Question 11 Mark
Without actual division, show that the following rational numbers is a terminating decimal. Express in decimal form:$\frac{19}{3125}$
Answer$\frac{19}{3125}=\frac{19}{5^5}=\frac{19\times2^5}{5^5\times2^5}$$=\frac{608}{100000}=0.00608$
We know either 2 or 5 is not a factor of 19, so it is in its simplest form.
Moreover, it is in the form of $\left(2^m \times 5^n\right)$.
Hence, the given rational is terminating.
View full question & answer→Question 21 Mark
Without actual division, show that the following rational numbers is a non-terminating repeating decimal:$\frac{32}{147}$
Answer$\frac{32}{147}=\frac{32}{3\times7^2}$We know either 3 or 7 is not a factor of 32, so it is in its simplest form.
Moreover, $\left(3 \times 7^2\right) \neq\left(2^m \times 5^n\right)$
Hence, the given rational is non-terminating repeating decimal.
View full question & answer→Question 31 Mark
Very-Short-Answer Questions:
State fundamental theorem of arithmatic.
AnswerThe fundamental theorem of arithmetic states that, every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique except for the order in which the prime factors occur.
View full question & answer→Question 41 Mark
Without actual division, show that the following rational numbers is a non-terminating repeating decimal:$\frac{129}{\big(2^2\times5^7\times7^5\big)}$
Answer$\frac{129}{\big(2^2\times5^7\times7^5\big)}$We know 2, 5 or 7 is not a factor of 129, so it is in its simplest form.
Moreover, $(2^2 \times 5^7 \times 7^5) \neq (2^m \times 5^n)$
Hence, the given rational is non-terminating repeating decimal.
View full question & answer→Question 51 Mark
Very-Short-Answer Questions:
Express 360 as product of its prime factors.
Answer$360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 = 2^3 \times 3^2 \times 5$
View full question & answer→Question 61 Mark
Define:
- Rational numbers.
- Irrational numbers.
- Real numbers.
Answer
- Rational numbers: The numbers of the form $\frac{\text{p}}{\text{q}}$ where p , q are integers and $\text{q}\neq0$ are called rational numbers.
Example: $\frac{2}{3}$
- Irrational numbers: The numbers which when expressed in decimal form are expressible as non-terminating and non-repeating decimals are called irrational numbers.
Example: $\sqrt{2}$
- Real numbers: The numbers which are positive or negative, whole numbers or decimal numbers and rational number or irrational number are called real numbers.
Example: $2,\ \frac{1}{3},\ \sqrt{2},\ -3$ etc. View full question & answer→Question 71 Mark
The following numbers are irrational?
$\sqrt{2}$
Answer$\sqrt{2}$ is irrational $(\because$ if p is prime, then $\sqrt{\text{p}}$ is irrational$).$
View full question & answer→Question 81 Mark
The following numbers are irrational?
$2.\bar3$
Answer$2.\bar3$ is rational because it is a non-terminating, repeating decimal.
View full question & answer→Question 91 Mark
Very-Short-Answer Questions:
Give an example of two irrationals whose sum is rational.
AnswerConsider the irrational numbers, $\big(3+\sqrt2\big)$ and $\big(3-\sqrt2\big)$
$\big(3+\sqrt2\big)+\big(3-\sqrt2\big)=6$
which is rational number.
View full question & answer→Question 101 Mark
Classify the following number as rational or irrational:$\sqrt[3]{3}$
Answer$\sqrt[3]{3}$ is an irrational number because 3 is a prime number. So, $\sqrt{3}$ is an irrational number.
View full question & answer→Question 111 Mark
Classify the following number as rational or irrational:5.636363...
Answer5.636363... is a rational number because it is a non-terminating, repeating decimal.
