[3 marks sum]

Question 13 Marks

Evaluate: $101^2 – 100^2$

Answer

$101^2 – 100^2$
Using property, for any natural number $n,$
$(n + 1)^2 – n^2 = (n + 1) + n$
$\Rightarrow (100 + 1)^2 – 100^2 = (100 + 1) + 100$
$\Rightarrow 101^2 – 100^2 = 101 + 100$
$\Rightarrow 101^2 – 100^2 = 201$

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Question 23 Marks

Evaluate: $85^2 – 84^2$

Answer

$85^2 – 84^2$
Using property, for any natural number $n$,
$(n + 1)^2 – n^2 = (n + 1) + n$
$\Rightarrow (84 + 1)^2 – 84^2 = (84 + 1) + 84$
$\Rightarrow 85^2 – 84^2= 85 + 84$
$\Rightarrow 85^2 – 84^2 = 169$

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Question 33 Marks

Evaluate: $37^2 – 36^2$

Answer

$37^2 – 36^2$
Using property, for any natural number $n$,
$(n + 1)^2 – n^2 = (n + 1) + n$
$\Rightarrow (36 + 1)^2 – 36^2 = (36 + 1) + 36$
$\Rightarrow 37^2 – 36^2= 37 + 36$
$\Rightarrow 37^2– 36^2 = 73$

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Question 43 Marks

Find the square root of: $\sqrt{108 \times 2028}$

Answer

$ \sqrt{108 \times 2028}$
$ =\sqrt{219024}$

$ $	$468$
$4$	$219024 16$
$86$	$590 516$
$928$	$7424 7424$
	$x$

Hence, $\sqrt{108 \times 2028}=468$ OR
$\sqrt{108 \times 2028}$

$2$	$108$
$2$	$54$
$3$	$27$
$3$	$9$
$3$	$3$
	$1$

$2$	$2028$
$2$	$1014$
$3$	$507$
$13$	$169$
$13$	$13$
	$1$

$ =\sqrt{2 \times 2 \times 3 \times 3 \times 3 \times 2 \times 2 \times 3 \times 13 \times 13}$
$ =\sqrt{2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 13 \times 13}$
$ =2 \times 2 \times 3 \times 13$
$=468$

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Question 53 Marks

Find the square root of the following correct to two decimal place : $6 \frac{7}{8} \mathrm{C}$

Answer

$6 \frac{7}{8}=6.875$

	$2.62$
$2$	$6.8750 4$
$46$	$287 276$
$522$	$1150 1044$
	$106$

Required. square root $= 2.62$

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Question 63 Marks

Find the square root of the following correct to two decimal place: $3 \frac{4}{5}$

Answer

$3 \frac{4}{5}=3.80$

	$1.949$
$1$	$3.80 1$
$29$	$280 261$
$384$	$1900 1536$
$3889$	$36400 35001$
	$1399$

Required. square root $= 1.949 = 1.95$ upto two places of demical.

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Question 73 Marks

Find the square root of:$\ 0.602$ correct to two places of decimal

Answer

$0.602$

	$0.775$
$0.7$	$.06020 .49$
$0.147$	$1120 1029$
$1545$	$9100 7725$
	$1375$

Required square root $= 0.78$ up to two places of decimals.

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Question 83 Marks

Find the square root of $82.6$ correct to two places of decimal.

Answer

$82.6$

	$9.088$
$9$	$82.60$ $81$
$1808$	$16000$ $14464$
$18168$	$153600$ $145324$

Required square root $= 9.088 = 9.09$ up to two places of demical.

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Question 93 Marks

Find the square root of $496$ correct to three places of decimal.

Answer

$496$

	$22.271$
$2$	$496$ $4$
$42$	$96$ $84$
$442$	$1200$ $884$
$4447$	$31600$ $31129$
$44541$	$47100$ $44541$

Required square root $= 22.2708 = 22.271$ upto three places of decimals.

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Question 103 Marks

Find the least number which must be added to $5483$ so that the resulting number is a perfect square.

Answer

Clearly, $5483$ is greater than $74^2$

	$74$
$9$	$5483$ $49$
$144$	$583$ $576$
	$7$

$∴$ On adding the required number to $5483,$ we shall be getting $75^2$ i.e. $5625.$
Hence, the required number $= 5625 - 5483$
$= 142$

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Question 113 Marks

Find the least number which must be subtracted from $2037$ so that the resulting number is a perfect square.

Answer

Clearly; if $12$ is subtracted from 2037, the remainder will be a perfect square. $\therefore 2037-12=2025 \text { and } \sqrt{2025}=45$

	$45$
$4$	$2037 49$
$85$	$437 425$
	$12$

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Question 123 Marks

Find the least number which must be added to $1205$ so that the resulting number is a perfect square.

Answer

Clearly, $1205$ is greater than $34^2$

$∴$ On adding the required number to $1205,$ we shall be getting $35^2$ i.e., $1225$
$∴$ The required number $= 1225 - 1205 = 20$

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Question 133 Marks

Find the square root of $7.832$ correct to :$2$ significant digits.

Answer

Square root of $7.832$

	$2.7985$
$4$	$7.832000 4$
$49$	$383 329$
$549$	$5420 4941$
$5588$	$47900 44704$
$5596$	$319600 279825$
	$39775$

$= 2.8$ upto two significant digits

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Question 143 Marks

Find the square root of $7.832$ correct to: $ 2$ decimal places

Answer

Square root of $7.832$

	$2.798$
$2$	$7.832000 4$
$47$	$383 329$
$549$	$5420 4941$
$5588$	$47900 44704$
	$3196$

$\sqrt{7.832}=2.80$ upto two decimal places

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Question 153 Marks

Out of $745$ students, maximum are to be arranged in the school field for a $\text{P.T}$. display, such that the number of rows is equal to the number of columns. Find the number of rows if $16$ students were left out after the arrangement.

