1 Marks Question

Question 11 Mark

In the following equations, find which variables x, y and z etc. represent rational or irrational numbers:
y² = 9

Answer

We have,
y² = 9
= 3
$=\frac{3}{1}$
can be expressed in the form of $=\frac{\text{a}}{\text{b}},$ so it a rational number.

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Question 21 Mark

In the following equations, find which variables x, y and z etc. represent rational or irrational numbers:
$\text{w}^2=27$

Answer

We have,
$\text{w}^2=27$
Taking square root on both sides, we get,
$\text{w}=3\sqrt{3}$
Product of a irrational and an irrational is an irrational number. So w is an irrational number.

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Question 31 Mark

In the following equations, find which variables x, y and z etc. represent rational or irrational numbers:
$\text{t}^2=0.4$

Answer

We have,
$\text{t}^2=0.4$
Taking square root on both sides, we get,
$\text{t}=\sqrt{\Big(\frac{4}{10}\Big)}$
$\text{t}=\frac{2}{\sqrt{10}}$
Since, quotient of a rational and an irrational number is irrational number. t² = 0.4 is an irrational number.

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Question 41 Mark

Give an example of two irrational numbers whose:
Sum is a rational number.

Answer

Let $\sqrt{5},\ \sqrt{5}$ are two irrational numbers and their sum is an rational number. That is $\sqrt{5}+\big(-\sqrt{5}\big)=0.$

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Question 51 Mark

Give an example of two irrational numbers whose:
Sum is an irrational number.

Answer

Let $2\sqrt{5},\ 3\sqrt{5}$ are two irrational numbers and their sum is an irrational number. That is $2\sqrt{5}+3\sqrt{5}=5\sqrt{5}$

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Question 61 Mark

Give an example of two irrational numbers whose:
Quotient is an rational number.

Answer

Let $\sqrt{8},\ \sqrt{2}$ are two irrational numbers and their quotient is an rational number. That is $\frac{\sqrt{8}}{\sqrt{2}}=\frac{2\sqrt{2}}{\sqrt{2}}=2$

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Question 71 Mark

Give an example of two irrational numbers whose:
Quotient is an irrational number.

Answer

Let $\sqrt{2},\ \sqrt{3}$ are two irrational numbers and their quotient is an irrational number. That is $\sqrt{2}\div\sqrt{3}=\frac{\sqrt{2}}{\sqrt{3}}$

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Question 81 Mark

Give an example of two irrational numbers whose:
Product is a rational number.

Answer

Let $\sqrt{8},\ \sqrt{2}$ are two irrational numbers and their product is a rational number. That is $\sqrt{8}\times\sqrt{2}=\sqrt{16}=4$

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Question 91 Mark

Give an example of two irrational numbers whose:
Product is an irrational number.

Answer

Let $\sqrt{2},\ \sqrt{3}$ are two irrational numbers and their product is an irrational number. That is $\sqrt{2}\times\sqrt{3}=\sqrt{6}$

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Question 101 Mark

Give an example of two irrational numbers whose:
Difference is a rational number.

Answer

Let $\sqrt{2},\ 1+\sqrt{2}$
So, $1+\sqrt{2}-\sqrt{2}=1$
Therefore, $\sqrt{2}$ and $1+\sqrt{2}$ are two irrational numbers and their difference is a rational number.

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Question 111 Mark

Give an example of two irrational numbers whose:
Difference is an irrational number.

Answer

Let $4\sqrt{3},\ 3\sqrt{3}$ are two irrational numbers and their difference is an irrational number. Because $4\sqrt{3}-3\sqrt{3}=\sqrt{3}$ is an irrational number.

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Question 121 Mark

Find two irrational numbers lying between 0.1 and 0.12.

Answer

Let
a = 0.1
b = 0.12
Here a and b are rational number. So we observe that in first decimal place a and b have same digit. So a < b.
Hence two irrational numbers are 0.1010010001... and 0.11010010001... lying between 0.1 and 0.12.