View full question & answer→Question 121 Mark
Without actual division, show that the following rational numbers is a terminating decimal. Express in decimal form:$\frac{15}{1600}$
Answer$\frac{15}{1600}=\frac{15}{2^6\times5^2}=\frac{15\times5^4}{2^6\times5^6}$$=\frac{9375}{1000000}=0.009375$
We know either 2 or 5 is not a factor of 15, so it is in its simplest form.
Moreover, it is in the form of $(2^m\times 5^n).$
Hence, the given rational is terminating.
View full question & answer→Question 131 Mark
The following numbers are irrational?
$5.27\overline{41}$
Answer$5.27\overline{41}$ is rational because it is a non-terminating, repeating decimal.
View full question & answer→Question 141 Mark
Classify the following number as rational or irrational:$\big(5+3\sqrt{2}\big)$
AnswerLet $5+3\sqrt2$ be rational.
Hence, $$5 and $5+3\sqrt2$ are rational.
$\therefore\big(5+3\sqrt{2}-2\big)=3\sqrt2=$ rational $[\because$ Difference of two rational is rational$]$
$\therefore\frac{1}3{}\times3\sqrt{2}=\sqrt{2}=$ rational $[\because$ Product of two rational is rational$]$
This contradicts the fact that $\sqrt2$ is irrational.
The contradiction arises by assuming $5+3\sqrt2$ is rational.
Hence, $5+3\sqrt2$ is irrational.
View full question & answer→Question 151 Mark
State whether the given statement is true of false:
The product of a rational and an irrational is irrational.
Question 161 Mark
Without actual division, show that each of the following rational numbers is a non-terminating repeating decimal:$\frac{29}{343}$
Answer$\frac{29}{343}=\frac{29}{7^3}$We know 7 is not a factor of 29, so it is in its simplest form.
Moreover, $7^3 \neq (2^m \times 5^n)$
Hence, the given rational is non-terminating repeating decimal.
View full question & answer→Question 171 Mark
Very-Short-Answer Questions:
If a and b are two prime numbers then find HCF (a, b).
AnswerIf a and b are prime numbers, it means they have no common factors other than 1So, the HCF (a, b) = 1
View full question & answer→Question 181 Mark
Is it possible to have two numbers whose HCF is 18 and LCM is 760? Give reason.
HINT: HCF always divides LCM completely.
AnswerNo it is, not possible since the LCM has to be a multiple of the HCF.760 is not a multiple of 18
View full question & answer→Question 191 Mark
Classify the following number as rational or irrational:$\big(2-\sqrt{3}\big)$
AnswerLet $2-\sqrt3$ be rational.
Hence, 2 and $2-\sqrt3$ are rational.
$\therefore\big(2-2+\sqrt{3}\big)=\sqrt3=$ rational $[\because$ Difference of two rational is rational$]$
This contradicts the fact that $\sqrt3$ is irrational.
The contradiction arises by assuming $2-\sqrt3$ is rational.
Hence, $2-\sqrt3$ is irrational.
View full question & answer→Question 201 Mark
A number when divided by 61 gives 27 as quotient and 32 as remainder. Find the number.
AnswerBy Euclid's Division algorithm we have:
Dividend = (divisor × quotient) + remainder
= (61 × 27) + 32
= 1647 + 32
= 1679
View full question & answer→Question 211 Mark
Very-Short-Answer Questions:
State Euclid's division lemma.
AnswerFor any two given positive integers 'a' and 'b' there exists unique whole numbers 'q' and 'r' such that a = bq + r, where 0 ≤ r < b
View full question & answer→Question 221 Mark
Classify the following number as rational or irrational:3.121221222...
Answer3.121221222... is an irrational number because it is a non-terminating and non-repeating decimal.
View full question & answer→Question 231 Mark
Classify the following number as rational or irrational:$\big(3+\sqrt{2}\big)$
AnswerLet $3+\sqrt2$ be rational.
Hence, 3 and $3+\sqrt2$ are rational.
