Selina solutions concise maths class 9 icse Solutions for Chapter 1 chapter 1 rational and irrational numbers

Exercises in Chapter 1 chapter 1 rational and irrational numbers Grade 9

Exercise 1(A)

Exercise 1(B)

Exercise 1(C)

Exercise 1(D)

Questions in Exercise 1(A)

Arrange

-5/9, 7/12 ,-2/3and11/18

in the ascending order of their magnitudes. Also,

find the difference between the largest and the smallest of these rational

numbers. Express this difference as a decimal fraction correct to one decimal

place.

Arrange

5/8 ,-3/16 ,-1/4 and 17/32

in the descending order of their magnitudes. Also, find the sum of the lowest and the largest of these rational numbers. Express the result obtained as a decimal fraction correct to two decimal places.

Without doing any actual division, find which of the following rational

numbers have terminating decimal representation:

(I) 7/16

(ii) 23/125

(iii) 9/14

(iv) 32/45

(v) 43/50

(vi) 17/40

(vii) 61/75

(viii) 123/250

Questions in Exercise 1(B)

State whether the following numbers are rational or not:

$(i) $(2+\sqrt{2})^{2}$ (ii) $(3-\sqrt{3})^{2}$ (iii) $(5+\sqrt{5})(5-\sqrt{5})$ (iv) $(\sqrt{3}-\sqrt{2})^{2}$ $(\mathbf{v})$ $\left(\frac{3}{2 \sqrt{2}}\right)^{2}$ $(\mathbf{v i})$ $\left(\frac{\sqrt{7}}{6 \sqrt{2}}\right)^{2}$$

Find the square of:

$\begin{aligned} &\text { (i) }\\ &\left(\frac{3 \sqrt{5}}{5}\right)^{2} \end{aligned}$

$\begin{aligned} &\text { (ii) }\\ &\sqrt{3}+\sqrt{2} \end{aligned}$

$\begin{aligned} &\text { (iii) }\\ &\sqrt{5}-2 \end{aligned}$

$\text { (iv) } \quad 3+2 \sqrt{5}$

State, in each case, whether true or false:

(i) √𝟐 + √𝟑 = √𝟓

(ii) 𝟐√𝟒 + 𝟐 = 𝟔

(iii) 𝟑√𝟕 − 𝟐√𝟕 = √𝟕

(iv) 2/7 is an irrational number.

(v) 5/11 is a rational number.

(vi) All rational numbers are real numbers.

(vii) All real numbers are rational numbers.

(viii) Some real numbers are rational numbers.

From the given set, find:

(i) Set of Rational numbers

(ii) Set of irrational numbers

(iii) Set of integers

(iv) Set of non-negative integers

Use method of contradiction to show that √𝟑 and √𝟓 are irrational.

Prove that each of the following numbers is irrational:

(i) √𝟑 + √𝟐

(ii) 𝟑 − √𝟐

(iii) √𝟓 − 𝟐

Write a pair of irrational numbers whose sum is irrational.

Write a pair of irrational numbers whose sum is rational.

Write a pair of irrational numbers whose difference is irrational.

Write a pair of irrational numbers whose difference is rational.

Write a pair of irrational numbers whose product is irrational.

Write a pair of irrational numbers whose product is rational.

Write in ascending order:

(i) 𝟑√𝟓 𝒂𝒏𝒅 𝟒√𝟑

(ii) 𝟐 √𝟓 𝟑 𝒂𝒏𝒅 𝟑 ³√𝟐

(iii) 𝟔√𝟓, 𝟕√𝟑 𝒂𝒏𝒅 𝟖√𝟐

Write in descending order:

(i) 𝟐 √𝟔 𝟒 𝒂𝒏𝒅 𝟑 √𝟐 𝟒

(ii) 𝟕√𝟑 𝒂𝒏𝒅 𝟑√𝟕 $^{^4}$

$\begin{array}{l} 7 \sqrt{3}=\sqrt{7^{2} \times 3}=\sqrt{147} \\ 8 \sqrt{2}=\sqrt{8^{2} \times 2}=\sqrt{128} \end{array}$

Compare:

(i) $\sqrt[6]{15}$ and $\sqrt[4]{12}$

(ii) √𝟐𝟒 𝒂𝒏𝒅 $\sqrt[3]{35}$

Insert two irrational numbers between 5 and 6.

