Question 14 Marks
Find the square root of the following decimal numbers.$51.84$
AnswerThe square root of $51.84$ can be calculated as follows.
$\begin{array}{c|c} & 7.2 \\ \hline7& {\overline{51.}\\{-49}} {\overline{84}}\\\hline142&{284\\284}\\\hline&0\end{array}$ $\therefore\sqrt{51.84}=7.2$
View full question & answer→Question 24 Marks
Find the square root of each of the following numbers by Division method.
$3249$
AnswerThe square root of $3249$ can be calculated as follows.
$\begin{array}{c|c} & 57 \\ \hline5 & {\overline{32}\\{-25}} {\overline{49}}\\\hline107&{749\\749}\\\hline&0\end{array}$
$\therefore\sqrt{3249}=57$
View full question & answer→Question 34 Marks
Find the square root of the following decimal numbers.
$2.56$
AnswerThe square root of $2.56$ can be calculated as follows.
$\begin{array}{c|c} & 16 \\ \hline1 & {\overline{2.}\\{-1}} {\overline{56}}\\\hline26&{156\\156}\\\hline&0\end{array}$
$\therefore\sqrt{2.56}=1.6$
View full question & answer→Question 44 Marks
Find the square root of each of the following numbers by Division method. $529$
AnswerThe square root of $529$ can be calculated as follows. $\begin{array}{c|c} & 23 \\ \hline2 & {\overline{5}\\{-4}} {\overline{29}}\\\hline43&{129\\129}\\\hline&0\end{array}$
$\therefore\sqrt{529}=23$
View full question & answer→Question 54 Marks
Find the square root of each of the following numbers by Division method.$1024$
AnswerThe square root of $1024$ can be calculated as follows.
$\begin{array}{c|c} & 32 \\ \hline3 & {\overline{10}\\{-9}} {\overline{24}}\\\hline62&{124\\124}\\\hline&0\end{array}$
$\therefore\sqrt{1024}=32$
View full question & answer→Question 64 Marks
Find the least number which must be subtracted from each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.
$402$
AnswerThe square root of $402 $can be calculated by long division method as follows.
+$\begin{array}{c|c} & 20 \\ \hline2 & {\overline{4}\\{-4}} {\overline{02}}\\\hline40&{02\\00}\\\hline&0\end{array}$
The reminder is $2.$ it represents that the square of $20$ is less than $402$ by $2.$
Therefore, a perfect square will be obtained by subtracting $2$ from the given number $402$.
Therefore, required perfect square $= 402 - 2 = 400$ And $\sqrt{400}=20$
View full question & answer→Question 74 Marks
Find the square root of each of the following numbers by Division method.$900$
AnswerThe square root of $900$ can be calculated as follows.
$\begin{array}{c|c} & 30 \\ \hline1 & {\overline{9}\\{-9}} {\overline{00}}\\\hline60&{00\\00}\\\hline&0\end{array}$
$\therefore\sqrt{900}=30$
View full question & answer→Question 84 Marks
Find the least number which must be subtracted from each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.$825$
AnswerThe square root of $825$ can be calculated by long division method as follows.
$\begin{array}{c|c} & 28 \\ \hline2 & {\overline{8}\\{-4}} {\overline{25}}\\\hline48&{425\\384}\\\hline&41\end{array}$
The remainder is $41.$
it represents that the square of $28$ is less than $825$ by $41$.
Therefore, a perfect square will be obtained by subtracting $41$ from the given number $825$.
Therefore, required perfect square $= 825 - 41 = 784$ And $\sqrt{784}=28$
View full question & answer→Question 94 Marks
Find the least number which must be subtracted from each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.$1989$
AnswerThe square root of $1989$ can be calculated by long division method as follows.