Answer

Total number of students $= 745$
Students left after standing in arrangement $= 16$
No. of students who were to be arranged $= 745 – 16 = 729$
The number of rows $=$ no. of students in each row
No. of rows $=\sqrt{729}$

$3$	$729$
$3$	$243$
$3$	$81$
$3$	$27$
$3$	$9$
$3$	$3$
	$1$

$ =\sqrt{\overline{3 \times 3} \times \overline{3 \times 3} \times \overline{3 \times 3}}$
$ =3 \times 3 \times 3=27$

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Question 163 Marks

Evaluate: $\sqrt{3^2 \times 6^3 \times 24}$

Answer

$\sqrt{3^2 \times 6^3 \times 24}$
$ =\sqrt{3^2 \times 6^3 \times 2 \times 2 \times 6}$
$=\sqrt{3^2 \times 6^3 \times 2 \times 2 \times 6}$
$=\sqrt{3^2 \times 6^4 \times 2^2}$
$=3 \times 6^2 \times 2$
$=3 \times 36 \times 2$
$=216$

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Question 173 Marks

Evaluate $\sqrt{\frac{0.225}{28.9}} x$

Answer

$\sqrt{\frac{0.225}{28.9}}$
$=\sqrt{\frac{0.225}{28.900}}$

$17$	$28900$
$17$	$1700$
$10$	$100$
	$10$

$=\sqrt{\frac{225}{28900}}$
$=\sqrt{\frac{15 \times 15}{17 \times 17 \times 10 \times 10}}$
$=\frac{15}{17 \times 10}$
$=\frac{15}{170}$
$=\frac{3 \times 5}{5 \times 34}$
$=\frac{3}{34}$

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Question 183 Marks

Evaluate $\sqrt{\frac{14.4}{22.5}}$

Answer

$\sqrt{\frac{14.4}{22.5}}$
$ =\sqrt{\frac{144}{225}}$
$ =\sqrt{\frac{12 \times 12}{15 \times 15}}$
$ =(\frac{12}{15} 15) \frac{0.8}{120}( \frac{120}{x x})$
$ =0.8$

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Question 193 Marks

Find the smallest number by which $12748$ be mutliplied so that the product is a perfect square?

Answer

$12748=\overline{2 \times 2} \times 3187$
On grouping the prime factors of $12748$ as shown; one factor i.e. $3187$ is left which cannot be paired with equal factor.

$2$	$12748$
$2$	$6374$
$3187$	$3187$
	$1$

The given number should be multiplied by $3187$.

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Question 203 Marks

By splitting into prime factors, find the square root of $194481$.

Answer

$\sqrt{194481}$
$=\sqrt{\overline{3 \times 3} \times \overline{3 \times 3} \times \overline{7 \times 7} \times \overline{7 \times 7}}$
$=3 \times 3 \times 7 \times 7$
$=441$

$3$	$194481$
$3$	$64827$
$3$	$21609$
$3$	$7203$
$7$	$2401$
$7$	$343$
$7$	$49$
	$7$

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Question 213 Marks

By splitting into prime factors, find the square root of $396900.$

Answer

$ \sqrt{396900}$
$ =\sqrt{\overline{2 \times 2} \times \overline{3 \times 3} \times \overline{3 \times 3} \times \overline{5 \times 5} \times \overline{7 \times 7}}$
$ =2 \times 3 \times 3 \times 5 \times 7$
$=630$

$2$	$396900$
$2$	$198450$
$3$	$99225$
$3$	$33075$
$3$	$11025$
$3$	$3675$
$5$	$1225$
$5$	$245$
$7$	$49$
	$7$

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Question 223 Marks

By splitting into prime factors, find the square root of $11025$.

Answer

$ \sqrt{11025}$
$ =\sqrt{\overline{5 \times 5} \times \overline{7 \times 7} \times \overline{3 \times 3}}$
$ =5 \times 7 \times 3$
$=105$

$5$	$11025$
$5$	$2205$
$7$	$441$
$7$	$63$
$3$	$9$
	$3$

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Question 233 Marks

If $\sqrt{784}=28$, find the value of: $\sqrt{7.84}+\sqrt{78400}$

Answer

$\sqrt{784}=28$
$\therefore \sqrt{7.84}=\sqrt{\frac{784}{100}}=\frac{28}{10}=2.8$
$\sqrt{78400}=\sqrt{28 \times 28 \times 10 \times 10}$
$=28 \times 10=280$
$\sqrt{7.84}+\sqrt{78400}=2.8+280=282.8$

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Question 243 Marks

Find the smallest perfect square divisible by $3, 4, 5$ and $6$.

Answer

$\text{L.C.M}$. of $3, 4, 5, 6 = 2 \times 2 \times 3 \times 5 = 60$

$2$	$3, 4, 5, 6$
$3$	$3, 2, 5, 3$
	$1, 2, 5, 1$

in which $3$ and $5$ are not in pairs $ \text{L.C.M}. = 2 \times 3 \times 2 \times 5 = 60$
We should multiple it by $3 \times 5$ i.e. by $15$
Required perfect square $= 60 \times 15 = 900$

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Question 253 Marks

Find the square root of: $\frac{507}{4107}$

Answer

$ \frac{507}{4107}$
$ =\frac{507+3}{4107+3}$
$ =\frac{169}{1369}$

	$13$
$1$	$169 1$
$23$	$69 69$
	$x$

	$37$
$3$	$1369 9$
$67$	$469 469$
	$x$

Hence, square root of $\sqrt{\frac{169}{1369}}=\frac{13}{37}$

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