that we can take atleast two common out of two numbers. So that the number is composite number.
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