Question 11 Mark
State whether the following statements are true or false? Justify your answer.
Number of rational numbers between 15 and 18 is finite.
AnswerFalse.
Solution:
Because between any two rational numbers there exist infinitely many rational numbers.
View full question & answer→Question 21 Mark
Insert a rational number and an irrational number between the following:
0 and 0.1
AnswerA rational number between 0 and 0.1 is 0.03.
An irrational number between 0 and 0.1 is 0.007000700007.........
View full question & answer→Question 31 Mark
State whether the following statements are true or false? Justify your answer.
There are infinitely many integers between any two integers.
AnswerFalse.
Solution:
Because between two consecutive integers (likel and 2), there does not exist any other integer.
View full question & answer→Question 41 Mark
Classify the following numbers as rational or irrational with justification:
1.010010001...
Answer1.010010001... is a number with non-terminating non-recurring decimal expansion. Hence, it is a irrational numbers.
View full question & answer→Question 51 Mark
Simplify the following:
$4\sqrt{28}\div3\sqrt{7}\div\sqrt[3]{7}$
Answer$4\sqrt{28}\div3\sqrt{7}\div\sqrt[3]{7}$ $=4\sqrt{2\times2\times7}\times\frac{1}{3\sqrt{7}}\div\sqrt[3]{7}$ $=\frac{8\sqrt{7}}{3\sqrt{7}}\div\sqrt[3]{7}=\frac{8}{3}\div\sqrt[3]{7}$ We know that, the cube root of 7 is 1.9129 and $\therefore\frac{8}{3}\div\sqrt[3]{7}=2.666\div1.9129$
$\Rightarrow\ 1.3936.$
View full question & answer→Question 61 Mark
Classify the following numbers as rational or irrational with justification:
0.5918
Answer0.5918, it is a number with terminating decimal, so it can be written in the form of $\frac{\text{p}}{\text{q}},$ where $\text{q}\neq0$, p and q are integer.Hence, it is a rational numbers.
View full question & answer→Question 71 Mark
Insert a rational number and an irrational number between the following:
$\sqrt{2}$ and $\sqrt{3}$
AnswerA rational number between $\sqrt{2}$ and $\sqrt{3}$ i.e., between 1.414..... and 1.7320…… is 1.5.
An irrational number between $\sqrt{2}$ and $\sqrt{3}$ is 1.585585558.........
View full question & answer→Question 81 Mark
Rationalise the denominator in each of the following and hence evaluate by taking $\sqrt{2}=1.414 ,$ $\sqrt{3}=1.732$ and $\sqrt{5}=2.236,$ upto three places of decimal.
$\frac{4}{\sqrt{3}}$
Answer$\frac{4}{\sqrt{3}}=\frac{4}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}=\frac{4\sqrt{3}}{3}$
$=\frac{4\times1.732}{3}=\frac{6.928}{3}=2.309$
View full question & answer→Question 91 Mark
Rationalise the denominator in each of the following and hence evaluate by taking $\sqrt{2}=1.414 ,$ $\sqrt{3}=1.732$ and $\sqrt{5}=2.236,$ upto three places of decimal.
$\frac{1}{\sqrt{3}+\sqrt{2}}$
Answer$\frac{1}{\sqrt{3}+\sqrt{2}}=\frac{1}{\sqrt{3}+\sqrt{2}}\times\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}$
$=\frac{\sqrt{3}-\sqrt{2}}{(\sqrt{3})^2-(\sqrt{2})^2}=\frac{\sqrt{3}-\sqrt{2}}{3-2}$
$=\frac{\sqrt{3}-\sqrt{2}}{1}=\sqrt{3}-\sqrt{2}$
$=1.732-1.414=0.318$
View full question & answer→Question 101 Mark
State whether the following statements are true or false? Justify your answer.
$\frac{\sqrt{2}}{3}$ is a rational number.
AnswerFalse.
Solution:
Here $\sqrt{2}$ is an irrational number and 3 is a rational number, we know that when we divide irrational number by non-zero rational number it will always give an irrational number.
View full question & answer→Question 111 Mark
Classify the following numbers as rational or irrational with justification:
$\sqrt{196}$
Answer$\sqrt{196}=\sqrt{(14)^2}=14$
Hence, it is a rational number.
