Question 513 Marks
The students of class VIII of a school donated Rs. $2401$ for PM'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.
AnswerLet $S$ be the number of students.
Let r be the amount in rupees donated by each student. The total donation can be expressed by, $S \times r = Rs. 2401$
Since the total amount in rupees is equal to the number of students, r is equal to $S.$
Substituting this in the first equation:
$S \times S = 2401$
$S^2= (7 \times 7) \times (7 \times 7)$
$S = 7 \times 7 = 49$
So, there are $49$ students in the class.
View full question & answer→Question 523 Marks
Show that the following numbers is a perfect square. Also, find the number whose square is the given number in case: $2025$
AnswerIn problem, factorise the number into its prime factors.
$2025 = 3 \times 3 \times 3 \times 3 \times 5 \times 5$
Grouping the factors into pairs of equal factors, we obtain,
$2025 = (3 \times 3) \times (3 \times 3) \times (5 \times 5)$
No factors are left over. Hence, $2025$ is a perfect square. Moreover, by grouping $2025$ into equal factors:
$2025 = (3 \times 3 \times 5) \times (3 \times 3 \times 5)$
$2025 = (3 \times 3 \times 5)^2$
Hence, $2025$ is the square of $45$, which is equal to $3 \times 3 \times 5$
View full question & answer→Question 533 Marks
Find the square root of the following correct to three places of decimal:
$\frac{5}{12}$
AnswerWe can find the square root up to four decimal places by expanding $\frac{5}{12}$ to decimal form up to eight digits to the right of the decimal point as shown below, $\frac{5}{2}=0.41666666$
Hence, we have,
So, the square root of $\frac{5}{12}$ up to three decimal places is $0.645$ View full question & answer→Question 543 Marks
Find the square root of the following by prime factorization.$1156$
AnswerResolving 1156 into prime factors, $1156 = 2 \times 2 \times 17 \times 17$
$\begin{array}{c|c}2& 1156 \\ \hline 2 & 578 \\\hline 17&289 \\\hline17&17 \\\hline&1\end{array}$
Grouping the factors into pairs of equal factors, $1156 = (2 \times 2) \times (17 \times 17)$
Taking one factor for each pair, we get the square root of $1764, 2 \times 17 = 34$
View full question & answer→Question 553 Marks
Find the squares of the following numbers using the identity $ (a+b)^2=a^2-a b+a b+b^2 : 702$
Answer$ (a+b)^2=a^2-a b+a b+b^2 $
$ (702)^2=(700+2)^2 $
$ =490000+1400+1400+4 $
$ =492804 $
View full question & answer→Question 563 Marks
Find the squares of the following numbers using column method. Verify the result by finding the square using the usual multiplication, $37$
Answer$(37)^2$ Here, $a = 3, b = 7$
|
$\text{a}^2$
|
$2\text{ab}$
|
$\text{b}^2$
|
|
$\ \ (3)^2\\=9\\+{4}$
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$2\times3\times7\\=42\\+\ \ 4$
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$\ \ \ (7)^2\\=49$
|
|
$13$
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$46$
|
|
$(37)^2= 37 × 37 = 1369$
$ (37)^2= 1369$ View full question & answer→Question 573 Marks
A welfare association collected Rs. $202500$ as donation from the residents. If each paid as many rupees as there were residents, find the number of residents.
AnswerLet R be the number of residents.
Let r be the money in rupees donated by each resident.
Total donation $= R \times r = 202500$
Since the money received as donation is the same as the number of residents:
$r = R$
Substituting this in the first equation, we get,
$R \times R = 202500$
$R^2= 202500$
$R^2= (2 \times 2) \times (5 \times 5) \times (5 \times 5) \times (3 \times 3)^2$
$R = 2 \times 5 \times 5 \times 3 \times 3 = 450$
So, the number of residents is $450$
View full question & answer→Question 583 Marks
Find the square root of the following correct to three places of decimal:$0.019$
AnswerWe can find the square root up to three decimal places by using long division until we get four decimal places and then rounding it to three decimal places.
Hence, the square root of $0.019$ up to three decimal places is $0.138$ View full question & answer→Question 593 Marks
Find the square root of the following correct to three places of decimal:$0.90$
AnswerWe can find the square root up to three decimal places by using long division until we get four decimal places and then rounding it to three decimal places.