$\begin{array}{c|c} & 44 \\ \hline1 & {\overline{19}\\{-16}} {\overline{89}}\\\hline84&{389\\336}\\\hline&53\end{array}$
The reminder is $53.$
it represents that the square of $44$ is less than $1989$ by $53.$
Therefore, a perfect square will be obtained by subtracting $53$ from the given number $1989.$
Therefore, required perfect square $= 1989 - 53 = 1936$ And $\sqrt{1936}=44$
View full question & answer→Question 104 Marks
Find the number of digits in the square root of each of the following numbers (without any calculation).$4489$
AnswerBy placing bars, we obtain $4489=\overline{44} \ \overline{89}$ Since there are two bars, the square root of $4489$ will have only $2$ digit in it.
View full question & answer→Question 114 Marks
Find the least number which must be subtracted from each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.$3250$
AnswerThe square root of 3250 can be calculated by long division method as follows.
$\begin{array}{c|c}& \ \ \ \ \ 57\\\hline5& \ \ \ \ \ \ \ \ \ \overline{32} \ \overline{50}\\ &-25\\ \hline107& \ \ \ \ \ 750\\& \ \ \ \ \ 749\\\hline& \ \ \ \ \ 1\end{array}$
The remainder is $1.$
it represents that the square of $57$ is less than $3250$ by $1.$
Therefore, a perfect square will be obtained by subtracting $1$ from the given number $3250$.
Therefore, required perfect square $= 3250 - 1 = 3249$ And $\sqrt{3249}=57$
View full question & answer→Question 124 Marks
Find the square root of the following decimal numbers.$31.36$
AnswerThe square root of $31.36$ can be calculated as follows.
$\begin{array}{c|c} & 5.6 \\ \hline5 & {\overline{31.}\\{-25}} {\overline{36}}\\\hline106&{636\\636}\\\hline&0\end{array}$
$\therefore\sqrt{31.36}=5.6$
View full question & answer→Question 134 Marks
Find the number of digits in the square root of each of the following numbers (without any calculation).$27225$
AnswerBy placing bars, we obtain $27225=\overline{2} \ \overline{72} \ \overline{25}$ Since there are three bars, the square root of $27225$ will have three digits in it.
View full question & answer→Question 144 Marks
Find the square root of each of the following numbers by Division method.$5776$
AnswerThe square root of $5776$ can be calculated as follows.
$\begin{array}{c|c} & 76 \\ \hline3 & {\overline{57}\\{-49}} {\overline{76}}\\\hline146&{876\\876}\\\hline&0\end{array}$
$\therefore\sqrt{5776}=76$
View full question & answer→Question 154 Marks
Find the square root of each of the following numbers by Division method.$576$
AnswerThe square root of 576 can be calculated as follows.
$\begin{array}{c|c} & 24\\ \hline1 & {\overline{5}\\{-4}} {\overline{76}}\\\hline44&{176\\176}\\\hline&0\end{array}$
$\therefore\sqrt{576}=24$
View full question & answer→Question 164 Marks
Find the number of digits in the square root of each of the following numbers (without any calculation).$144$
AnswerBy placing bars, we obtain $144=\overline{1} \ \overline{44}$ Since there is only one bars, the square root of $144$ will have only $2$ digit in it.
View full question & answer→Question 174 Marks
Find the square root of each of the following numbers by Division method. $2304$
AnswerThe square root of $2304$ can be calculated as follows.
$\begin{array}{c|c}& \ \ \ \ \ 48\\\hline4& \ \ \ \ \ \ \ \ \ \overline{23} \ \overline{04}\\ &-16\\ \hline88& \ \ \ \ \ 704\\& \ \ \ \ \ 704\\\hline& \ \ \ \ \ 0\end{array}$
$\therefore\sqrt{2304}=48$
View full question & answer→Question 184 Marks
Find the number of digits in the square root of each of the following numbers (without any calculation).$390625$
AnswerBy placing bars, we obtain $390625=\overline{39} \ \overline{06} \ \overline{25}$
Since there are three bars, the square root of $390625$ will have $3$ digits in it.
View full question & answer→Question 194 Marks
Find the square root of each of the following numbers by Division method.$1369$
AnswerThe square root of $1369$ can be calculated as follows.