View full question & answer→Question 121 Mark
Insert a rational number and an irrational number between the following:
.0001 and .001
AnswerA rational number between 0.0001 and 0.001 is 0.00011.
An irrational number between 0.0001 and 0.001 is 0.0001131331333...........
View full question & answer→Question 131 Mark
Rationalise the denominator in each of the following and hence evaluate by taking $\sqrt{2}=1.414 ,$ $\sqrt{3}=1.732$ and $\sqrt{5}=2.236,$ upto three places of decimal.
$\frac{\sqrt{10}-\sqrt{5}}{2}$
Answer$\frac{\sqrt{10}-\sqrt{5}}{2}=\frac{\sqrt{2}\times\sqrt{5}-\sqrt{5}}{2}$
$=\frac{\sqrt{5}(\sqrt{2}-1)}{2}=\frac{2.236(1.414-1)}{2}$
$=1.118\times0.414=0.463$
View full question & answer→Question 141 Mark
Insert a rational number and an irrational number between the following:
3.623623 and 0.484848
AnswerA rational number between 3.623623 and 0.484848 is 1.
An irrational number between 3.623623 and 0.484848 is 1.909009000........
View full question & answer→Question 151 Mark
Find three rational numbers between:
$\frac{5}{7}$ and $\frac{6}{7}$
Answer$\frac{5}{7}=\frac{5}{7}\times\frac{10}{10}=\frac{50}{70}$and $\frac{6}{7}=\frac{6}{7}\times\frac{10}{10}=\frac{60}{70}$
$\Rightarrow\frac{51}{70}, \frac{52}{70}, \frac{53}{70}$ are three rational numbers lying and between $\frac{50}{70}$ and $\frac{60}{70}$ and therefore lie between $\frac{5}{7}$ and $\frac{6}{7}.$
View full question & answer→Question 161 Mark
Rationalise the denominator in each of the following and hence evaluate by taking $\sqrt{2}=1.414 ,$ $\sqrt{3}=1.732$ and $\sqrt{5}=2.236,$ upto three places of decimal.
$\frac{6}{\sqrt{6}}$
Answer$\frac{6}{\sqrt{6}}=\frac{6}{\sqrt{6}}\times\frac{\sqrt{6}}{\sqrt{6}}=\frac{6\sqrt{6}}{6}$
$=\sqrt{6}=\sqrt{2\times3}=\sqrt{2}\times\sqrt{3}$
$=1.414\times1.732=2.44909=2.448\text{ (approx.)}$
View full question & answer→Question 171 Mark
State whether the following statements are true or false? Justify your answer.
$\frac{\sqrt{12}}{\sqrt{3}}$ is not a rational number as $\sqrt{12}$ and $\sqrt{3}$ are not integers.
AnswerFalse.
Solution:
$\frac{\sqrt{12}}{\sqrt{3}}=\frac{\sqrt{4\times3}}{\sqrt{3}}=\frac{\sqrt{4}\times\sqrt{3}}{\sqrt{3}}=2\times1=2,$ which is a rational number.
View full question & answer→Question 181 Mark
Find which of the variables x, y, z and u represent rational numbers and which irrational numbers:
$\text{u}^2=\frac{17}{4}$
Answer$\text{u}^2=\frac{17}{4}\Rightarrow\text{u}=\sqrt{\frac{17}{4}}=\frac{\sqrt{17}}{2},$ which is of the form $\frac{\text{p}}{\text{q}}$ where $\text{p}=\sqrt{17}$ is not an integer. Hence, u is an irrational number.
View full question & answer→Question 191 Mark
Insert a rational number and an irrational number between the following:
2 and 3
AnswerA rational number between 2 and 3 is 2.1.
To find an irrational number between 2 and 3.
Find a number which is non-terminating non-recurring lying between them.
Such number will be 2.040040004.......
View full question & answer→Question 201 Mark
Insert a rational number and an irrational number between the following:
6.375289 and 6.375738
AnswerA rational number between 6.375289 and 6.375738 is 6.3753.
An irrational number between 6.375289 and 6.375738 is 6.375414114111.........
View full question & answer→Question 211 Mark
State whether the following statements are true or false? Justify your answer.
