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

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

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) √? + √? = √?

(ii) ?√? + ? = ?

(iii) ?√? − ?√? = √?

(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 √? and √? are irrational.

Prove that each of the following numbers is irrational:

(i) √? + √?

(ii) ? − √?

(iii) √? − ?

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) ?√? ??? ?√?

(ii) ? √? ? ??? ? ³√?

(iii) ?√?, ?√? ??? ?√?

Write in descending order:

(i) ? √? ? ??? ? √? ?

(ii) ?√? ??? ?√? ^{^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) √?? ??? \sqrt[3]{35}

Insert two irrational numbers between 5 and 6.

Insert five irrational numbers between ?√? and ?√?.

Write two rational numbers between √? ??? √?.

Write three rational numbers between √? ??? √?.

Simplify each of the following:

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

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

(iii) (? + √?)(? + √?)

(iv) (√? − √?)²

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

(i) √???

(ii) \sqrt[4]{27}

(iii) \sqrt[5]{128}

(iv) \sqrt[3]{64}

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

(vi) \sqrt{-125}

(vii) √?

(viii) √? + √?

Write the lowest rationalizing factor of:

(i) ?√?

(ii) √??

(iii) √? − ?

(iv) ? − √?

(v) √?? − √??

(vi) √? − √?

(vii) √?? + ?

(viii) ?? − ?√?

(ix) ?√? + ?√?

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 ? = ?√? + ?√?, find:

(I) \frac{1}{x}

(ii) ? + \frac{1}{x}

(iii) (? + \frac{1}{x} )²

If ? = ? − √?, 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 ? = ? − ?√?, find: ?? + \frac{1}{^{x^2}}

If √? = ?. ? and √? = ?. ?, 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.

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 √??=3.2.

If x =√? − √?. 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 √? cm.

Draw a line segment of length √? cm.

Draw a line segment of length √? 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=√?.