KC Sinha: Exercise 1.2 - Mathematics Solution Class 9 Chapter 1 Real numbers

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Exercise 1.2

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Type 1

Question 1 

Are the square roots of all positive integers irrational ? If not, then give an example of the square root of a positive integer which is a rational number.
Sol :
No; Example: √9=3, which is a rational number

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Question 2

Identify two Statements in the following statements :
(i) √5 is a rational number
Sol : False

(ii) \sqrt{\dfrac{25}{169}} is a rational number
Sol : False
\dfrac{\sqrt{25}}{\sqrt{169}}=\sqrt{\dfrac{5times5}{13\times 13}} =\sqrt{\dfrac{5^2}{13^2}}=\dfrac{5}{13} = rational number

(iii) Every number which is not an integer, is an irrational number.
Sol : False, =\dfrac{3}{2} is not an integer but it is not an irrational number.

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Type 2

Question 3

Fill in the blanks :
(i) Decimal representation of a rational number \dfrac{8}{27} is __
Sol : 0.\overline{296}

(ii) Decimal representation of a rational number is either __ or __
Sol : Terminating, Recurring

(iii) Decimal representation of an irrational number is neither __ nor __
Sol : Terminating, Recurring

(iv) Corresponding to every point on the number line , there is a __ which is either __ or __
Sol : Real number, Rational, Irrational

(v) 0 is a/an __ number. (rational number)
Sol : Rational

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Question 4

Find, which of the variables x,y,z represent rational numbers and which represent irrational numbers ?
(i) x²=15
Sol: x represents an irrational number

(ii) y²=16
Sol: y represents a rational number

(iii) z²=0.09
Sol: z represents a rational number

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Question 5

State, which of the following numbers are rational numbers and which are irrational numbers ?
(i) \sqrt{47}
Sol : Irrational

(ii) \sqrt{19}
Sol : Irrational

(iii) \sqrt{576}
Sol : Rational

(iv) 1.232332333...
Sol : Irrational

(v) 0.98107
Sol : Rational

(vi) 2.201220012220001...
Sol : Irrational

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Question 6

Which of the following number is irrational ?
(i) \sqrt{\dfrac{36}{64}}
Sol:

(ii) √3
Sol:

(iii)  -\dfrac{9}{\sqrt{49}}
Sol:

(iv)  \dfrac{9}{81}
Sol:

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Question 7

Which of the following number is not irrational ?
(i) \dfrac{2}{\sqrt{3}},\dfrac{\sqrt{3}}{4},\sqrt{\dfrac{25}{49}},7\sqrt{2}
Sol: =\sqrt{\dfrac{25}{49}}=\dfrac{5}{7}

(ii) \dfrac{3}{\sqrt{2}},\dfrac{1}{2},\sqrt{3},\sqrt{11}
Sol: =\dfrac{1}{2}

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Question 8

State, which of the following does not belong to the same class :
(i) \sqrt{\dfrac{81}{625}} , \sqrt{\dfrac{625}{325}},\sqrt{\dfrac{5}{196}}
Sol : =\sqrt{\dfrac{81}{625}}=\dfrac{9}{25}

(ii) \sqrt{7},3\sqrt{7},\sqrt[4]{\dfrac{25}{256}},\sqrt{33}
Sol : \sqrt[4]{\dfrac{25}{256}}

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Category A

Question 9

(i) Insert an irrational number between \dfrac{1}{3}\text{ and }\dfrac{1}{2}
Sol : 0.4010010001...

(ii) Find a rational and an irrational number between 2 and 3. How many rational and irrational numbers are there between 2 and 3 ?
Sol : \dfrac{5}{2} ,
2.010020003..infinite

(iii) Find three irrational numbers between 0.2 and 0.32
Sol :
0.251010010001.. ,
0.252020020002.. ,
0.1030030003..

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Category B

Question 10

(i) Write two rational numbers between √3 and √5
Sol : 1.81 , 1.82

(ii) Write three rational numbers between √5 and √7
Sol : 2.51 , 2.52 , 2.53

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Category C

Question 11

(i) Find two irrational numbers between √3 and √5
Sol : √3.1 , √3.2

(ii) Write three irrational numbers between √5 and √7
Sol : √5.1 , √5.2 , √5.3

(iii) Find three irrational numbers between 0.1 and 0.1101
Sol : 0.1010010001...,
0.1020020002... ,
0.1030030003.. ,

(iv) Find two rational numbers between 0.01001000100001.. and 0.1001000100001..
Sol : 0.02 , 0.03