$\therefore3+\sqrt{2}-3=\sqrt2=$ rational $[\because$ Difference of two rational is rational$]$
This contradicts the fact that $\sqrt2$ is irrational.
The contradiction arises by assuming $3+\sqrt2$ is rational.
Hence, $3+\sqrt2$ is irrational.
View full question & answer→Question 241 Mark
Classify the following number as rational or irrational:$\big(\sqrt3+\sqrt5\big)$
AnswerLet $\sqrt3+\sqrt5$ be rational. $\therefore\sqrt3+\sqrt5= \text{a},$ where aa is rational $\therefore\sqrt3=\text{a}-\sqrt5\dots(1)$ On squaring both sides of equation (1), we get$3=\big(\text{a}-\sqrt5\big)^2$
$=\text{a}^2+5-2\sqrt5\text{a}$ $\Rightarrow\sqrt5=\frac{\text{a}^2+2}{\text{2a}}$ This is impossible because right-hand side is rational, whereas the left-hand side is irrational. This is a contradiction. Hence, $\sqrt3+\sqrt5$ is irrational.
View full question & answer→Question 251 Mark
Without actual division, show that the following rational numbers is a non-terminating repeating decimal:$\frac{77}{210}$
Answer$\frac{77}{210}=\frac{77\div\text{7}}{210\div7}$$=\frac{11}{30}=\frac{11}{2\times3\times5}$
We know 2, 3 or 5 is not a factor of 11, so $\frac{11}{30}$ is in its simplest form.
Moreover, $(2 \times 3 \times 5) \neq (2^m \times 5^n)$
Hence, the given rational is non-terminating repeating decimal.
View full question & answer→Question 261 Mark
Very-Short-Answer Questions:
Write the decimal expansion of $\frac{73}{\big(2^4\times5^3\big)}$
AnswerThe given number is $\frac{73}{\big(2^4\times5^3\big)}$Clearly, none of 2 and 5 is a factor of 73
So, the given rational is in its simplest form.
So, the given number is a terminating decimal.
Now, $\frac{73}{\big(2^4\times5^3\big)}=\frac{73\times5}{\big(2^4\times5^3\big)}$
$=\frac{365}{10000}=0.0365$
0.0365 is the decimal expansion.
View full question & answer→Question 271 Mark
The HCF of two numbers is 18 and their product is 12960. Find their LCM.
AnswerLet the two numbers be a and b.
HCF × LCM = ab
⇒ 18 × LCM = 12960
⇒ LCM = 720
View full question & answer→Question 281 Mark
The following numbers are irrational?
0.232332333...
Answer0.232332333... is irrational because it is a non-terminating, non-repeating decimal.
View full question & answer→Question 291 Mark
Without actual division, show that the following rational numbers is a terminating decimal. Express in decimal form:$\frac{17}{320}$
Answer$\frac{17}{320}=\frac{17}{2^6\times5}=\frac{17\times5^5}{2^6\times5^6}$$=\frac{53125}{1000000}=0.053125$
We know either 2 or 5 is not a factor of 17, so it is in its simplest form.
Moreover, it is in the form of $(2^m\times 5^n).$
Hence, the given rational is terminating.
View full question & answer→Question 301 Mark
The following numbers are irrational?
$\pi$
Answer$\pi$ is irrational because it is a non-repeating, non-terminating decimal.
View full question & answer→Question 311 Mark
State whether the given statement is true of false:
The sum of two rationals is always rational.
Question 321 Mark
Classify the following number as rational or irrational:2.040040004...
Answer2.040040004... is an irrational number because it is a non-terminating and non-repeating decimal.
View full question & answer→Question 331 Mark
Very-Short-Answer Questions:
What is a composite number?
AnswerA whole number that can be divided evenly by numbers other than 1 or itself.
View full question & answer→Question 341 Mark
Without actual division, show that the following rational numbers is a terminating decimal. Express in decimal form:$\frac{24}{125}$
Answer$\frac{24}{125}=\frac{24}{5^3}=\frac{24\times2^3}{5^3\times2^3}$$=\frac{192}{1000}=0.192$
We know 5 is not a factor of 24, so it is in its simplest form.
