Question 11 Mark
Express the following decimals in the form $\frac{\text{p}}{\text{q}}: 7.010$
AnswerGiven 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, $7.010=\frac{701}{100}$
View full question & answer→Question 21 Mark
Express the following decimals in the form $\frac{\text{p}}{\text{q}}: 9.90$
AnswerGiven 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, $9.90=\frac{99}{10}$
View full question & answer→Question 31 Mark
Give an example of two irrational numbers whose: Quotient is an irrational number.
AnswerLet $\sqrt{2},\ \sqrt{3}$ are two irrational numbers and their quotient is an irrational number. That is $\sqrt{2}\div\sqrt{3}=\frac{\sqrt{2}}{\sqrt{3}}$
View full question & answer→Question 41 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.
View full question & answer→Question 51 Mark
In the following equations, find which variables $x, y$ and $z$ etc. represent rational or irrational numbers:
$y^2=9$
AnswerWe have,
$y^2=9$
$=3$
$=\frac{3}{1}$
can be expressed in the form of $=\frac{a}{b}$, so it a rational number.
View full question & answer→Question 61 Mark
Give an example of two irrational numbers whose: Product is a rational number.
AnswerLet $\sqrt{8},\ \sqrt{2}$ are two irrational numbers and their product is a rational number. That is $\sqrt{8}\times\sqrt{2}=\sqrt{16}=4$
View full question & answer→Question 71 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.
View full question & answer→Question 81 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.
View full question & answer→Question 91 Mark
Give an example of two irrational numbers whose: Sum is a rational number.
AnswerLet $\sqrt{5},\ \sqrt{5}$ are two irrational numbers and their sum is an rational number. That is $\sqrt{5}+\big(-\sqrt{5}\big)=0.$
View full question & answer→Question 101 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.
View full question & answer→Question 111 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.
View full question & answer→Question 121 Mark
Express the following decimals in the form $\frac{\text{p}}{\text{q}}:$
$0.39$
AnswerGiven 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, $0.39=\frac{39}{100}$
View full question & answer→Question 131 Mark
Give an example of two irrational numbers whose: Quotient is an rational number.
AnswerLet $\sqrt{8},\ \sqrt{2}$ are two irrational numbers and their quotient is an rational number. That is $\frac{\sqrt{8}}{\sqrt{2}}=\frac{2\sqrt{2}}{\sqrt{2}}=2$
View full question & answer→Question 141 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.
View full question & answer→Question 151 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.
View full question & answer→Question 161 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.
View full question & answer→Question 171 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.
View full question & answer→Question 181 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, $(a+b)(a-b)=a^2-b^2$]
$4-2=\frac{2}{1}$
Since, $2$ is a rational number.
$\big(2-\sqrt{2}\big)\big(2+\sqrt{2}\big)$ is a rational number.
View full question & answer→Question 191 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 $\sqrt{2}\times\sqrt{3}=\sqrt{6}$
View full question & answer→Question 201 Mark
Express the following decimals in the form $\frac{\text{p}}{\text{q}}: 1.0001$
AnswerGiven 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, $1.0001=\frac{10001}{10000}$
View full question & answer→Question 211 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, $(a+b)^2=a^2+2 a b+b^2$]
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.
View full question & answer→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.
View full question & answer→Question 231 Mark
In the following equations, find which variables $x, y$ and $z$ etc. represent rational or irrational numbers: $\text{w}^2=27$
AnswerWe 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.
View full question & answer→Question 241 Mark
Express the following decimals in the form $\frac{\text{p}}{\text{q}}:$
$2.15$
AnswerGiven 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, $2.15=\frac{43}{20}$
View full question & answer→Question 251 Mark
Find two irrational numbers lying between $0.1$ and $0.12.$
AnswerLet $a = 0.1 , b = 0.12$ 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 $0.12.$
View full question & answer→Question 261 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.
View full question & answer→Question 271 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.
View full question & answer→Question 281 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.
View full question & answer→Question 291 Mark
Give an example of two irrational numbers whose: Difference is an irrational number.
AnswerLet $4\sqrt{3},\ 3\sqrt{3}$ are two irrational numbers and their difference is an irrational number. Because $4\sqrt{3}-3\sqrt{3}=\sqrt{3}$ is an irrational number.
View full question & answer→Question 301 Mark
Examine, whether the following numbers are rational or irrational:
$\sqrt{225}$
View full question & answer→Question 311 Mark
In the following equations, find which variables $x, y$ and $z$ etc. represent rational or irrational numbers:
$\text{t}^2=0.4$
AnswerWe 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. $t^2 = 0.4$ is an irrational number.
View full question & answer→Question 321 Mark
Express the following decimals in the form $\frac{\text{p}}{\text{q}}: 0.750$
AnswerGiven 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, $0.750=\frac{3}{4}$
View full question & answer→Question 331 Mark
Give an example of two irrational numbers whose: Sum is an irrational number.
AnswerLet $2\sqrt{5},\ 3\sqrt{5}$ are two irrational numbers and their sum is an irrational number. That is $2\sqrt{5}+3\sqrt{5}=5\sqrt{5}$
View full question & answer→