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Question 131 Mark

Express the following decimals in the form $\frac{\text{p}}{\text{q}}:$
9.90

Answer

Given decimal is 9.90
Now we have to convert given decimal number into the $\frac{\text{p}}{\text{q}}$ form
Let $\frac{\text{p}}{\text{q}}=9.90$
$\Rightarrow\frac{\text{p}}{\text{q}}=\frac{990}{100}$
$\Rightarrow\frac{\text{p}}{\text{q}}=\frac{99}{10}$
Hence, $9.90=\frac{99}{10}$

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Question 141 Mark

Express the following decimals in the form $\frac{\text{p}}{\text{q}}:$
7.010

Answer

Given decimal is 7.010
Now we have to convert given decimal number into the $\frac{\text{p}}{\text{q}}$ form
Let $\frac{\text{p}}{\text{q}}=7.010$
$\Rightarrow\frac{\text{p}}{\text{q}}=\frac{7010}{1000}$
$\Rightarrow\frac{\text{p}}{\text{q}}=\frac{701}{100}$
Hence, $7.010=\frac{701}{100}$

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Question 151 Mark

Express the following decimals in the form $\frac{\text{p}}{\text{q}}:$
2.15

Answer

Given decimal is 2.15
Now we have to convert given decimal number into the $\frac{\text{p}}{\text{q}}$ form
Let $\frac{\text{p}}{\text{q}}=2.15$
$\Rightarrow\frac{\text{p}}{\text{q}}=\frac{215}{1000}$
$\Rightarrow\frac{\text{p}}{\text{q}}=\frac{43}{100}$
Hence, $2.15=\frac{43}{20}$

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Question 161 Mark

Express the following decimals in the form $\frac{\text{p}}{\text{q}}:$
1.0001

Answer

Given decimal is 1.0001
Now we have to convert given decimal number into the $\frac{\text{p}}{\text{q}}$ form
Let $\frac{\text{p}}{\text{q}}=1.0001$
$\Rightarrow\frac{\text{p}}{\text{q}}=\frac{10001}{10000}$
Hence, $1.0001=\frac{10001}{10000}$

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Question 171 Mark

Express the following decimals in the form $\frac{\text{p}}{\text{q}}:$
0.750

Answer

Given decimal is 0.750
Now we have to convert given decimal number into the $\frac{\text{p}}{\text{q}}$ form
Let $\frac{\text{p}}{\text{q}}=0.750$
$\Rightarrow\frac{\text{p}}{\text{q}}=\frac{750}{1000}$
$\Rightarrow\frac{\text{p}}{\text{q}}=\frac{75}{100}$
$\Rightarrow\frac{\text{p}}{\text{q}}=\frac{3}{4}$
Hence, $0.750=\frac{3}{4}$

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Question 181 Mark

Express the following decimals in the form $\frac{\text{p}}{\text{q}}:$
0.39

Answer

Given decimal is 0.39
Now we have to convert given decimal number into the $\frac{\text{p}}{\text{q}}$ form
Let $\frac{\text{p}}{\text{q}}=0.39$
$\Rightarrow\frac{\text{p}}{\text{q}}=\frac{39}{100}$
Hence, $0.39=\frac{39}{100}$

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Question 191 Mark

Examine, whether the following numbers are rational or irrational:
$\sqrt{7}$

Answer

$\sqrt{7}$ is not a perfect square root so it is an irrational number.

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Question 201 Mark

Examine, whether the following numbers are rational or irrational:
$\sqrt{5}-2$

Answer

$\sqrt{5}-2$
The difference of an irrational number and a rational number is an irrational number. $\big(\sqrt{5}-2\big)$ is an irrational number.

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Question 211 Mark

Examine, whether the following numbers are rational or irrational:
$\sqrt{4}$

Answer

$\sqrt{4}$ is a perfect square root so it is an rational number.
We have,
$\sqrt{4}$ can be expressed in the form of
$\frac{\text{a}}{\text{b}},$ so it is a rational number. The decimal representation of $\sqrt{9}$ is 3.0. 3 is a rational number.

