[4 marks sum]

Question 14 Marks

Write three sets of Pythagorean triplets such that each set has numbers less than $30.$

Answer

The three sets of Pythagorean triplets such that each set has numbers less than $30$ are $3, 4$ and $5; 6, 8$ and $10; 5, 12$ and $13$
Proof:
In $3, 4$ and $5$
$3^2 + 4^2 = 5^2$
$\Rightarrow 9 + 16 = 25$
$\Rightarrow 25 = 25$
In $6, 8$ and $10$
$6^2 + 8^2 = 10^2$
$\Rightarrow 36 + 64 = 100$
$\Rightarrow 100 = 100$
In $5, 12,$ and $13$
$5^2+ 12^2 = 13^2$
$\Rightarrow 25 + 144 = 169$
$\Rightarrow 169 = 169$

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Question 24 Marks

Find the square root of $0.065$ correct to three places of decimal.

Answer

	$0.2549$
$2$	$0.065$
	$4$
$45$	$250$
	$225$
$504$	$2500$
	$2016$
$5089$	$48400$
	$45801$
	$2599$

$\sqrt{0.065}$
$=0.2549$
$ =0.255$
Required square root $=0.255$ up to three places of decimal.

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Question 34 Marks

A man, after a tour, finds that he had spent every day as many rupees as the number of days he had been on tour. How long did his tour last, if he had spent in all $₹ 1,296$

Answer

Let the number of days he had spent $= x$
Number of rupees spent in each day $= x$
Total money spent $= x\ \times\ x = x^2 = 1,296 ($given$)$
$\therefore x=\sqrt{1296}$

$4$	$1296$
$4$	$324$
$9$	$81$
	$9$

$ \Rightarrow \mathrm{x}=\sqrt{4 \times 4 \times 9 \times 9} $
$ \mathrm{x}=4 \times 9 $
$ \Rightarrow \mathrm{x}=36$
Hence required number of days $=36$

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Question 44 Marks

Evaluate: $\sqrt{248+\sqrt{52+\sqrt{144}}}$

Answer

$\sqrt{248+\sqrt{52+\sqrt{144}}} $
$ =\sqrt{248+\sqrt{52+\sqrt{12}}} \ldots(\because \sqrt{144}=12)$
$ =\sqrt{248+\sqrt{64}}$
$ =\sqrt{248+8} \quad \ldots(\because \sqrt{64}=8)$
$=\sqrt{256}=16 \quad \ldots(\because \sqrt{256}=\sqrt{16 \times 16}=16)$

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Question 54 Marks

Evaluate: $\sqrt{5\left(2 \frac{3}{4}-\frac{3}{10}\right)}$

Answer

$\sqrt{5\left(2 \frac{3}{4}-\frac{3}{10}\right)}$
$ =\sqrt{5\left(\frac{11}{4}-\frac{3}{10}\right)}$
$ =\sqrt{5\left(\frac{55-6}{20}\right)}$
$ =\sqrt{5\left(\frac{49}{20}\right)}$
$ =\sqrt{\frac{5 \times 49}{20}}$
$ =\sqrt{\frac{79}{4}}$
$=\frac{7}{2}$
$=3 \frac{1}{2}$

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Question 64 Marks

Evaluate: $\sqrt{\left(5+2 \frac{21}{25}\right) \times \frac{0.169}{1.6}}$

Answer

$\sqrt{\left(5+2 \frac{21}{25}\right) \times \frac{0.169}{1.6}}$
$ =\sqrt{\left(5+\frac{71}{25}\right) \times \frac{0.169}{1.600}}$
$ =\sqrt{\frac{196}{25} \times \frac{169}{1600}}$
$ =\sqrt{\frac{14 \times 14}{5 \times 5} \times \frac{13 \times 13}{40 \times 40}}$
$ =\frac{14 \times 13}{5 \times 40}$
$ =\frac{7 \times 13}{5 \times 20}$
$ =\frac{91}{100}$
$=0.91$

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Question 74 Marks

Evaluate: $\sqrt{(0.5)^3 \times 6 \times 3^5}$

Answer

$\sqrt{(0.5)^3 \times 6 \times 3^5}$
$ =\sqrt{(0.5)^2 \times 0.5 \times 3 \times 2 \times 3^5}$
$ =\sqrt{(0.5)^2 \times 0.5 \times 2 \times 3 \times 3^5}$
$ =\sqrt{(0.5)^2 \times 1.0 \times 3^6} \ldots[0.5 \times 2=1.0]$
$ =\sqrt{(0.5)^2 \times 1 \times 3^6}$
$ =0.5 \times 3^3$
$ =0.5 \times 27$
$ =13.5$

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Question 84 Marks

Find the smallest number by which $10368$ be divided, so that the result is a perfect square. Also, find the square root of the resulting numbers.

Answer

$10368=\overline{2 \times 2} \times \overline{2 \times 2} \times \overline{2 \times 2} \times 2 \times \overline{3 \times 3} \times\overline{3 \times 3}$
On grouping the prime factors of $10368$ as shown; one factors i.e. $2$ is left which cannot be paired with equal factor.

