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MATHEMATICS [3 marks sum]

Irrational Numbers · ICSE STD 9 (ICSE - English Medium). 25 questions.

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25 questions · timed · auto-graded

Question 13 Marks
Simplify the following:$\frac{4+\sqrt{5}}{4-\sqrt{5}}+\frac{4-\sqrt{5}}{4+\sqrt{5}}$
Answer
$\frac{4+\sqrt{5}}{4-\sqrt{5}}+\frac{4-\sqrt{5}}{4+\sqrt{5}}$
$=\frac{(4+\sqrt{5})^2+(4-\sqrt{5})^2}{(4-\sqrt{5})(4+\sqrt{5})} $
$=\frac{16+5+8 \sqrt{5}+16+5-8 \sqrt{5}}{16-5}$
$=\frac{42}{11}$
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Question 23 Marks
Simplify by rationalising the denominator in the following.$\frac{\sqrt{5}-\sqrt{7}}{\sqrt{3}}$
Answer
$\frac{\sqrt{5}-\sqrt{7}}{\sqrt{3}}$
$=\frac{\sqrt{5}-\sqrt{7}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} $
$=\frac{\sqrt{5} \times \sqrt{3}-\sqrt{7} \times \sqrt{3}}{(\sqrt{3})^2}$
$ =\frac{\sqrt{15}-\sqrt{21}}{3}$
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Question 33 Marks
Simplify by rationalising the denominator in the following.$\frac{3 \sqrt{2}}{\sqrt{5}}$
Answer
$\frac{3 \sqrt{2}}{\sqrt{5}}$
$ =\frac{3 \sqrt{2}}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} $
$ =\frac{3 \sqrt{2} \times \sqrt{5}}{(\sqrt{5})^2}$
$=\frac{3 \sqrt{10}}{5}$
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Question 43 Marks
Insert two irrational numbers between $3$ and $4.$
Answer
Since $3$ and $4$ are rational numbers and $3 \times 4=12$ is not a perfect square.
$\therefore$ One irrational number between $3$ and $4=\sqrt{3 \times 4}=\sqrt{12}$
And , an irrational number between $3$ and $\sqrt{12}=\sqrt{3 \times \sqrt{12}}=\sqrt{3 \sqrt{12}}$
$\therefore$ Required irrational numbers between $3$ and $4$ are : $\sqrt{12}$ and $\sqrt{3 \sqrt{12}}$
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Question 53 Marks
Write the following in ascending order:$2 \sqrt{5}, \sqrt{3}$ and $5 \sqrt{2}$
Answer
$2 \sqrt{5}=\sqrt{2^2 \times 5}=\sqrt{4 \times 5}=\sqrt{20}$
$\sqrt{3}=\sqrt{3} $
$5 \sqrt{2}=\sqrt{5^2 \times 2}=\sqrt{25 \times 2}=\sqrt{50}$
Since, $3<20<50$,
we have $\sqrt{3}<\sqrt{20}<\sqrt{50}$.
Hence, $\sqrt{3}<\sqrt{2}<5 \sqrt{2}$.
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Question 63 Marks
Write a pair of irrational numbers whose product is rational.
Answer
$(\sqrt{3}+\sqrt{2})$ and $(\sqrt{3}-\sqrt{2})$ are irrational numbers whose product is rational.
Thus, we have
$(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})$
$=(\sqrt{3})^2-(\sqrt{2})^2$
$=3-2$
$=1$, which is a rational number.
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Question 73 Marks
Write a pair of irrational numbers whose product is irrational.
Answer
Consider two irrational numbers $(5+\sqrt{2})$ and $(\sqrt{5}-2)$.
Thus, we have
$(5+\sqrt{2})(\sqrt{5}-2) $
$=5(\sqrt{5}-2)+\sqrt{2}(\sqrt{5}-2)$
$=5 \sqrt{5}-10+\sqrt{10}-2 \sqrt{2}$ , which is irrational.
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Question 83 Marks
Write a pair of irrational numbers whose difference is rational.
Answer
$(\sqrt{5}-3)$ and $(\sqrt{5}+3)$ are irrational numbers whose difference is rational.
Thus, we have
$(\sqrt{5}-3)-(\sqrt{5}+3) $
$=\sqrt{5}-3-\sqrt{5}-3$
$=-6$,
which is a rational number.
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Question 93 Marks
Write a pair of irrational numbers whose difference is irrational.
Answer
$(\sqrt{3}+2)$ and $(\sqrt{2}-3)$ are irrational numbers whose difference is irrational.
Thus, we have
$(\sqrt{3}+2)-(\sqrt{2}-3)$
$=\sqrt{3}+2-\sqrt{2}+3$
$=\sqrt{3}-\sqrt{2}+5,$
which is irrational.
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Question 103 Marks
Write a pair of irrational numbers whose sum is rational.
Answer
$(\sqrt{3}+5)$ and $(4-\sqrt{3})$ are two irrational numbers whose sum is rational.
Thus, we have
$(\sqrt{3}+5)+(4-\sqrt{3}) $
$=\sqrt{3}+5+4-\sqrt{3}$
$=9$,
which is a rational number.
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Question 113 Marks
Write a pair of irrational numbers whose sum is irrational.
Answer
$(\sqrt{3}+5)$ and $(\sqrt{5}-3)$ are irrational numbers whose sum is irrational.
Thus, we have
$(\sqrt{3}+5)+(\sqrt{5}-3) $
$=\sqrt{3}+5+\sqrt{5}-3$
$=\sqrt{3}+\sqrt{5}+2,$
which is irrational.
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Question 123 Marks
Check whether the square of the following is rational or irrational:$\sqrt{2}+\sqrt{3}$
