Question 11 Mark
Show that $x$ is irrational, if $x^2 = 6.$
Answer$x^2=6$
$x=\sqrt{6}$
It is irrational.
View full question & answer→Question 21 Mark
If $m=\frac{1}{3-2 \sqrt{2}}$ and $n=\frac{1}{3+2 \sqrt{2}}$, find $mn$
Answer$mn = ( 3 + 2\sqrt2 )( 3 - 2\sqrt2 )$
$= (3)^2 - (2\sqrt2)^2$
$= 9 - 8$
$= 1$
View full question & answer→Question 31 Mark
If $x=\frac{\sqrt{5}-2}{\sqrt{5}+2}$ and $y=\frac{\sqrt{5}+2}{\sqrt{5}-2}$; find : $x^2+y^2+x y$
Answer$x^2+ y^2+xy$
$= 161 - 72\sqrt5 + 161 +72\sqrt5 + 1$
$= 322 + 1 = 323$
View full question & answer→Question 41 Mark
If $x=\frac{\sqrt{5}-2}{\sqrt{5}+2}$ and $y=\frac{\sqrt{5}+2}{\sqrt{5}-2} ;$ find : $x y$
Answer$x y=\frac{(\sqrt{5}-2)(\sqrt{5}+2)}{(\sqrt{5}+2)(\sqrt{5}-2)}=1$
View full question & answer→Question 51 Mark
Write the lowestrationalising factor of :$3\sqrt2 + 2\sqrt3$
Answer$3\sqrt2 + 2\sqrt3$
$= ( 3\sqrt2 + 2\sqrt3 )( 3\sqrt2 - 2\sqrt3 )$
$= ( 3\sqrt2)^2 - (2\sqrt3)^2$
$= 9 x 2 - 4 x 3$
$= 18 - 12$
$= 6$
its lowest rationalizing factor is $3\sqrt2 - 2\sqrt3.$
View full question & answer→Question 61 Mark
Write the lowestrationalising factor of : $\sqrt5 - \sqrt2$
Answer$\sqrt5 - \sqrt2$
$( \sqrt5 - \sqrt2 )( \sqrt5 + \sqrt2 )$
$= ( \sqrt5 )^2 - ( \sqrt2 )^2$
$= 3$
Therefore lowestrationalizing factor is $\sqrt5 + \sqrt2$
View full question & answer→Question 71 Mark
Write the lowestrationalising factor of :$\sqrt5 - 3$
Answer$( \sqrt5 - 3 )( \sqrt5 + 3 )$
$= ( \sqrt5 )^2 - (3)^2$
$= 5 - 9$
$= -4$
∴ lowest rationalizing factor is $( \sqrt5 + 3 )$
View full question & answer→Question 81 Mark
Write the lowest rationalising factor of : $\sqrt24$
Answer$\sqrt{24 } $
$=\sqrt{2 \times 2 \times 2 \times 3}$
$=2 \sqrt{6 } $
$\therefore$ lowest rationalizing factor is $\sqrt{ 6} $.
View full question & answer→Question 91 Mark
Write the lowest rationalising factor of : $5\sqrt2$
Answer$5\sqrt2 \times 5\sqrt2$
$= 5 \times 2$
$= 10$ which is rational.
$\therefore $ lowest rationalizing factor is $\sqrt2$
View full question & answer→Question 101 Mark
State, with reason, of the following is surd or not : $\sqrt{3+\sqrt{2}}$
Answer$\sqrt{3+\sqrt{2}}$ is not a surds because $3+\sqrt{2 } $ is irrational.
View full question & answer→Question 111 Mark
State, with reason, of the following is surd or not: $\sqrt\pi$
Answer We observe that $\sqrt{\pi}$ is an irrational number.
$\therefore \pi$ is not a surd.
View full question & answer→Question 121 Mark
State, with reason, of the following is surd or not :$\sqrt[3]{-125}$
Answer$\sqrt[3]{-125}$
$=\sqrt[3]{-5 \times-5 \times-5}$
$=-5$
$\therefore \sqrt[3]{-125}$ is not a surds.
View full question & answer→Question 131 Mark
State, with reason, of the following is surd or not : $\sqrt[3]{25} \cdot \sqrt[3]{40}$
Answer$\sqrt[3]{25} \cdot \sqrt[3]{40}$
$=\sqrt[3]{5 \times 5 \times 2 \times 2 \times 2 \times 5}$
$=2 \times 5 $
$=10$
$\therefore \sqrt[3]{25} \cdot \sqrt[3]{40}$ is not a surds.
View full question & answer→Question 141 Mark
State, with reason, of the following is surd or not : $\sqrt[3]{64}$
Answer$\sqrt[3]{64}$
$=\sqrt[3]{2 \times 2 \times 2 \times 2 \times 2 \times 2}$
$=4$ which is rational.
$\therefore \sqrt[3]{64}$ is not a surds.