$\frac{\sqrt{15}}{\sqrt{3}}$ is written in the form $\frac{\text{p}}{\text{q}}, \text{ q}\neq0$ and so it is a rational number.
AnswerFalse.
Solution:
$\frac{\sqrt{15}}{\sqrt{3}}=\frac{\sqrt{5\times3}}{\sqrt{3}}=\frac{\sqrt{5}\times\sqrt{3}}{\sqrt{3}}=\sqrt{5},$ which is an irrational number.
View full question & answer→Question 221 Mark
Insert a rational number and an irrational number between the following:
$\frac{1}{3}$ and $\frac{1}{2}$
AnswerA rational number between $\frac{1}{3}$ and $\frac{1}{2}$ is $\frac{5}{12}.$
An irrational number between $\frac{1}{3}$ and $\frac{1}{2}$ i.e., between 0-3 and 0.5 is 0.4141141114.........
View full question & answer→Question 231 Mark
Simplify:
$(625)^{-\frac{1}{2}^{-\frac{1}{4}^{2}}}$
Answer$\Bigg[\Big((625)^{-\frac{1}{2}}\Big)^{-\frac{1}{4}}\Bigg]^{2}=(25^{-1})^{-\frac{1}{4}\times2}=[(5^2)^{-1}]^{-\frac{1}{4}\times2}$ $[\because(\text{a}^\text{m})^\text{n}=\text{a}^\text{mn}]$
$=5^{-2\times-\frac{1}{4}\times2}=5^1=5$
View full question & answer→Question 241 Mark
Find three rational numbers between:
-1 and -2
Answer-1.1, -1.2, -1.3 (terminating decimals) are three rational numbers lying between -1 and -2.
View full question & answer→Question 251 Mark
Simplify:
$\big(1^3+2^3+3^3\big)^{\frac{1}{2}}$
Answer$\big(1^3+2^3+3^3\big)^{\frac{1}{2}}=(1+8+27)^{\frac{1}{2}}$
$=(36)^\frac{1}{2}=(6^2)^\frac{1}{2}=6^{2\times\frac{1}{2}}=6$ $[\because(\text{a}^\text{m})^\text{n}=\text{a}^\text{mn}]$
View full question & answer→Question 261 Mark
Find three rational numbers between:
$\frac{1}{4}$ and $\frac{1}{5}$
Answer$\frac{1}{4}=\frac{1}{4}\times\frac{20}{20}=\frac{20}{80}$ and $\frac{1}{5}=\frac{1}{5}\times\frac{16}{16}=\frac{16}{80}$
Now, $\sqrt{2}\times\sqrt{3}\frac{18}{80}\Big(\frac{9}{40}\Big),\frac{19}{80}$ are three rational numbers lying between $\frac{1}{4}$ and $\frac{1}{5}.$
View full question & answer→Question 271 Mark
Rationalise the denominator in each of the following and hence evaluate by taking $\sqrt{2}=1.414 ,$ $\sqrt{3}=1.732$ and $\sqrt{5}=2.236,$ upto three places of decimal.
$\frac{\sqrt{2}}{2+\sqrt{2}}$
Answer$\frac{\sqrt{2}}{2+\sqrt{2}}=\frac{\sqrt{2}}{2+\sqrt{2}}\times\frac{2-\sqrt{2}}{2-\sqrt{2}}$
$=\frac{\sqrt{2}(2-\sqrt{2})}{(2)^2-(\sqrt{2})^2}=\frac{\sqrt{2}(2-\sqrt{2})}{4-2}$
$=\frac{\sqrt{2}(2-\sqrt{2})}{2}=\frac{2\sqrt{2}-2}{2}$
$=\sqrt{2}-1=1.414-1=0.414$
View full question & answer→Question 281 Mark
Simplify the following:
$\sqrt{45}-3\sqrt{20}+4\sqrt{5}$
Answer$\sqrt{45}-3\sqrt{20}+4\sqrt{5}$
$=\sqrt{9\times5}-3\sqrt{4\times5}+4\sqrt{5}$
$=3\sqrt{5}-3\times2\sqrt{5}+4\sqrt{5}$
$=(3-6+4)\sqrt{5}=\sqrt{5}$
View full question & answer→Question 291 Mark
Insert a rational number and an irrational number between the following:
2.357 and 3.121
AnswerA rational number between 2.357 and 3.121 is 3.