Answer
$(\sqrt{2}+\sqrt{3})^2$
$ =(\sqrt{2})^2+(\sqrt{3})^2+2 \times \sqrt{2} \times \sqrt{3} $
$=2+3+2 \sqrt{6} $
$ =5+2 \sqrt{6}$
which is irrational.
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Question 133 Marks
Check whether the square of the following is rational or irrational:$3 + \sqrt{2}$
Answer
$(3+\sqrt{2})^2 $
$=(3)^2+(\sqrt{2})^2+2 \times 3 \times \sqrt{2} $
$ =9+2+6 \sqrt{2} $
$ =11+6 \sqrt{2},$
which is irrational.
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Question 143 Marks
State if the following is a surd. Give reasons.$\sqrt[12]{8} \div \sqrt[6]{6}$
Answer
$\sqrt[12]{8} \div \sqrt[6]{6} $
$=\frac{\sqrt[12]{8}}{\sqrt[6]{6}}$
Numerator and Denominator, both are irrational numbers.
Hence, $\sqrt[12]{8} \div \sqrt[6]{6}$ is a surd.
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Question 153 Marks
State whether the following number is rational or irrational:$(2+\sqrt{2})(2-\sqrt{2})$
Answer
$(2+\sqrt{2})(2-\sqrt{2}) $
$=(2)^2-(\sqrt{2})^2 $
$=4-2$
$ =$2  which is rational.
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Question 163 Marks
State whether the following number is rational or irrational:$(5-\sqrt{5})^2$
Answer
$(5-\sqrt{5})^2 $
$ =(5)^2+(\sqrt{5})^2-2 \times 5 \times \sqrt{5}$
$=25+5-10 \sqrt{5}$
$=30-10 \sqrt{5},$
which is irrational.
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Question 173 Marks
State whether the following number is rational or irrational:$(3+\sqrt{3})^2$
Answer
$(3+\sqrt{3})^2$
$=(3)^2+(\sqrt{3})^2+2 \times 3 \times \sqrt{3}$
$=9+3+6 \sqrt{3}$
$=12+6 \sqrt{3},$
which is irrational .
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Question 183 Marks
Insert five irrational number's between $2 \sqrt{3}$ and $3 \sqrt{5}$.
Answer
We know that $2 \sqrt{3}=\sqrt{4 \times 3}=\sqrt{12}$ and $3 \sqrt{5}=\sqrt{9 \times 5}=\sqrt{45}$.
Thus, we have $\sqrt{12}<\sqrt{13}<\sqrt{14}<\sqrt{17}<\ldots \ldots<\sqrt{43}<\sqrt{44}<\sqrt{45}$
So, any five irrational numbers between $2 \sqrt{3}$ and $3 \sqrt{5}$ are:
$\sqrt{13}, \sqrt{14}, \sqrt{23}, \sqrt{37}, \sqrt{41}$
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Question 193 Marks
Insert a rational number between:$\frac{5}{9}$ and $\frac{6}{7}$
Answer
A rational number lying between $\frac{5}{9}$ and $\frac{6}{7}$
$=\frac{\frac{5}{9}+\frac{6}{7}}{2} $
$=\frac{\frac{35+54}{63}}{2} $
$=\frac{\frac{89}{63}}{2}$
$=\frac{89}{126}$
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Question 203 Marks
Insert a rational number between:$\frac{4}{3}$ and $\frac{7}{5}$
Answer
A rational number lying between $\frac{4}{3}$ and $\frac{7}{5}$
$=\frac{\frac{4}{3}+\frac{7}{5}}{2} $
$=\frac{\frac{20+21}{15}}{2} $
$=\frac{\frac{41}{15}}{2} $
$=\frac{41}{30}$
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Question 213 Marks
Insert a rational number between:$\frac{3}{4}$ and $\frac{5}{7}$
Answer
A rational number lying between $\frac{3}{4}$ and $\frac{5}{7}$
$=\frac{\frac{3}{4}+\frac{5}{7}}{2} $
$=\frac{\frac{21+20}{28}}{2} $
$=\frac{\frac{41}{28}}{2} $
$=\frac{41}{56}$
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Question 223 Marks
Insert a rational number between:$\frac{2}{5}$ and $\frac{3}{4}$
Answer
A rational number lying between $\frac{2}{5}$ and $\frac{3}{4}$
$=\frac{\frac{2}{5}+\frac{3}{4}}{2} $
$=\frac{\frac{8+15}{20}}{2}$
$=\frac{\frac{23}{20}}{2} $
$=\frac{23}{40}$
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Question 233 Marks
Express the following decimal as a rational number.$0.7$
Answer
Let $x=0.7$
Then, $x=0.7777 \ldots$
Here, the number of digits recurring is only $1,$
so we multiply both sides of the equation $(1)$ by $10 .$
$\therefore 10 x=10 \times 0.7777 \ldots . $
$=7.777 \ldots \ldots(2)$
On subtracting $(1)$ from $(2),$ we get
$9 x=7 $
$\therefore x=\frac{7}{9} $
$\therefore 0.7=\frac{7}{9}$
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Question 243 Marks
Express the following decimal as a rational number.$21.025$
Answer
$21.025 $
$=\frac{21.025}{1000} $
$ =\frac{21025 \div 25}{1000 \div 25} $
$ =\frac{841}{40}$
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Question 253 Marks
Express the following decimal as a rational number.$0.614$
Answer
$=\frac{614}{1000} $
$=\frac{614 \div 2}{1000 \div 2}$
$ =\frac{307}{500}$
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[3 marks sum] - MATHEMATICS STD 9 Questions - Vidyadip