$\begin{array}{c|c} & 37 \\ \hline3 & {\overline{13}\\{-9}} {\overline{69}}\\\hline67&{469\\469}\\\hline&0\end{array}$
$\therefore\sqrt{1369}=37$
View full question & answer→Question 204 Marks
Find the square root of each of the following numbers by Division method.$3136$
AnswerThe square root of 3136 can be calculated as follows.
$\begin{array}{c|c} & 56 \\ \hline5& {\overline{31}\\{-25}} {\overline{36}}\\\hline106&{636\\636}\\\hline&0\end{array}$
$\therefore\sqrt{3136}=56$
View full question & answer→Question 214 Marks
Find the square root of the following decimal numbers.$7.29$
AnswerThe square root of $7.29$ can be calculated as follows.
$\begin{array}{c|c} & 2.7 \\ \hline12& {\overline{7.}\\{-4}} {\overline{29}}\\\hline47&{329\\329}\\\hline&0\end{array}$
$\therefore\sqrt{7.29}=2.7$
View full question & answer→Question 224 Marks
Find the square root of the following decimal numbers.$42.25$
AnswerThe square root of $42.25$ can be calculated as follows.
$\begin{array}{c|c} & 6.5 \\ \hline6 & {\overline{42.}\\{-36}} {\overline{25}}\\\hline125&{625\\625}\\\hline&0\end{array}$
$\therefore\sqrt{42.25}=6.5$
View full question & answer→Question 234 Marks
Find the square root of each of the following numbers by Division method.$4489$
AnswerThe square root of $4489$ can be calculated as follows.
$\begin{array}{c|c} & 67 \\ \hline6 & {\overline{44}\\{-36}} {\overline{89}}\\\hline127&{889\\889}\\\hline&0\end{array}$
$\therefore\sqrt{4489}=67$
View full question & answer→Question 244 Marks
Find the number of digits in the square root of each of the following numbers (without any calculation). $64$
AnswerBy placing bars, we obtain $64=\overline{64}$ Since there is only one bar, the square root of $64$ will have only one digit in it.
View full question & answer→Question 254 Marks
Find the square root of each of the following numbers by Division method.$7921$
AnswerThe square root of $7921$ can be calculated as follows.
$\begin{array}{c|c} & 89 \\ \hline8 & {\overline{79}\\{-64}} {\overline{21}}\\\hline169&{1521\\1521}\\\hline&0\end{array}$
$\therefore\sqrt{7921}=89$
View full question & answer→Question 264 Marks
Find the square root of each of the following numbers by Division method.$3481$
AnswerThe square root of $3481$ can be calculated as follows.
$\begin{array}{c|c} & 59 \\ \hline5 & {\overline{34}\\{-25}} {\overline{81}}\\\hline109&{981\\981}\\\hline&0\end{array}$
$\therefore\sqrt{3481}=59$
View full question & answer→Question 274 Marks
Find the least number which must be added to each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.$1825$
AnswerThe square root of $1825$ can be calculated by long division method as follows.
$\begin{array}{c|c} & 42 \\ \hline4& {\overline{18}\\{-16}} {\overline{25}}\\\hline82&{225\\164}\\\hline&61\end{array}$
The remainder is $61.$
it represents that the square of $42$ is less than $1825$
Next number is $43$ and $43^2= 1849$
Hence, number to be added to $1825 = 43^2- 1825 = 1849 - 1825 = 24$
The required perfect square is 1849 and $\sqrt{1849}=43$
View full question & answer→Question 284 Marks
In a right triangle $ABC, ÐB = 90^\circ $. If $AB = 6 \ cm, BC = 8 \ cm$, find $AC$
Answer$\triangle\text{ABC}$ is right - angled at $B.$
Therefore, by applying pythagoras theoram, we obtain
$ A C^2=A B^2+B C^2 $
$ A C^2=(6 \mathrm{~cm})^2+(8 \mathrm{~cm})^2 $
$ A C^2=(36+64) \mathrm{cm}^2=100 \mathrm{~cm}^2 $
$\text{AC}=(\sqrt{100})\text{cm} = (\sqrt{10\times10})\text{cm}$
$\text{Ac} = 10\text{ cm}$
View full question & answer→Question 294 Marks
There are $500$ children in a school. For a $P.T$. drill they have to stand in such a manner that the number of rows is equal to number of columns. How many children would be left out in this arrangement.