Insert five irrational numbers between 𝟐√𝟓 and 𝟑√𝟑.

Write two rational numbers between √𝟐 𝒂𝒏𝒅 √𝟑.

Write three rational numbers between √𝟑 𝒂𝒏𝒅 √𝟓.

Simplify each of the following:

(i) $\sqrt[5]{16} \times \sqrt[5]{2}$

(ii) $\frac{\sqrt[4]{243}}{\sqrt[4]{3}}$

(iii) (𝟑 + √𝟐)(𝟒 + √𝟕)

(iv) (√𝟑 − √𝟐)²

Questions in Exercise 1(C)

State, with reason, which of the following are surds and which are not:

(i) √𝟏𝟖𝟎

(ii) $\sqrt[4]{27}$

(iii) $\sqrt[5]{128}$

(iv) $\sqrt[3]{64}$

(v) $\sqrt[3]{23}.\sqrt[3]{40}$

(vi) $\sqrt{-125}$

(vii) √𝝅

(viii) √𝟑 + √𝟐

Write the lowest rationalizing factor of:

(i) 𝟓√𝟐

(ii) √𝟐𝟒

(iii) √𝟓 − 𝟑

(iv) 𝟕 − √𝟕

(v) √𝟏𝟖 − √𝟓𝟎

(vi) √𝟓 − √𝟐

(vii) √𝟏𝟑 + 𝟑

(viii) 𝟏𝟓 − 𝟑√𝟐

(ix) 𝟑√𝟐 + 𝟐√𝟑

Rationalize the denominators of:

Find the values of ‘a’ and ‘b’ in each of the following:

$\begin{aligned} &(i)\\ &\frac{2+\sqrt{3}}{2-\sqrt{3}}=a+b \sqrt{3} \end{aligned}$

$\text { (ii) } \quad \frac{\sqrt{7}-2}{\sqrt{7}+2}=a \sqrt{7}+b$

$\text { (iii) } \frac{3}{\sqrt{3}-\sqrt{2}}=a \sqrt{3}+b \sqrt{2}$

$\text { (iv) } \frac{5+3 \sqrt{2}}{5-3 \sqrt{2}}=a+b \sqrt{2}$

Simplify:

(i) $\frac{22}{2 \sqrt{3}+1}+\frac{17}{2 \sqrt{3}-1}$

(ii) $\frac{\sqrt{2}}{\sqrt{6}-\sqrt{2}}+\frac{\sqrt{3}}{\sqrt{6}+\sqrt{2}}$

If x= $=\frac{\sqrt{5}-2}{\sqrt{5}+2} \text { and } y=\frac{\sqrt{5}+2}{\sqrt{5}-2}$ Find:

(i) x²

(ii) y²

(iii) xy

(iv) x² +y² =xy

If m $=\frac{1}{3-2 \sqrt{2}} \text { and } n=\frac{1}{3+2 \sqrt{2}}$ find:

(i) m²

(ii) n²

(iii) mn

If 𝒙 = 𝟐√𝟑 + 𝟐√𝟐, find:

(I) $\frac{1}{x}$

(ii) 𝒙 + $\frac{1}{x}$

(iii) (𝒙 + $\frac{1}{x}$ )²

If 𝒙 = 𝟏 − √𝟐, find the value of $\left(x+\frac{1}{x}\right)^{3}$

Rationalize the denominator of:

$\frac{1}{\sqrt{3}-\sqrt{2}+1}$

Show that:

$\frac{1}{3-2 \sqrt{2}}-\frac{1}{2 \sqrt{2}-\sqrt{7}}+\frac{1}{\sqrt{7}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{5}-2}=5$

If 𝒙 = 𝟓 − 𝟐√𝟔, find: 𝒙𝟐 + $\frac{1}{^{x^2}}$

If √𝟐 = 𝟏. 𝟒 and √𝟑 = 𝟏. 𝟕, find the value of each of the following, correct to

one decimal place:

(I) $\frac{1}{\sqrt{3}-\sqrt{2}}$

(ii) $\frac{1}{\sqrt{3}+\sqrt{2}}$

(iii) $\frac{2-\sqrt{3}}{\sqrt{3}}$

Evaluate:

$\frac{4-\sqrt{5}}{4+\sqrt{5}}+\frac{4+\sqrt{5}}{4-\sqrt{5}}$

.If $\frac{2+\sqrt{5}}{2-\sqrt{5}}=x \text { and } \frac{2-\sqrt{5}}{2+\sqrt{5}}=y$ find the value of x²-y².