Moreover, it is in the form of $(2^m\times 5^n).$
Hence, the given rational is terminating.
View full question & answer→Question 351 Mark
Classify the following number as rational or irrational:$3.\overline{142857}$
Answer$3.\overline{142857}$ is a rational number because it is a repeating decimal.
View full question & answer→Question 361 Mark
Prove that $\big(2+\sqrt3\big)$ is irrational.
AnswerIf possible, let $\big(2+\sqrt3\big)$ be rational.
Then 2 and $\sqrt3$ are rational.
$\Rightarrow2+\sqrt3-2$ is rational $\dots(\because$ difference of two rationals is rational$)$
$\Rightarrow\sqrt3$ is rational
This contradicts the fact that $\sqrt3$ is irrational.
The contradiction arises by asuming that $\big(2+\sqrt3\big)$ is rational.
Hence, $\big(2+\sqrt3\big)$ is irrational.
View full question & answer→Question 371 Mark
The following numbers are irrational?
3.142857
Answer3.142857 is rational because it is a terminating decimal.
View full question & answer→Question 381 Mark
State whether the given statement is true of false:
The product of two irrationals is an irrational.
AnswerFalse.
Counter example:
$2\sqrt3$ and $4\sqrt3$ are two irrational numbers. But their product is 24, which is a rational number.
View full question & answer→Question 391 Mark
Classify the following number as rational or irrational:$\frac{3}{\sqrt{5}}$
AnswerLet $\frac{3}{\sqrt5}$ be rational.
$\therefore\frac{1}3{}\times\frac{3}{\sqrt5}=\frac{1}{\sqrt5}=$ rational $[\because$ Product of two rational is rational$]$
This contradicts the fact that $\frac{1}{\sqrt5}$ is irrational.
$\therefore\frac{1\times\sqrt5}{\sqrt5\times\sqrt5}=\frac{1}5{}\sqrt5$
So, if $\frac{1}{\sqrt5}$ is rational, then $\frac{1}{5}\sqrt5$ is rational.
$\therefore5\times\frac{1}{5}\sqrt5=\sqrt5=$ rational $[\because$ Product of two rational is rational$]$
Hence, $\frac{1}{\sqrt5}$ is irrational.
The contradiction arises by assuming $\frac{3}{\sqrt5}$ is rational.
Hence, $\frac{3}{\sqrt5}$ is irrational.
View full question & answer→Question 401 Mark
What do you mean by Euclid's division algorithm.
AnswerFor any two given positive integers a and b there exist unique whole numbers q and r such that
$\text{a}=\text{bq}+\text{r},$ where $0\le\text{r}<\text{b}$
Here, we call 'a' as dividend, 'b' as divisor, 'q' as quotient and 'r' as remainder.
Dividend = (divisor × quotient) + remainder
View full question & answer→Question 411 Mark
Classify the following number as rational or irrational:$\frac{22}{7}$
Answer$\frac{22}{7}$ is a rational number because it is of the form of $\frac{\text{p}}{\text{q}},\ \text{q}\neq0.$
View full question & answer→Question 421 Mark
Find the simplest form of:$\frac{69}{92}$
AnswerPrime factorisation of 69 and 92 is:
$69 = 3 \times 23$
$92 = 2^2 \times 23$
Therefore, $\frac{69}{92}=\frac{3\times\text{23}}{2^2\times23}=\frac{3}{2^2}=\frac{3}{4}$
Thus, simplest form of $\frac{69}{92}$ is $\frac{3}{4}.$
View full question & answer→Question 431 Mark
Very-Short-Answer Questions:
If the rational number $\frac{\text{a}}{\text{b}}$ has a terminating decimal expansion, what is the condition to be satisfied by b?