Hence, the square root of $0.9$ up to three decimal places is $0.949$ View full question & answer→Question 603 Marks
Find the squares of the following numbers using column method. Verify the result by finding the square using the usual multiplication, $54$
Answer$(54)^2$
| $\text{a}^2$ |
$\text{2ab}$ |
$\text{b}^2$ |
| $\ \ \ (5)^2\\=25\\ +\ { 4}$ |
$2\times5\times4\\=40\\+\ \ 1$ |
$\ \ \ (4)^2\\=16$ |
| $29$ |
$41$ |
|
$(54)^2= 54 × 54 = 2961$
$\therefore (54)^2= 2961$ View full question & answer→Question 613 Marks
Find the squares of the following numbers using diagonal method: $273$
Answer$(273)^2$
$\therefore\ (273)^2=74529$ View full question & answer→Question 623 Marks
Find the squares of the following numbers using diagonal method: $295$
Answer$(295)^2$
$\therefore\ (295)^2=87025$ View full question & answer→Question 633 Marks
Find the square root of $11$ correct to five decimal places.
AnswerUsing the long division method,
$\therefore\ \sqrt{11}=3.31662$ View full question & answer→Question 643 Marks
Using square root table, find the square root: $4192$
Answer$\sqrt{4192}=\sqrt{2\times2\times2\times2\times2\times131}$
$=2\times2\sqrt{2}\times\sqrt{131}$ The square root of $131$ is not listed in the table.
Hence, we have to apply long division to find it.
Substituting the values, $=2\times2\times11.4455$
$\big($Using the table to find $\sqrt{2}\big)$
$=64.75$ View full question & answer→Question 653 Marks
Find the squares of the following numbers using column method. Verify the result by finding the square using the usual multiplication, $25$
Answer$(25)^2$
Here, $a = 2, b = 5$
| $\text{a}^2$ |
$2\text{ab}$ |
$\text{b}^2$ |
| $\ \ (2)^2\\=4\\+{2}\\ \ \ \ \ \overline{6}$ |
$2\times2\times5\\=20\\+\ \ 2\\ \ \ \ \ \overline{22}$ |
$\ \ \ (5)^2\\=25$ |
$(25)^2= 25 \times 25 = 625$
$(25)^2= 625$ View full question & answer→Question 663 Marks
Show that the following numbers is a perfect square. Also, find the number whose square is the given number in case:$4761$
AnswerIn problem, factorise the number into its prime factors.
$4761 = 3 \times 3 \times 23 \times 23$
Grouping the factors into pairs of equal factors, we obtain,
$4761 = (3 \times 3) \times (23 \times 23)$
No factors are left over. Hence, $4761$ is a perfect square. The above expression is already grouped into equal factors,
$4761 = (3 \times 23) \times (3 \times 23)$
$4761 = (3 \times 23)^2$
Hence, $4761$ is the square of $69,$ which is equal to $3 \times 23.$
View full question & answer→Question 673 Marks
Write five numbers for which you cannot decide whether they are squares.
AnswerA number whose unit digit is $2, 3, 7$ or $8$ cannot be a perfect square.
On the other hand, a number whose unit digit is $1, 4, 5, 6, 9$ or $0$ might be a perfect square (although we will have to verify whether it is a perfect square or not).
Applying the above two conditions,
we cannot quickly decide whether the following numbers are squares of any numbers, $1111, 1444, 1555, 1666, 1999$
View full question & answer→Question 683 Marks
Write the prime factorization of the following numbers and hence find their square roots. $7056$
AnswerThe prime factorisation of $9604, 7056 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7 \times 7$
Grouping them into pairs of equal factors, we get: $7056 = (2 \times 2) \times (2 \times 2) \times (3 \times 3) \times (7 \times 7)$
Taking one factor from each pair, we get, $\sqrt{7056}=2\times2\times3\times7=84$
View full question & answer→Question 693 Marks
Find the square root of the following correct to three places of decimal: $\frac{7}{8}$
AnswerWe can find the square root up to four decimal places by expanding $\frac{7}{8}$ to decimal form up to eight digits to the right of the decimal point as shown below, $\frac{7}{8}=0.875$
Hence, we have,
So, the square root of $\frac{7}{8}$ up to three decimal places is $0.935$ View full question & answer→Question 703 Marks
Find the square root of the following correct to three places of decimal:$0.00064$
AnswerWe can find the square root up to three decimal places by using long division until we get four decimal places and then rounding it to three decimal places.