AnswerIt is given that there are $500$ children in the school.
They have to stand for a $P.T$ drill such that the number of rows is equal to the number of colunms.
The number of children who will be left out in this arrangement has to be calculated.
That is the number which should be substracted from $500$ to make it a perfect square has to be calculated.
The square root of $500$ can be calculated by long division method as follows.
$\begin{array}{c|c} & 22 \\ \hline2 & {\overline{5}\\{-4}} {\overline{00}}\\\hline42&{100\\84}\\\hline&16\end{array}$
The remainder is $16$.
it shows that the square of $22$ is less than $500$ by $16$.
Therefore, if we substract $16$ from $500$, we will obtain a perfect square.
Required perfect square $= 500 - 16 = 484$
Thus, the number of children who will be left out is $16.$
View full question & answer→Question 304 Marks
For each of the following numbers, find the smallest whole number by which it should
be divided so as to get a perfect square. Also find the square root of the square
number so obtained.
$252$
Answer$\begin{array}{c|c}2&252\\\hline2&126\\\hline2&63\\\hline3&21\\\hline7&7\\\hline&1\end{array}$
$252 = 2 \times 2 \times 3 \times 3 \times 7$
Here, prime factor $7$ has no pair.
Therefore $252$ must be divided by $7$ to make it a perfect square.
$\therefore 252 ÷ 7 = 36$
$\sqrt{36}=2\times3=6$
View full question & answer→Question 314 Marks
For each of the following numbers, find the smallest whole number by which it should be divided so as to get a perfect square. Also find the square root of the square number so obtained.$2800$
Answer$\begin{array}{c|c}2&2800\\\hline2&1400\\\hline2&700\\\hline2&350\\\hline5&175\\\hline5&35\\\hline7&7\\\hline&1\end{array}$
$2800 = 2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 7$
Here, prime factor $5$ has no pair.
Therefore $2645$ must be divided by $5$ to make it a perfect square.
$\therefore 2800 ÷ 7 = 400$
$\sqrt{400}=2\times2\times5=20$
View full question & answer→Question 324 Marks
Find the least number which must be added to each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.$6412$
AnswerThe square root of $6412$ can be calculated by long division method as follows.
$\begin{array}{c|c} & 80 \\ \hline8 & {\overline{64}\\{-64}} {\overline{12}}\\\hline160&{012\\0}\\\hline&12\end{array}$
The remainder is $12.$
it represents that the square of $80$ is less than $6412.$
Next number is 81 and $81^2= 6561$
Hence, number to be added to $6412 = 81^2- 6412 = 6561 - 6412 = 149$
The required perfect square is 6561 and $\sqrt{6561}=81$
View full question & answer→Question 334 Marks
Find the least number which must be added to each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.$252$
AnswerThe square root of $252$ can be calculated by long division method as follows.
$\begin{array}{c|c} & 15 \\ \hline1 & {\overline{2}\\{-1}} {\overline{52}}\\\hline25&{152\\125}\\\hline&27\end{array}$
The remainder is $27.$
it represents that the square of $15$ is less than $252.$
Next number is $16$ and $16^2= 256$
Hence,number to be added to $252 = 16^2- 252 = 256 - 252 = 4$
The required perfect square is $256$ and $\sqrt{256}=16$
View full question & answer→Question 344 Marks
Find the length of the side of a square whose area is $441 m^2$.
AnswerLet the legnth of the side of the square be $x m.$
Area of square $= (x)^2= 441\ m^2$
$\text{x}=\sqrt{441}$
The square root of $441$ can be calcuted as follows.
