Question 13 Marks
Find a rational number between $\frac{3}{5}$ and $\frac{7}{9}$.
View full question & answer→Question 23 Marks
Rationalise the denominator of the following:
$\frac{1}{(2 \sqrt{5}-\sqrt{3})}$
Answer$\frac{(2 \sqrt{5}+\sqrt{3})}{17}$
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Rationalise the denominator of the following:
$\frac{3-2 \sqrt{2}}{3+2 \sqrt{2}}$
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Rationalise the denominator of the following:
$\frac{\sqrt{3}-1}{\sqrt{3}+1}$
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Rationalise the denominator of the following:
$\frac{1}{(\sqrt{6}-\sqrt{3})}$
Answer$\frac{\sqrt{6}+\sqrt{3}}{3}$
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Rationalise the denominator of the following:
$\frac{1}{(4+2 \sqrt{3})}$
Answer$\frac{2-\sqrt{3}}{2}$
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Rationalise the denominator of the following:
$\frac{1}{(\sqrt{3}-1)}$
Answer$\frac{\sqrt{3}+1}{2}$
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Rationalise the denominator of the following:
$\frac{1}{(3+\sqrt{5})}$
Answer$\frac{3-\sqrt{5}}{4}$
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If $x=(4-\sqrt{15})$, find the values of $\left(x^2+\frac{1}{x^2}\right)$.
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Rationalise the denominator of the following:
$\frac{\sqrt{2}}{2 \sqrt{3}}$
Answer$\frac{\sqrt{6}}{6}$
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If $x=(3+\sqrt{8})$, find the values of $\left(x^2+\frac{1}{x^2}\right)$.
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Show that : \[\frac{1}{(3-\sqrt{8})}+\frac{1}{(\sqrt{7}-\sqrt{6})}+\frac{1}{(\sqrt{5}-2)}-\frac{1}{\sqrt{8}-\sqrt{7}}-\frac{1}{(\sqrt{6}-\sqrt{5})}=5\]
Question 133 Marks
Simplify : $\frac{7+3 \sqrt{5}}{3+\sqrt{5}}-\frac{7-3 \sqrt{5}}{3-\sqrt{5}}$
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Simplify : $\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$
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If $\frac{5+2 \sqrt{3}}{7+4 \sqrt{3}}=a-b \sqrt{3}$, find the values of $a$ and $b$.
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If $\frac{5-\sqrt{6}}{5+\sqrt{6}}=a-b \sqrt{6}$, find the values of $a$ and $b$.
Answer$a=\frac{31}{19}, b=\frac{10}{19}$
View full question & answer→Question 173 Marks
If $\frac{3+\sqrt{2}}{3-\sqrt{2}}=a+b \sqrt{2}$, find the values of $a$ and $b$.
Answer$a=\frac{11}{7}, b=\frac{6}{7}$
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If $\frac{\sqrt{3}+1}{\sqrt{3}-1}=a+b \sqrt{3}$, find the values of $a$ and $b$.
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Rationalise the denominator of the following:
$\frac{\sqrt{2}}{\sqrt{2}+\sqrt{3}-\sqrt{5}}$
Answer$\frac{\sqrt{6}+3+\sqrt{15}}{6}$
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Rationalise the denominator of the following:
$\frac{1}{(1+\sqrt{5}+\sqrt{3})}$
Answer$\frac{(7-\sqrt{5}+3 \sqrt{3}-2 \sqrt{15})}{11}$
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Rationalise the denominator of the following:
$\frac{2}{\sqrt{6}}$
Answer$\frac{\sqrt{6}}{3}$
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Prove that $(\sqrt{3}+\sqrt{7})$ is irrational.
Answer
[Hint. If possible, let $(\sqrt{3}+\sqrt{7})$ be rational.
Then, $(\sqrt{3}+\sqrt{7})^2$ is rational.
But, $(\sqrt{3}+\sqrt{7})^2=(3+7+2 \sqrt{21})=(10+2 \sqrt{21})$, which is clearly irrational.
So, our supposition is wrong.]
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Write down the examples of 4 distinct irrational numbers.
Answer$\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{6}$.
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Prove that $\sqrt{5}$ is an irrational number
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What are irrational numbers? Give ten examples.
Question 263 Marks
What are rational numbers? Give ten examples.