Hence, the square root of $0.00064$ up to three decimal places is $0.025$ View full question & answer→Question 713 Marks
Find the squares of the following numbers using diagonal method: $98$
Answer$(98)^2$
$\therefore\ (98)^2=9604$ View full question & answer→Question 723 Marks
Write five numbers which you cannot decide whether they are square just by looking at the unit's digit.
AnswerA number whose unit digit is $2, 3, 7$ or $8$ cannot be a perfect square.
On the other hand, a number whose unit digit is $1, 4, 5, 6, 9$ or $0$ might be a perfect square although we have to verify that. Applying these two conditions,
we cannot determine whether the following numbers are squares just by looking at their unit digits, $1111, 1001, 1555, 1666$ and $1999$
View full question & answer→Question 733 Marks
Write the prime factorization of the following numbers and hence find their square roots. $9604$
AnswerThe prime factorisation of $9604, 9604 = 2 \times 2 \times 7 \times 7 \times 7 \times 7$
Grouping them into pairs of equal factors,
we get: $9604 = (2 \times 2) \times (7 \times 7) \times (7 \times 7)$ Taking one factor from each pair,
we get, $\sqrt{9604}=2\times7\times7=98$
View full question & answer→Question 743 Marks
Find the square root of the following by prime factorization.$7056$
AnswerResolving 7056 into prime factors, $7056 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7 \times 7$
$\begin{array}{c|c}2& 7056 \\ \hline 2 & 3528 \\\hline 2&1764 \\\hline2&882\\\hline3&441\\\hline3&147\\\hline7&49\\\hline7&7\\\hline&1\end{array}$
Grouping the factors into pairs of equal factors, $7056 = (2 \times 2) \times (2 \times 2) \times (3 \times 3) \times (7 \times 7)$
Taking one factor for each pair, we get the square root of $705 2 \times 2 \times 3 \times 7 = 84$
View full question & answer→Question 753 Marks
Find the squares of the following numbers using column method. Verify the result by finding the square using the usual multiplication, $71$
Answer$(71)^2$
Here, $a = 7, b = 1$
| $\text{a}^2$ |
$2\text{ab}$ |
$\text{b}^2$ |
| $\ \ \ (7)^2\\=49\\+\ \ {1}$ |
$2\times7\times1\\=14$ |
$\ \ \ (1)^2\\=\ 1$ |
| $50$ |
|
|
$(71)^2= 71 \times 71 = 5041$
$\therefore (71)^2= 5041$ View full question & answer→Question 763 Marks
Find the squares of the following numbers using the identity $ (a+b)^2=a^2-a b+a b+b^2 : 505$
Answer$ (a+b)^2=a^2-a b+a b+b^2 $
$ (505)^2=(500+5)^2 $
$ =250000+2500+2500+25 $
$ =255025 $
View full question & answer→Question 773 Marks
Using square root table, find the square root: $21.97$
Answersolution We have to find $\sqrt{21.97}$ From the square root table,
we have, $\sqrt{21}=\sqrt{3} \times \sqrt{7}=4.583$ and $\sqrt{22}=\sqrt{2} \times \sqrt{11}=4.690$
Their difference is $0.107$ Thus, for the difference of $1(22-21)$,
the difference in the values of the square roots is $0.107$ For the difference of $0.97 ,$
the difference in the values of the values of their square roots is, $0.107 \times 0.97=0.104 \therefore \sqrt{21.97}=4.583+0.104 \approx 4.687$
View full question & answer→Question 783 Marks
Find the square root of the following by prime factorization.$1764$
AnswerResolving $1764$ into prime factors,
$1764 = 2 \times 2 \times 3 \times 3 \times 7 \times 7$
$\begin{array}{c|c}2& 1764 \\ \hline 2 & 882 \\\hline 3&441 \\\hline3&147 \\\hline7&49\\\hline7&7\\\hline&1 \end{array}$
Grouping the factors into pairs of equal factors,
$1764 = (2 \times 2) \times (3 \times 3) \times (7 \times 7)$
Taking one factor for each pair, we get the square root of $1764,$
$2 \times 3 \times 7 = 42$
View full question & answer→Question 793 Marks
Find the square root of the following correct to three places of decimal:$0.1$
AnswerWe can find the square root up to three decimal places by using long division until we get four decimal places and then rounding it to three decimal places.