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Question 221 Mark

Examine, whether the following numbers are rational or irrational:
$\sqrt{3}+\sqrt{5}$

Answer

$\sqrt{3}$ is not a perfect square and it is an irrational number and $\sqrt{5}$ is not a perfect square and is an irrational number. The sum of an irrational number and an irrational number is an irrational number, so $\sqrt{3}+\sqrt{5}$ is an irrational number.

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Question 231 Mark

Examine, whether the following numbers are rational or irrational:
$\sqrt{3}+\sqrt{2}$

Answer

$\sqrt{3}$ is not a perfect square and it is an irrational number and $\sqrt{2}$ is not a perfect square and is an irrational number. The sum of an irrational number and an irrational number is an irrational number, so $\sqrt{3}+\sqrt{2}$ is an irrational number.

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Question 241 Mark

Examine, whether the following numbers are rational or irrational:
$\sqrt{23}$

Answer

$\sqrt{23}$
$\sqrt{23}=4.79583152331\ ...$
As decimal expansion of this number is non-terminating, non-recurring so it is an irrational number.

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Question 251 Mark

Examine, whether the following numbers are rational or irrational:
$\sqrt{225}$

Answer

$\sqrt{225}$
$\sqrt{225}=15=\frac{15}{1}$
$\sqrt{225}$ is rational number as it can be represented in $\frac{\text{p}}{\text{q}}$ form.

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Question 261 Mark

Examine, whether the following numbers are rational or irrational:
$\big(\sqrt{2}+\sqrt{3}\big)^2$

Answer

$\big(\sqrt{2}+\sqrt{3}\big)^2$
We have,
$\big(\sqrt{2}+\sqrt{3}\big)^2=2+2\sqrt{6}+3=5+\sqrt{6}$ [Since, (a + b)² = a² + 2ab + b²]
The sum of a rational number and an irrational number is an irrational number, so $\big(\sqrt{2}+\sqrt{3}\big)^2$ is an irrational number.

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Question 271 Mark

Examine, whether the following numbers are rational or irrational:
$\big(\sqrt{2}-2\big)^2$

Answer

$\big(\sqrt{2}-2\big)^2$
We have, $\big(\sqrt{2}-2\big)^2$
$=2+4-4\sqrt{2}$
$=6+4\sqrt{2}$
6 is a rational number but $4\sqrt{2}$ is an irrational number.
The sum of a rational number and an irrational number is an irrational number, so $\big(\sqrt{2}+\sqrt{4}\big)^2$ is an irrational number.

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Question 281 Mark

Examine, whether the following numbers are rational or irrational:
$\big(2-\sqrt{2}\big)\big(2+\sqrt{2}\big)$

Answer

$\big(2-\sqrt{2}\big)\big(2+\sqrt{2}\big)$
We have,
$\big(2-\sqrt{2}\big)\big(2+\sqrt{2}\big)=(2)^2-(\sqrt{2})^2$[Since, (a + b)(a - b) = a² - b²]
$4-2=\frac{2}{1}$
Since, 2 is a rational number.
$\big(2-\sqrt{2}\big)\big(2+\sqrt{2}\big)$ is a rational number.

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Question 291 Mark

Examine, whether the following numbers are rational or irrational:
$2+\sqrt{3}$

Answer

$2+\sqrt{3}$
Here, 2 is a rational number and $\sqrt{3}$ is an irrational number. So, the sum of a rational and an irrational number is an irrational number.

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Question 301 Mark

Examine, whether the following numbers are rational or irrational:
7.478478...

Answer

7.478478... = 7.478, decimal expansion of this number is non-terminating recurring so it is a rational number.

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Question 311 Mark

Examine, whether the following numbers are rational or irrational:
1.101001000100001...

Answer

1.101001000100001..., as decimal expansion of this number is non-terminating, non-recurring so it is an irrational number.

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Question 321 Mark

Examine, whether the following numbers are rational or irrational:
0.3796

Answer

0.3796, as decimal expansion of this number is terminating, so it is a rational number.

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Question 331 Mark

Define an irrational number.

Answer

An irrational number is a real number which can be written as a decimal but not as a fraction i.e. it cannot be expressed as a ratio of integers. It cannot be expressed as terminating or repeating decimal.

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