Page 1.35
Type 1
Sol :
No; Example: √9=3, which is a rational number
(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.
Type 2
(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
(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
(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
(i) $\sqrt{\dfrac{36}{64}}$
Sol:
(ii) √3
Sol:
(iii) $-\dfrac{9}{\sqrt{49}}$
Sol:
(iv) $\dfrac{9}{81}$
Sol:
(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: $=\dfrac{1}{2}$
(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 : $\sqrt[4]{\dfrac{25}{256}}$
Category A
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..
Category B
Sol : 1.81 , 1.82
(ii) Write three rational numbers between √5 and √7
Sol : 2.51 , 2.52 , 2.53
Category C
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
Type 4
(i) -√2
Sol :
(ii) √5
Sol :
(iii) $-\dfrac{\sqrt{3}}{2}$
Sol :
Exercise 1.2
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
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.
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
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
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
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:
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: $=\dfrac{1}{2}$
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 : $\sqrt[4]{\dfrac{25}{256}}$
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..
Category B
Question 10
(i) Write two rational numbers between √3 and √5Sol : 1.81 , 1.82
(ii) Write three rational numbers between √5 and √7
Sol : 2.51 , 2.52 , 2.53
Category C
Question 11
(i) Find two irrational numbers between √3 and √5Sol : √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
Type 4
Question 12
Represent following irrational numbers on the number line :(i) -√2
Sol :
(ii) √5
Sol :
(iii) $-\dfrac{\sqrt{3}}{2}$
Sol :
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