Hence, the square root of $0.1$ up to three decimal places is $0.316$ View full question & answer→Question 803 Marks
Using square root table, find the square root:
$13.21$
AnswerFrom the square root table, we have,
$\sqrt{13}=3.606$ and $\sqrt{14}=\sqrt{2}\times\sqrt{7}=3.742$
Their difference is $0.136$
Thus, for the difference of $1 (14 - 13)$, the difference in the values of the square roots is $0.136$
For the difference of $0.21$, the difference in the values of their square roots is,
$0.136 \times 0.21 = 0.02856$
$\therefore\sqrt{13.21}=3.606+0.02856\approx3.635$
View full question & answer→Question 813 Marks
Find the square root of the following by prime factorization.$586756$
AnswerResolving 586756 into prime factors, $586756 = 2 \times 2 \times 383 \times 383$
$\begin{array}{c|c}2&586756 \\ \hline 2 & 293378 \\\hline 383&146689 \\\hline383&383 \\\hline&1\end{array}$
Grouping the factors into pairs of equal factors, $586756 = (2 \times 2) \times (383 \times 383)$ Taking one factor for each pair, we get the square root of $586756 2 \times 383 = 766$
View full question & answer→Question 823 Marks
Find the square root of the following by prime factorization.$4096$
AnswerResolving 4096 into prime factors, $4096 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$
$\begin{array}{c|c}2& 4096 \\ \hline 2 & 2048 \\\hline 2&1024 \\\hline2&512 \\\hline2&256\\\hline2&128\\\hline2&64\\\hline2&32\\\hline2&16\\\hline2&8\\\hline2&4\\\hline2&2\\\hline&1\end{array}$ Grouping the factors into pairs of equal factors, $4096 = (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2)$
Taking one factor for each pair, we get the square root of $4096, (2 \times 2) \times (2 \times 2) \times (2 \times 2) = 64$
View full question & answer→Question 833 Marks
Find the squares of the following numbers using the identity $(a+b)^2=a^2-a b+a b+b^2: 52$
Answer$(a+b)^2=a^2-a b+a b+b^2$
$(52)^2=(50+2)^2$
$=2500+100+100+4$
$=2704$
View full question & answer→Question 843 Marks
Find the smallest number by which $3645$ must be divided so that it becomes a perfect square. Also, find the square root of the resulting number.
AnswerThe prime factorisation of $3645,$
$3645 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 5$
Grouping the factors into pairs of equal factors, we get,
$3645 = (3 \times 3) \times (3 \times 3) \times (3 \times 3) \times 5$
The factor, 5 does not have a pair. Therefore, we must divide $3645$ by $5$ to make a perfect square. The new number is,
$(3 \times 3) \times (3 \times 3) \times (3 \times 3) = 729$
Taking one factor from each pair on the $L.H.S$, the square root of the new number is $3 \times 3 \times 3$, which is equal to $27.$
View full question & answer→Question 853 Marks
Find the square root of the following by prime factorization.$11664$
AnswerResolving 11664 into prime factors,$11664 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3$
$\begin{array}{c|c}2& 11644 \\ \hline 2 & 5832 \\\hline 2&2916 \\\hline2&1458 \\\hline3&729\\\hline3&243\\\hline3&81\\\hline3&27\\\hline3&9\\\hline3&3\\\hline&1\end{array}$
Grouping the factors into pairs of equal factors,
$11664 = (2 \times 2) \times (2 \times 2) \times (3 \times 3) \times (3 \times 3) \times (3 \times 3)$
Taking one factor for each pair, we get the square root of $11664$
$2 \times 2 \times 3 \times 3 \times 3 = 108$
View full question & answer→Question 863 Marks
Find the smallest number by which $1152$ must be divided so that it becomes a perfect square. Also, find the square root of the number so obtained.