AnswerThe condition for $\frac{\text{a}}{\text{b}}$ to be a terminating decimal is that b should be of form $(2^m \times 5^n)$ for some non-negative integers m and n .
View full question & answer→Question 441 Mark
Find the simplest form of $\frac{148}{185}.$
Answer$\frac{148}{185}=\frac{2\times2\times37}{5\times37}$$=\frac{4}{5}$
which is in simplest form.
View full question & answer→Question 451 Mark
Without actual division, show that the following rational numbers is a non-terminating repeating decimal:$\frac{11}{\big(2^3\times3\big)}$
Answer$\frac{11}{\big(2^3\times3\big)}$We know either 2 or 3 is not a factor of 11, so it is in its simplest form.
Moreover, $(2^3 \times 3) \neq (2^m \times 5^n)$
Hence, the given rational is non-terminating repeating decimal.
View full question & answer→Question 461 Mark
Give an examples of two irrationals whose product is rational.
HINT: Take $\big(3+\sqrt2\big)$ and $\big(3-\sqrt2\big).$
AnswerLet $2\sqrt3,\ 3\sqrt3$ be two irrationals.
$\therefore2\sqrt3\times3\sqrt3=18=$ rational number
View full question & answer→Question 471 Mark
Without actual division, show that the following rational numbers is a terminating decimal. Express in decimal form:$\frac{171}{800}$
Answer$\frac{171}{800}=\frac{171}{2^5\times5^2}=\frac{171\times5^3}{2^5\times5^5}$$=\frac{21375}{100000}=0.21375$
We know either 2 or 5 is not a factor of 171, so it is in its simplest form.
Moreover, it is in the form of ($2^m\times 5^n).$
Hence, the given rational is terminating.
View full question & answer→Question 481 Mark
Classify the following number as rational or irrational:$\frac{22}{7}$
Answer$\frac{22}{7}$ is a rational number because it is of the form of $\frac{\text{p}}{\text{q}},\ \text{q}\neq0.$
View full question & answer→Question 491 Mark
Classify the following number as rational or irrational:$\big(2+\sqrt{5}\big)$
AnswerLet $3+\sqrt2$ be rational.
Hence, $2+\sqrt5$ and $\sqrt5$ are rational.
$\therefore\big(2+\sqrt{5}\big)-2=2+\sqrt5-2=\sqrt{5}=$ rational $[\because$ Difference of two rational is rational$]$
This contradicts the fact that $\sqrt5$ is irrational.
The contradiction arises by assuming $2-\sqrt5$ is rational.
Hence, $2-\sqrt5$ is irrational.
View full question & answer→Question 501 Mark
Very-Short-Answer Questions:
If a and b are two prime numbers then find LCM(a, b).
AnswerIf a and b are prime numbers, it means rthey have no common multiples other than their product.
So, the LCM(a, b) = ab.
View full question & answer→Question 511 Mark
Without actual division, show that the following rational numbers is a non-terminating repeating decimal:$\frac{73}{\big(2^2\times3^3\times5\big)}$
Answer$\frac{73}{\big(2^2\times3^3\times5\big)}$We know 2, 3 or 5 is not a factor of 73, so it is in its simplest form.
Moreover,$ (2^2 \times 3^3 \times 5) \neq (2^m \times 5^n)$
Hence, the given rational is non-terminating repeating decimal.
View full question & answer→Question 521 Mark
Classify the following number as rational or irrational:$\big(2-3\sqrt5\big)$
AnswerLet $2-3\sqrt5$ be rational.
Hence 2 and $2-3\sqrt5$ are rational.
$\therefore2-\big(2-3\sqrt5\big)=2-2+3\sqrt5$
$=3\sqrt5=$ rational $[\because$ Difference of two rational is rational$]$
$\therefore\frac{1}{3}\times3\sqrt5=\sqrt5=$ rational $[\because$ Product of two rational is rational$]$
This contradicts the fact that $\sqrt5$ is irrational.