$\begin{array}{c|c} & 21 \\ \hline1 & {\overline{4}\\{-4}} {\overline{41}}\\\hline41&{041\\41}\\\hline&0\end{array}$
$\therefore \text{x} = 21\text{m}$
Hence, the legnth of the side of the square is $21 m.$
View full question & answer→Question 354 Marks
The students of Class VIII of a school donated $₹ 2401$ in all, for Prime Minister’s National Relief Fund. Each student donated as many rupees as the number of students in the class. Find the number of students in the class.
AnswerIt is given that each student donated as many rupees as the number of students of the class.Number of students in the class will be the square root of the amount donated by the students of the class.
The total amount of donation is Rs $2401.$
number of students in the class = $\sqrt{2401}$
$2401 = 7 \times 7 \times 7 \times 7$
$\therefore\sqrt{2401}=7\times7=49$
Hence, the number of students in the class is $49.$
View full question & answer→Question 364 Marks
In a right triangle $ABC, ÐB = 90^\circ $. If $AC = 13 \ cm, BC = 5 \ cm$, find $AB$
Answer$\triangle\text{ABC}$is right−angled at $B.$
Therefore, by applying pythagoras theoram, we obtain
$ A C^2=A B^2+B C^2 $
$ (13 \mathrm{~cm})^2=(A B)^2+(5 \mathrm{~cm})^2 $
$ A B^2=\left(13 \mathrm{~cm}^2\right)-(5 \mathrm{~cm})^2=(169-25) \mathrm{cm}^2 $
$ =144 \mathrm{~cm}^2 $
$\text{AB}=(\sqrt{144})\text{cm} = (\sqrt{12\times12})\text{cm}$
$\text{AB} = 12 \text{cm} $
View full question & answer→Question 374 Marks
For each of the following numbers, find the smallest whole number by which it should be divided so as to get a perfect square. Also find the square root of the square number so obtained.$1620$
Answer$\begin{array}{c|c}2&1620\\\hline2&405\\\hline3&135\\\hline3&45\\\hline3&15\\\hline5&5\\\hline&1\end{array}$
$2800 = 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 5$
Here, prime factor $5$ has no pair.
Therefore $1620$ must be divided by $5$ to make it a perfect square.
$\therefore 1620 ÷ 5 = 324$
$\sqrt{32}=2\times2\times3=18$
View full question & answer→Question 384 Marks
Find the least number which must be added to each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.$1750$
AnswerThe square root of $1750$ can be calculated by long division method as follows.
$\begin{array}{c|c} & 41 \\ \hline4 & {\overline{17}\\{-16}} {\overline{50}}\\\hline81&{150\\81}\\\hline&69\end{array}$
The remainder is $69.$
it represents that the square of $41$ is less than $1750.$
Next number is $42$ and $42^2= 1764$
Hence, number to be added to $1750 = 42^2- 1750= 1764 - 1750 = 14$
The required perfect square is $1764$ and $\sqrt{1764}=42$
View full question & answer→Question 394 Marks
Find the least number which must be added to each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained. $525$
AnswerThe square root of $525$ can be calculated by long division method as follows.
$\begin{array}{c|c} & 22 \\ \hline1 & {\overline{5}\\{-4}} {\overline{25}}\\\hline42&{125\\84}\\\hline&41\end{array}$
The remainder is $41.$
it represents that the square of $22$ is less than $525$
Next number is $23$ and $23^2= 529$
Hence, number to be added to $525 = 23^2- 525 = 529 - 525 = 4$
The required perfect square is $529$ and $\sqrt{529}=23$
View full question & answer→Question 404 Marks
$2025$ plants are to be planted in a garden in such a way that each row contains as many plants as the number of rows. Find the number of rows and the number of plants in each row.
AnswerIt is given that in the garden, each row contains as many plants as the number of rows.