AnswerThe prime factorisation of $1152, 1152 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3$
Grouping the factors into pairs of equal factors, we get, $1152 = (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (3 \times 3) \times 2$
The factor, $2$, at the end, does not have a pair.
Therefore, we must divide $1152$ by $2$ to make a perfect square.
The new number is, $(2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (3 \times 3) = 576$
Taking one factor from each pair on the $LHS$, the square root of the new number is $2 \times 2 \times 2 \times 3$, which is equal to $24$
View full question & answer→Question 873 Marks
Find the square root of the following correct to three places of decimal:$15.3215$
AnswerWe can find the square root up to three decimal places by using long division until we get four decimal places and then rounding it to three decimal places.
Hence, the square root of $15.3215$ up to three decimal places is $3.914$ View full question & answer→Question 883 Marks
Find the smallest number by which $28812$ must be divided so that the quotient becomes a perfect square.
AnswerPrime factorisation of $28812 28812 = 2 \times 2 \times 3 \times 7 \times 7 \times 7 \times 7$
$\begin{array}{c|c} 2& 28812 \\ \hline 2 & 14406 \\\hline 3&7203 \\\hline 7 &2401\\\hline7&343\\\hline7&49\\\hline7&7\\\hline&1 \end{array}$
Grouping them into pairs of equal factors, $28812 = (2 \times 2) \times (7 \times 7) \times (7 \times 7) \times 3$
The factor, $3$ is not paired. The smallest number by which $28812$ must be multiplied such that the resulting number is a perfact square is $3$
View full question & answer→Question 893 Marks
Simplify: $\frac{\sqrt{0.2304}\ +\sqrt{0.1764}}{\sqrt{0.2304}\ -\sqrt{0.1764}}$
AnswerWe have, $\sqrt{0.2304}=\sqrt{\frac{2304}{10000}}$
$=\frac{\sqrt{2\times2\times2\times2\times2\times2\times3\times3}}{\sqrt{10000}}$
$\frac{2\times2\times2\times2\times3}{100}$
$=0.48$
$\sqrt{0.1764}=\sqrt{\frac{1764}{10000}}$
$=\frac{\sqrt{2\times2\times3\times3\times7\times7}}{\sqrt{10000}}$
$=\frac{2\times3\times7}{100}$
$=0.42$
$\frac{\sqrt{0.2304}\ +\sqrt{0.1764}}{\sqrt{0.2304}\ -\sqrt{0.1764}}=\frac{0.48+0.42}{0.48-0.42}$
$=\frac{0.9}{0.06}$
$=15$
View full question & answer→Question 903 Marks
Find the square root of the following by prime factorization.$24336$
AnswerResolving 24336 into prime factors, $24336 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 13 \times 13$
$\begin{array}{c|c}2& 24336 \\ \hline 2 & 12168 \\\hline 2&6084 \\\hline2&3042 \\\hline3&1521 \\\hline 3&507\\\hline13&169\\\hline13&13\\\hline&1\end{array}$
Grouping the factors into pairs of equal factors, $24336 = (2 \times 2) \times (2 \times 2) \times (3 \times 3) \times (13 \times 13)$
Taking one factor for each pair,
we get the square root of $24336 2 \times 2 \times 3 \times 13 = 156$
View full question & answer→Question 913 Marks
Find the square root of the following correct to three places of decimal:
$427$
AnswerWe can find the square root up to three decimal places by using long division until we get four decimal places and then rounding it to three decimal places.
Hence, the square root of $427$ up to three decimal places is $20.664$ View full question & answer→Question 923 Marks
A $PT$ teacher wants to arrange maximum possible number of $6000$ students in a field such that the number of rows is equal to the number of columns. Find the number of rows if $71$ were left out after arrangement.
AnswerSince $71$ students were left out, there are only $5929 (6000 - 71)$ students remaining.
Hence, the number of rows or columns is simply the square root of $5929.$ Factorising $5929$ into its prime factors, $5929 = 7 \times 7 \times 11 \times 11$
Grouping them into pairs of equal factors, $5929 = (7 \times 7) \times (11 \times 11)$
The square root of $5929 =\sqrt{5929}=7\times11=77$
Hence, in the arrangement, there were $77$ rows of students.
View full question & answer→