The contradiction arises by assuming $2-3\sqrt5$ is rational.
Hence, $2-3\sqrt5$ is irrational.
View full question & answer→Question 531 Mark
State whether the given statement is true of false:
The sum of a rational and and irrational is irrational.
Question 541 Mark
Very-Short-Answer Questions:
Is it possible to have two numbers whose HCF is 25 and LCM is 520?
AnswerNo, it is not possible since the LCM has to be a multiple of the HCF.
520 is not a multiple of 25
View full question & answer→Question 551 Mark
Without actual division, show that the following rational numbers is a terminating decimal. Express in decimal form:$\frac{23}{\big(2^3\times5^2\big)}$
Answer$\frac{23}{2^3\times5^2}=\frac{23\times5}{2^3\times5^3}$$=\frac{115}{1000}=0.115$
We know either 2 or 5 is not a factor of 23, so it is in its simplest form.
Moreover, it is in the form of $(2^m\times 5^n).$
Hence, the given rational is terminating.
View full question & answer→Question 561 Mark
Without actual division, show that the following rational numbers is a terminating decimal. Express in decimal form:$\frac{15}{1600}$
Answer$\frac{15}{1600}=\frac{15}{2^6\times5^2}=\frac{15\times5^4}{2^6\times5^6}$$=\frac{9375}{1000000}=0.009375$
We know either 2 or 5 is not a factor of 15, so it is in its simplest form.
Moreover, it is in the form of $(2^m\times 5^n).$
Hence, the given rational is terminating.
View full question & answer→Question 571 Mark
Classify the following number as rational or irrational:$\pi$
Answer$\pi$ is an irrational number because it is a non-repeating and non-terminating decimal.
View full question & answer→Question 581 Mark
Classify the following number as rational or irrational:$\sqrt{21}$
Answer$\sqrt{21}=\sqrt{3}\times\sqrt{7}$ is an irrational number because $\sqrt{3}$ and $\sqrt{7}$ are irrational and prime numbers.
View full question & answer→Question 591 Mark
Very-Short-Answer Questions:
Give an example of two irrationals whose product is rational.
AnswerConsider the irrational numbers, $\frac{1}{\sqrt2}$ and $\sqrt2$
$\frac{1}{\sqrt2}\times\sqrt{2}=1$
which is rational number.
View full question & answer→Question 601 Mark
The following numbers are irrational?
$\sqrt[3]{6}$
Answer$\sqrt[3]{6}=\sqrt[3]{2}\times\sqrt[3]{3}$ is irrational.
View full question & answer→Question 611 Mark
Find the simplest form of:$\frac{473}{645}$
AnswerPrime factorisation of 473 and 645 is:
473 = 11 × 43
645 = 3 × 5 × 43
Therefore, $\frac{473}{645}=\frac{11\times\text{43}}{3\times5\times43}=\frac{11}{15}$
Thus, simplest form of $\frac{473}{645}$ is $\frac{11}{15}.$
View full question & answer→Question 621 Mark
Without actual division, show that the following rational numbers is a non-terminating repeating decimal:$\frac{9}{35}$
Answer$\frac{9}{35}=\frac{9}{5\times7}$We know either 5 or 7 is not a factor of 9, so it is in its simplest form.
Moreover, (5 × 7) ≠ $(2^m\times 5^n).$
Hence, the given rational is non-terminating repeating decimal.
View full question & answer→Question 631 Mark
The following numbers are irrational?
$\frac{22}{7}$
Answer$\frac{22}{7}$ is rational because it is in the form of $\frac{\text{p}}{\text{q}},\ \text{q}\neq0$
View full question & answer→Question 641 Mark
Classify the following number as rational or irrational:3.1416
Answer3.1416 is a rational number because it is a terminating decimal.
View full question & answer→Question 651 Mark
Very-Short-Answer Questions:
Show that there is no value of n for which $(2^m\times 5^n).$ends in 5.