An irrational number between 2.357 and 3.121 is 3.101101110......
View full question & answer→Question 301 Mark
State whether the following statements are true or false? Justify your answer.
There are numbers which cannot be written in the form $\frac{\text{p}}{\text{q}},\text{ q}\neq0, \text{ p, q}$ both are integers.
AnswerTrue.
Solution:
Because there are infinitely many numbers which cannot be written in the form $\frac{\text{p}}{\text{q}},\text{ q}\neq0, \text{ p, q}$ both are integers and these numbers are called irrational numbers.
View full question & answer→Question 311 Mark
Let x and y be rational and irrational numbers, respectively. Is x + y necessarily an irrational number? Give an example in support of your answer.
AnswerYes, (x + y) is necessarily an irrational number.
e.g., Let $\text{x}=2, \text{ y}=\sqrt{3}$
Then, $\text{x+y}=2+\sqrt{3}$
If possible, let $\text{x+y}=2+\sqrt{3}$ be a rational number.
Consider $\text{a}=2+\sqrt{3}$
On squaring both sides, we get
$\text{a}^2=(2+\sqrt{3})^2$
$ [\text{using identity (a+b)}^2=\text{a}^2+\text{b}^2+2\text{ab}]$
$\Rightarrow\text{a}^2=2^2+(\sqrt{3})^2+2(2)(\sqrt{3})$
$\Rightarrow\text{a}^2=4+3+4\sqrt{3}\Rightarrow\frac{\text{a}^2-7}{4}=\sqrt{3}$
So, a is rational $\Rightarrow\frac{\text{a}^2-7}{4}$ is rational $\Rightarrow\sqrt{3}$ is rational.
View full question & answer→Question 321 Mark
Classify the following numbers as rational or irrational with justification:
$\sqrt{\frac{\sqrt{28}}{\sqrt{343}}}$
Answer$\frac{\sqrt{28}}{\sqrt{343}}=\frac{\sqrt{2\times2\times7}}{\sqrt{7\times7\times7}}=\frac{2\sqrt{7}}{7\sqrt{7}}=\frac{2}{7}$
Hence, it is a rational number.
View full question & answer→Question 331 Mark
Insert a rational number and an irrational number between the following:
$\frac{-2}{5}$ and $\frac{1}{2}$
AnswerA rational number between $\frac{-2}{5}$ and $\frac{1}{2}$ is 0.
An irrational number between $\frac{-2}{5}$ and $\frac{1}{2}$ i.e., between –0.4 and 0.5 is 0.151151115.........
View full question & answer→Question 341 Mark
Simplify:
$\frac{1}{27}^{\frac{-2}{3}}$
Answer$\Big(\frac{1}{27}\Big)^{\frac{-2}{3}}=\Big(\frac{1}{3^3}\Big)^{\frac{-2}{3}}=(3^{-3})^{-\frac{2}{3}}$ $\Big[\because\frac{1}{\text{a}}=\text{a}^{-1}\Big]$
$=3^{-3\times-\frac{2}{3}}=3^2=9$ $[\because(\text{a}^\text{m})^\text{n}=\text{a}^\text{mn}]$
View full question & answer→Question 351 Mark
Express the following in the form $\frac{\text{p}}{\text{q}},$ where p and q are integers and $\text{q}\neq0$:
0.2
AnswerLet $\text{x}=0.2=\frac{2}{10}=\frac{1}{5}$
View full question & answer→Question 361 Mark
Let x be rational and y be irrational. Is xy necessarily irrational? Justify your answer by an example.
AnswerNo, (xy) is necessarily an irrational only when x ≠ 0.
Let x be a non-zero rational and y be an irrational. Then, we have to show that xy be an irrational. If possible, let xy be a rational number.
Since, quotient of two non-zero rational number is a rational number.
So, (xy/ x) is a rational number ⇒ y is a rational number.
But, this contradicts the fact that y is an irrational number. Thus, our supposition is wrong.
Hence, xy is an irrational number. But, when x = 0, then xy = 0, a rational number.