Question 273 Marks
Write three irrational numbers between $\sqrt{2}$ and $\sqrt{7}$
Answer[Hint. $\sqrt{2}<\sqrt{3}<\sqrt{5}<\sqrt{6}<\sqrt{7}$ ]
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Write two irrational numbers between $\sqrt{14}$ and $\sqrt{19}$.
Answer[Hint. $\sqrt{14}<\sqrt{15}<\sqrt{17}<\sqrt{19}$.]
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Prove that $(\sqrt{2}+\sqrt{3})$ is irrational.
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Find six rational numbers between 3 and 4 .
Answer$3.1,3.2,3.3,4.4,3.5,3.6$
[Hint. Here $a=3, b=4$. Take $n=9$. Use $d=\frac{(b-a)}{10}=0.1$.
View full question & answer→Question 313 Marks
Find four rational numbers between 4 and 4.5.
Answer$4.1,4.2,4.3,4.4$
[Hint. Here $a=4, b=4.5$ and $n=4$. Use $d=\frac{(b-a)}{(n+1)}=\frac{(4.5-4)}{(4+1)}=\frac{0.5}{5}=0.1$
So, the required numbers are $a+d, a+2 d, a+3 d, a+4 d$, i.e., 4.1, 4.2, 4.3 and 4.4]
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Find three rational numbers between: $-\frac{1}{3}$ and $\frac{1}{4}$
Answer$\frac{-3}{16}, \frac{-1}{24}, \frac{5}{48}$
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Find three rational numbers between: $-\frac{1}{2}$ and $\frac{1}{3}$
Answer$\frac{-7}{24}, \frac{-1}{12}, \frac{1}{8}$
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Find three rational numbers between: $2 \frac{1}{3}$ and $3 \frac{2}{3}$
Answer$\frac{8}{3}, 3, \frac{10}{3}$
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Find three rational numbers between: -1 and 1
Answer$\frac{-1}{2}, 0, \frac{1}{2}$
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Find three rational numbers between: $\frac{1}{2}$ and $\frac{3}{5}$
Answer$\frac{21}{40}, \frac{11}{20}, \frac{23}{40}$
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Find three rational numbers between: 4 and 5
Answer$\frac{17}{4}, \frac{9}{2}, \frac{19}{4}$
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Find two rational numbers between: -2 and 1
Answer$\frac{-5}{4}, \frac{-1}{2}$
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Find two rational numbers between: $\frac{3}{4}$ and $1 \frac{1}{5}$
Answer$\frac{39}{40}, \frac{87}{80}$
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Find two rational numbers between: $\frac{1}{3}$ and $\frac{2}{5}$
Answer$\frac{11}{30}, \frac{23}{60}$
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Find two rational numbers between: 2 and 3
Answer$\frac{5}{2}, \frac{11}{4}$
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Represent each of the following on the number line :
(i) $\frac{3}{7}$$\quad$(ii) $\frac{16}{5}$$\quad$(iii) $-\frac{4}{9}$$\quad$(iv) $-\frac{18}{11}$$\quad$(v) $-3 \frac{1}{6}$
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Express $\frac{13}{34}$ as a decimal, correct to three decimal places.
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Express $\frac{15}{56}$ as a decimal, correct to four decimal places.
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If $x=(4-\sqrt{15})$, find the values of $\left(x^2+\frac{1}{x^2}\right)$.
View full question & answer→Question 463 Marks
If $x=(3+\sqrt{8})$, find the values of $\left(x^2+\frac{1}{x^2}\right)$.
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Show that:$
\frac{1}{(3-\sqrt{8})}+\frac{1}{(\sqrt{7}-\sqrt{6})}+\frac{1}{(\sqrt{5}-2)}-\frac{1}{\sqrt{8}-\sqrt{7}}-\frac{1}{(\sqrt{6}-\sqrt{5})}=5
$
View full question & answer→Question 483 Marks
Simplify : $\frac{7+3 \sqrt{5}}{3+\sqrt{5}}-\frac{7-3 \sqrt{5}}{3-\sqrt{5}}$
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Simplify : $\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$
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If $\frac{5+2 \sqrt{3}}{7+4 \sqrt{3}}=a-b \sqrt{3}$, find the values of $a$ and $b$.
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