Answer$2 \times 5 = 10$
$\Rightarrow (2 \times 5) = 10^n$
$\Rightarrow (2^n \times 5^n) = 10^n$
Since $(2^N \times 5^n).$ when combined will give 0 as the last digit, for value of n, it can never end in 5
View full question & answer→Question 661 Mark
Prove that $\frac{1}{\sqrt3}$ is irrational.Hint: $\frac{1}{\sqrt3} =\frac{1}{\sqrt3}\times\frac{\sqrt3}{\sqrt3}=\frac{1}{3}.\sqrt3$
AnswerLet $\frac{1}{\sqrt3}$ be rational.
$\therefore\frac{1}{\sqrt3}=\frac{\text{a}}{\text{b}},$ where a, b are positive integers having no common factor other than 1
$\therefore\sqrt3=\frac{\text{b}}{\text{a}}\dots(1)$
Since a, b are non-zero integers, $\frac{\text{b}}{\text{a}}$ is rational.
Thus, equation (1) shows that $\sqrt3$ is rational.
This contradicts the fact that $\sqrt3$ is rational.
The contradiction arises by assuming $\sqrt3$ is rational.
Hence, $\frac{1}{\sqrt3}$ is irrational.
View full question & answer→Question 671 Mark
Give an example of two irrationals whose sum is rational.
HINT: Take $\big(2+\sqrt3\big)$ and $\big(2-\sqrt3\big).$
AnswerLet $\big(2+\sqrt3\big),\big(2-\sqrt3\big)$ be two irrationals.
$\therefore\big(2+\sqrt3\big)+\big(2-\sqrt3\big)=4=$ rational number
View full question & answer→Question 681 Mark
Classify the following number as rational or irrational:1.535335333...
Answer1.535335333... is an irrational number because it is a non-terminating and non-repeating decimal.
View full question & answer→Question 691 Mark
Classify the following number as rational or irrational:$\sqrt{6}$
AnswerLet $\sqrt6=\sqrt2\times\sqrt3$ be rational.
Hence, $\sqrt2,\ \sqrt{3}$ are both rational.
This contradicts the fact that $\sqrt2,\ \sqrt{3}$ are irrational.
The contradiction arises by assuming $\sqrt6$ is rational.
Hence, $\sqrt6$ is irrational.
View full question & answer→Question 701 Mark
Very-Short-Answer Questions:
If a and b are relatively prime then what is their HCF?
AnswerIf a and b are relatively prime, it means they have no common factor other than 1.So, the HCF(a, b) = 1
View full question & answer→Question 711 Mark
State whether the given statement is true of false:
The sum of two irrationals is an irrational.
AnswerFalse.
Counter example:
$2+\sqrt3$ and $2+\sqrt3$ are two irrational numbers. But their sum is 4, which is a rational number.
View full question & answer→Question 721 Mark
State whether the given statement is true of false:
The product of two rationals is always rational.
Question 731 Mark
Without actual division, show that the following rational numbers is a non-terminating repeating decimal:$\frac{64}{455}$
Answer$\frac{64}{455}=\frac{64}{5\times7\times13}$We know 5, 7 or 13 is not a factor of 64, so it is in its simplest form.
Moreover, (5 × 7 × 13) ≠ $(2^m\times 5^n).$
Hence, the given rational is non-terminating repeating decimal.
View full question & answer→Question 741 Mark
Classify the following number as rational or irrational:$3\sqrt{7}$
AnswerLet $3\sqrt7$ be rational.
$\therefore\frac{1}3{}\times3\sqrt{7}=\sqrt{7}=$ rational $[\because$ Product of two rational is rational$]$
This contradicts the fact that $\sqrt7$ is irrational.
The contradiction arises by assuming $3\sqrt7$ is rational.
Hence, $3\sqrt7$ is irrational.
View full question & answer→