View full question & answer→Question 371 Mark
Classify the following numbers as rational or irrational with justification: $\Big(1+\sqrt{5}\Big)-\Big(4+\sqrt{5}\Big)$
Answer$\Big(1+\sqrt{5}\Big)-\Big(4+\sqrt{5}\Big)$
$=1-4+\sqrt{5}-\sqrt{5}=-3$
Hence, it is a rational numbers.
View full question & answer→Question 381 Mark
Classify the following numbers as rational or irrational with justification:
$\sqrt{\frac{9}{27}}$
Answer$\sqrt{\frac{9}{27}}=\sqrt{\frac{9}{9\times3}}=\frac{1}{\sqrt{3}}$
Hence, it is an irrational number, because $\sqrt{3}$ is an irrational number.
View full question & answer→Question 391 Mark
Classify the following numbers as rational or irrational with justification:
$\sqrt[3]{18}$
Answer$\sqrt[3]{18}=\sqrt[3]{(3)^2\times2}=3\times3\sqrt{2}=9\sqrt{2}$
Hence, it is an irrational number.
View full question & answer→Question 401 Mark
State whether the following statements are true or false? Justify your answer.
The square of an irrational number is always rational.
AnswerFalse. Solution: e.g., Let an irrational number be $\sqrt{2}$ and $\sqrt[4]{2}$ - $(\sqrt{2})^2=2,$ which is a rational number.
- $(\sqrt[4]{2})^2=\sqrt{2},$ which is not a rational number.
Hence, square of an irrational number is not always a rational number.
View full question & answer→Question 411 Mark
Simplify the following:
$\frac{2\sqrt{3}}{3}-\frac{\sqrt{3}}{6}$
Answer$\frac{2\sqrt{3}}{3}-\frac{\sqrt{3}}{6}$
$=\sqrt{3}\Big(\frac{2}{3}-\frac{1}6{}\Big)=\sqrt{3}\Big(\frac{4-1}{6}\Big)$
$\sqrt{3}\times\frac{3}{6}=\frac{\sqrt{3}}{2}$
View full question & answer→Question 421 Mark
Find which of the variables x, y, z and u represent rational numbers and which irrational numbers:
z2 = .04
Answer$\text{z}^2=.04\Rightarrow\text{z}=\sqrt{.04}=0.2,$ which is a terminating decimal. Hence, it is rational number.
View full question & answer→Question 431 Mark
Classify the following numbers as rational or irrational with justification:
$-\sqrt{0.4}$
Answer$-\sqrt{0.4}=-\sqrt{\frac{4}{10}}=-\sqrt{\frac{2}{10}}$
Hence, it is a quotient of rational and irrational numbers, so it is an irrational number.
View full question & answer→Question 441 Mark
Find three rational numbers between:
0.1 and 0.11
AnswerSolution:
0.101, 0.102, 0.103 (terminating decimals) are three rational numbers which lie between 0.1 and 0.11.
View full question & answer→Question 451 Mark
Find which of the variables x, y, z and u represent rational numbers and which irrational numbers:
x2 = 5
Answer$\text{x}^2=5\Rightarrow\text{x}=\sqrt{5},$ which is an irrational number.
View full question & answer→Question 461 Mark
Insert a rational number and an irrational number between the following:
0.15 and 0.16
AnswerA rational number between 0.15 and 0.16 is 0.151.
An irrational number between 0.15 and 0.16 is 0.1515515551..........
View full question & answer→Question 471 Mark
Classify the following numbers as rational or irrational with justification:
10.124124...
Answer10.124124... is a number with non-terminating decimal expansion. Hence, it is a rational numbers.
View full question & answer→Question 481 Mark
Find which of the variables x, y, z and u represent rational numbers and which irrational numbers:
y2 = 9
Answer$\text{y}^2=5\Rightarrow\text{y}=\sqrt{9}=3,$ which is a rational number.
View full question & answer→Question 491 Mark
Classify the following numbers as rational or irrational with justification:
$\frac{\sqrt{12}}{\sqrt{75}}$
Answer$\frac{\sqrt{12}}{\sqrt{75}}=\frac{\sqrt{4\times3}}{\sqrt{25\times3}}=\frac{\sqrt{4}\sqrt{3}}{\sqrt{25}\sqrt{3}}=\frac{2}{5}$
Hence, it is a rational numbers.
View full question & answer→