Questions in Exercise 1(D)

Simplify: $\frac{\sqrt{18}}{5 \sqrt{18}+3 \sqrt{72}+2 \sqrt{162}}$

Simplify:

$\frac{\sqrt{x^{2}+y^{2}}-y}{x-\sqrt{x^{2}-y^{2}}} \div \frac{\sqrt{x^{2}-y^{2}}+x}{\sqrt{x^{2}+y^{2}}+y}$

Evaluate, correct to one place of decimal. The expression $\frac{5}{\sqrt{20}-\sqrt{10}} \text { if } \sqrt{5}=2.2$ and √𝟏𝟎=3.2.

If x =√𝟑 − √𝟐. Find the value of:

(I) $x+\frac{1}{x}$

(ii) $x^{2}+\frac{1}{x^{2}}$

(iii) $x^{3}+\frac{1}{x^{3}}$

(iv) $x^{3}+\frac{1}{x^{3}}-3\left(x^{2}+\frac{1}{x^{2}}\right)+x+\frac{1}{x}$

Show that:

(i) Negative of an irrational number is irrational.

(ii) The product of a non-zero rational number and an irrational number is an irrational number.

Draw a line segment of length √𝟓 cm.

Draw a line segment of length √𝟑 cm.

Draw a line segment of length √𝟖 cm.

Show that:

$\frac{4-\sqrt{5}}{4+\sqrt{5}}+\frac{2}{5+\sqrt{3}}+\frac{4+\sqrt{5}}{4-\sqrt{5}}+\frac{2}{5-\sqrt{3}}=\frac{52}{11}$

Show that:

(i) $x^{3}+\frac{1}{x^{3}}=52, \text { if } x=2+\sqrt{3}$

(ii) $x^{2}+\frac{1}{x^{2}}=34, \text { if } x=3+2 \sqrt{2}$

(iii) $\frac{3 \sqrt{2}-2 \sqrt{3}}{3 \sqrt{2}+2 \sqrt{3}}+\frac{2 \sqrt{3}}{\sqrt{3}-\sqrt{2}}=11$

Show that x is irrational if:

(i) x² =6

(ii) x² =0.009

(iii)x² =27

Show that x is rational if:

(i) x²=16

(ii) x2=0.0004

(iii) x2=

$1\frac{7}{9}$

Using the following figure, show that BD=√𝒙.

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Chapters in this book

Chapter 1- Rational and Irrational Numbers

Chapter 2- Compound Interest [Without Using Formula

Chapter 3- Compound Interest [Using Formula

Chapter 4- Expansions

Chapter 5- Factorisation

Chapter 6- Simultaneous Equations

Chapter 7- Indices [exponents]

Chapter 8- Logarithms

Chapter 9- Triangles [Congruency in Triangles]

Chapter 10- Isosceles Triangle

Chapter 11- Inequalities

Chapter 12- Mid-Point and Its Converse

Chapter 13- Pythagoras Theorem

Chapter 14- Rectilinear Figures

Chapter 15- Construction of Polygons

Chapter 16- Area Theorems

Chapter 17- Circles

Chapter 18- Statistics

Chapter 19- Mean and Median

Chapter 20- Area and Perimeter of Plane Figures

Chapter 21- Solids

Chapter 22- Trigonometrical Ratios

Chapter 23- Trigonometrical Ratios of Standard Angles

Chapter 24- Solution Of Right Triangles

Chapter 25- Complementary angles

Chapter 26- Co-ordinate Geometry

Chapter 27- Graphical Solution

Chapter 28- Distance Formula