$2$	$10368$
$2$	$5184$
$2$	$2592$
$2$	$1296$
$2$	$648$
$2$	$324$
$2$	$162$
$3$	$81$
$3$	$27$
$3$	$9$
	$3$

$\therefore$ The given number should be divided by $2 $.
Now $\sqrt{\frac{10368}{2}}$
$=\sqrt{\frac{\overline{2 \times 2} \times \overline{2 \times 2} \times \overline{2 \times 2} \times 2 \times \overline{3 \times 3} \times \overline{3 \times 3}}{2}}$
$ =2 \times 2 \times 2 \times 3 \times 3=72$

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Question 94 Marks

If $\sqrt{784}=28$, find the value of: $\sqrt{0.0784}+\sqrt{0.000784}$

Answer

$\sqrt{784}=28$
$\therefore \sqrt{0.0784}=\sqrt{\frac{784}{10000}}=\frac{28}{100}=0.28$
and $\sqrt{0.000784}=\sqrt{\frac{784}{1000000}}$
$=\sqrt{\frac{28 \times 28}{10 \times 10 \times 10 \times 10 \times 10 \times 10}}$
$=\frac{28}{10 \times 10 \times 10}$
$=\frac{28}{1000}=0.028$
Now,
$\sqrt{0.0784}+\sqrt{0.000784}$
$=0.28+0.028=0.308$

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Question 104 Marks

$13$ and $31$ is a strange pair of numbers such that their squares $169$ and $961$ are also mirror images of each other. Find two more such pairs.

Answer

$(13)^2 = 169$ and $(31)^2 = 961$
Similarly, two such number can be $12$ and $21$
$\therefore (12)^2 = 144$ and $(21)^2 = 441$
And $102, 201$
$(102)^2 = 102 \times 102 = 10404$
And $(201)^2= 201 \times 201 = 40401$
$102\times102=204=1020 $
$10404 $
$201\times 201=201$
$4020 $
$40401$

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Question 114 Marks

Evaluate $\sqrt{1 \frac{4}{5} \times 14 \frac{21}{22} \times 2 \frac{7}{55}}$

Answer

$\sqrt{1 \frac{4}{5} \times 14 \frac{21}{22} \times 2 \frac{7}{55}}$
$=\sqrt{\frac{9}{5} \times \frac{637}{44} \times \frac{117}{55}}$
$=\sqrt{\frac{9 \times 637 \times 117}{5 \times 44 \times 5}}$
$=\sqrt{\frac{9 \times 7 \times 7 \times 13 \times 13 \times 9}{5 \times 11 \times 2 \times 2 \times 11 \times 55}}$

$7$	$637$
$7$	$91$
	$13$

$9$	$117$
	$13$

$=\frac{9 \times 7 \times 13}{5 \times 11 \times 2}$
$=\frac{819}{110}$
$=7 \frac{49}{110}$

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Question 124 Marks

Evaluate $\sqrt{\frac{25}{32} \times 2 \frac{13}{18} \times 0.25}$

Answer

$\sqrt{\frac{25}{32} \times 2 \frac{13}{18} \times 0.25}$
$=\sqrt{\frac{25}{32} \times \frac{49}{18} \times 0.25}$
$=\sqrt{\frac{25}{32} \times \frac{49}{18} \times \frac{25}{100}}$
$=\sqrt{\frac{25 \times 49 \times 25}{32 \times 18 \times 100}}$
$=\sqrt{\frac{25 \times 49 \times 1}{32 \times 18 \times 4}}$
$=\sqrt{\frac{5 \times 5 \times 7 \times 7}{(2 \times 2 \times 2 \times 2 \times 2) \times(2 \times 3 \times 3) \times(2 \times 2)}}$
$=\sqrt{\frac{\overline{5 \times 5} \times \overline{7 \times 7}}{\overline{2 \times 2} \times \overline{2 \times 2} \times \overline{2 \times 2} \times \overline{3 \times 3} \times \overline{2 \times 2}}}$
$=\frac{5 \times 7}{2 \times 2 \times 2 \times 3 \times 2}$
$=\frac{35}{48}$

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Question 134 Marks

Find the smallest number by which $2592$ be multiplied so that the product is a perfect square.

Answer

$2592=\overline{2 \times 2} \times \overline{2 \times 2} \times 2 \times \overline{3 \times 3} \times \overline{3 \times 3}$
On grouping the prime factors of $2592$ as shown; on factor
i.e. $2$ is left which cannot be paired with equal factor.

$2$	$2592$
$2$	$1296$
$2$	$648$
$2$	$324$
$2$	$162$
$3$	$81$
$3$	$27$
$3$	$9$
$3$	$3$
$ $	$1$

$2$ is the smallest number that must be multiplied by $2592$ to get a perfect square.
$\sqrt{2592 \times 2}=\sqrt{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3}$
$\sqrt{5184}=2 \times 2 \times 2 \times 3 \times 3$
$=8 \times 9$
$=72$

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MATHS [4 marks sum]

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