View full question & answer→Question 151 Mark
State, with reason, of the following is surd or not :$\sqrt[5]{128}$
Answer$\sqrt[5]{128}$
$=\sqrt[5]{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2}$
$=2 \sqrt[5]{4}$
$\therefore \sqrt[5]{128}$ is a surds.
View full question & answer→Question 161 Mark
State, with reason, of the following is surd or not: $\sqrt[4]{27}$
Answer$\sqrt[4]{27}$
$=\sqrt[4]{3 \times 3 \times 3}$ which is irrational.
$\therefore \sqrt[4]{27}$ is a surds.
View full question & answer→Question 171 Mark
State, with reason, of the following is surd or not : $\sqrt180$
Answer$\sqrt{ 180}$
$=\sqrt{2 \times 2 \times 5 \times 3 \times 3}$
$=6 \sqrt{5 } $ which is irrational.
$\therefore \sqrt{180 }$ is a surds.
View full question & answer→Question 181 Mark
Write a pair of irrational numbers whose difference is irrational.
Answer$\sqrt3 + 2$ and $\sqrt2 - 3$ are two irrational numbers whose difference is irrational.
$( \sqrt3 + 2 ) - ( \sqrt2 - 3 )$
$= \sqrt3 - \sqrt2 + 2 + 3$
$= \sqrt3 - \sqrt2 + 5$ which is irrational.
View full question & answer→Question 191 Mark
Write a pair of irrational numbers whose sum is rational.
Answer$\sqrt3 + 5$ and $4 - \sqrt3$ are two irrational numbers whose sum is rational.
$( \sqrt3 + 5 ) + ( 4 - \sqrt3 )$
$= \sqrt3 + 5+ 4 - \sqrt3$
$= 9$
View full question & answer→Question 201 Mark
Write a pair of irrational numbers whose sum is irrational.
Answer$\sqrt3 + 5$ and $\sqrt5 - 3 $ are irrational numbers whose sum is irrational.
$( \sqrt3 + 5 ) + ( \sqrt5 - 3 )$
$= \sqrt3 + \sqrt5 + 5 - 3$
$= \sqrt3 + \sqrt5 + 2$ which is irrational.
View full question & answer→Question 211 Mark
State, whether the following numbers is rational or not : $\left(\frac{3}{2 \sqrt{2}}\right)^2$
Answer$\left(\frac{3}{2 \sqrt{2}}\right)^2$
$=\frac{(3)^2}{(2 \sqrt{2})^2}$
$=\frac{9}{4 \times 2}$
$ =\frac{9}{8}$ Rational Number.
View full question & answer→Question 221 Mark
State, whether the following numbers is rational or not : $( \sqrt3 - \sqrt2 )^2$
Answer$( \sqrt3 - \sqrt2 )^2$
$= ( \sqrt3 )^2- 2 ( \sqrt3 )( \sqrt2 ) + ( \sqrt2 )^2$
$= 3 - 2\sqrt6 + 2$
$= 5 - 2\sqrt6$
Irrational Number
View full question & answer→Question 231 Mark
State, whether the following numbers is rational or not : $( 5 + \sqrt5 )( 5 - \sqrt5 )$
Answer$( 5 + \sqrt5 )( 5 - \sqrt5 )$
$= ( 5 )^2 - ( \sqrt5 )^2$
$ = 25 - 5$
$= 20$
Rational Number
View full question & answer→Question 241 Mark
Write a pair of irrational numbers whose product is rational.
AnswerConsider $ \sqrt2$ as an irrational number.
$\sqrt2\times \sqrt2$
$= \sqrt4$
$= 2 $ which is a rational number.
View full question & answer→Question 251 Mark
Write a pair of irrational numbers whose product is irrational.
AnswerConsider two irrational numbers $( 5 + \sqrt2 )$ and $( \sqrt5 - 2 )$
Thus, the product,$ ( 5 + \sqrt2 ) \times ( \sqrt5 - 2 )$$= 5\sqrt5 - 10 + \sqrt10 - 2\sqrt2$ is irrational.
View full question & answer→Question 261 Mark
Write a pair of irrational numbers whose difference is rational.
Answer$\sqrt5 - 3$ and $\sqrt5 + 3$ are irrational numbers whose difference is rational.
$( \sqrt5 - 3 ) - ( \sqrt5 + 3 ) <$
$= \sqrt5 - 3 - > \sqrt5 - 3$
$= -6$ which is rational.
View full question & answer→Question 271 Mark
Is zero a rational number ? Can it be written in the form $\frac{P}{q}$, where $\mathrm{p}$ and $q$ are integers and $q \neq 0$ ?
AnswerYes, zero is a rational number.As it can be written in the form of, where $p$ and $q$ are integers and $q \neq 0$ ?
$
\Rightarrow 0=\frac{0}{1}
$
View full question & answer→