Hence,
Number of rows = Number of plants in each row
Total number of plants = Number of rows $\times $ Number of plants in each row
Number of rows $\times $ Numbers of plants in each row $= 2025$
(Number of rows)$^2= 2025$
Number of row $=\sqrt{2025}$
$2025 = 5 \times 5 \times 3 \times 3 \times 3 \times 3$
$\sqrt{2025}=5\times3\times3=45$
Thus, the number of rows and the number of plans in each row is $45.$
View full question & answer→Question 414 Marks
For each of the following numbers, find the smallest whole number by which it should
be divided so as to get a perfect square. Also find the square root of the square
number so obtained.$396$
Answer$\begin{array}{c|c}2&396\\\hline2&198\\\hline3&99\\\hline3&33\\\hline11&11\\\hline&1\end{array}$
$396 = 2 \times 2 \times 3 \times 3 \times 11$
Here, prime factor $11$ has no pair.
Therefore $396$ must be divided by $11$ to make it a perfect square.
$\therefore 396 ÷ 11 = 36$
$\sqrt{36}=2\times3=6$
View full question & answer→Question 424 Marks
Find the smallest square number that is divisible by each of the numbers $8, 15$ and $20.$
AnswerThe number that is perfectly divisible by each one of $8, 15,$ and $20$ is thir $LCM$.
$\begin{array}{c|c}2&8,5,10\\\hline2&4,15,10\\\hline2&2,15,5,\\\hline3&1,15,5\\\hline5&1,5,5\\\hline&1,1,1\end{array}$
$LCM$ of $8, 15$, and $20 = 2 \times 2 \times 2 \times 3 \times 5 = 120$
Here, prime factor $2, 3,$ and $5$ do not have There respective pairs. $120$ is not a perfect square.
Therefore, $120$ should be multiplied by $2 \times 3 \times 5$
i.e. to obtain a perfect square. Hence, the required square number is $120 \times 2 \times 3 \times 5 = 3600$
View full question & answer→Question 434 Marks
Find the smallest square number that is divisible by each of the numbers $4, 9$ and $10.$
AnswerThe number that will be perfectly divisible by each one of $4, 9,$ and $10$ is thir $LCM$ of these numbers is as follows.
$\begin{array}{c|c}2&4,9,10\\\hline2&2,9,5\\\hline3&1,9,5,\\\hline3&1,3,5\\\hline5&1,1,5\\\hline&1,1,1\end{array}$
$LCM$ of $4, 9, 10 = 2 \times 2 \times 3 \times 3 \times 5 = 180$
Here, prime factor $5$ does not have its pair.
Therefore, $180$ is not a perfect square. if we multiply $180$ with $5$, then the number will become a perfect square.
Therefore, 180 should be multiplied with $5$ to obtain a perfect square.
Hence, the required square number is $180 \times 5 = 900$
View full question & answer→Question 444 Marks
For each of the following numbers, find the smallest whole number by which it should be divided so as to get a perfect square. Also find the square root of the square number so obtained.$2925$
Answer$\begin{array}{c|c}3&2925\\\hline3&975\\\hline3&325\\\hline5&65\\\hline13&13\\\hline&1\end{array}$
$2925 = 3 \times 3 \times 5 \times 5 \times 13$
Here, prime factor $13$ has no pair.
Therefore $2925$ must be divided by $13$ to make it a perfect square.
$\therefore 2925 ÷ 13 = 225$
$\sqrt{225}=3\times5=15$
View full question & answer→Question 454 Marks
A gardener has $1000$ plants. He wants to plant these in such a way that the number of rows and the number of columns remain same. Find the minimum number of plants he needs more for this.
AnswerIt is given that the gardener has $1000$ plants. The number of rows and the number of columns is the same.
We have to find the number of more plants that should be there, so that when the gardener plants them, the number of rows and columns are same.
That is; the number which should be added to make it a perfect square has to be calculated.
The square root of 1000 can be calculated by long division method as follows.
$\begin{array}{c|c} & 31 \\ \hline3 & {\overline{10}\\{-9}} {\overline{00}}\\\hline61&{100\\61}\\\hline&39\end{array}$
The remainder is $39.$
it represents that the square of $31$ is less than $1000.$
The next number is $32$ and $32^2= 1024$
Hence, number to be added to $1000$ to make it perfect square
$= 32^2- 1000 = 1024 - 1000 = 24$
Thus, the required number of plants is $24.$
View full question & answer→