3 Mark Question - Page 2 - Maths STD 8 Questions - Vidyadip

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3 Mark Question

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.

Answer

Let $S$ be the number of students.
Let $r$ be the amount in rupees donated by each student. The total donation can be expressed by, $S \times r=R s .2401$ Since the total amount in rupees is equal to the number of students, $r$ is equal to S .
Substituting this in the first equation:
$S \times S=2401$
$S^2=(7 \times 7) \times(7 \times 7)$
$S=7 \times 7=4 S$
So, there are 49 students in the class.

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Question 523 Marks

Show that the following numbers is a perfect square. Also, find the number whose square is the given number in case:
$2025$

Answer

In 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 $3 \times 3 \times 5$

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Question 533 Marks

Find the square root of the following correct to three places of decimal:
$\frac{5}{12}$

Answer

We 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

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Question 543 Marks

Find the square root of the following by prime factorization.1156

Answer

Resolving 1156 into prime factors,
1156 = 2 × 2 × 17 × 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 × 2) × (17 × 17)
Taking one factor for each pair, we get the square root of 1764,
2 × 17 = 34

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Question 553 Marks

Find the squares of the following numbers using the identity $(a-b)^2=a^2-2 a b+b^2$ :
$702$

Answer

$(a+b)^2=a^2-a b+a b+b^2$
$(702)^2=(700+2)^2$
$=490000+1400+1400+4$
$=492804$

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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

$\text{a}^2$	$2\text{ab}$	$\text{b}^2$
$\ \ (3)^2\\=9\\+{4}$	$2\times3\times7\\=42\\+\ \ 4$	$\ \ \ (7)^2\\=49$
$13$	$46$

$(37)^2=37 \times 37=1369$
$(37)^2=1369$

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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.

Answer

Let 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

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Question 583 Marks

Find the square root of the following correct to three places of decimal:0.019

Answer

We 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

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Question 593 Marks

Find the square root of the following correct to three places of decimal:0.90

Answer

We 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

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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

$(54)^2$

$\text{a}^2$	$\text{2ab}$	$\text{b}^2$
$\ \ \ (5)^2\\=25\\ +\ { 4}$	$2\times5\times4\\=40\\+\ \ 1$	$\ \ \ (4)^2\\=16$
$29$	$41$

$(54)^2=54 \times 54=2961$
$\therefore(54)^2=2961$

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Question 613 Marks

Find the squares of the following numbers using diagonal method:
$273$

Answer

$(273)^2$

$\therefore\ (273)^2=74529$

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Question 623 Marks

Find the squares of the following numbers using diagonal method:
$295$

Answer

$(295)^2$

$\therefore\ (295)^2=87025$

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Question 633 Marks

Find the square root of 11 correct to five decimal places.

Answer

Using the long division method,

$\therefore\ \sqrt{11}=3.31662$

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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 $\sqrt{2}\big)$
$=64.75$

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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, $a = 2, b = 5$

$\text{a}^2$	$2\text{ab}$	$\text{b}^2$
$\ \ (2)^2\\=4\\+{2}\\ \ \ \ \ \overline{6}$	$2\times2\times5\\=20\\+\ \ 2\\ \ \ \ \ \overline{22}$	$\ \ \ (5)^2\\=25$

$(25)^2=25 \times 25=625$
$(25)^2=625$

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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

Answer

In 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 $3 \times 23$.

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Question 673 Marks

Write five numbers for which you cannot decide whether they are squares.

Answer

A 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

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Question 683 Marks

Write the prime factorization of the following numbers and hence find their square roots.
7056

Answer

The prime factorisation of 9604,
7056 = 2 × 2 × 2 × 2 × 3 × 3 × 7 × 7
Grouping them into pairs of equal factors, we get:
7056 = (2 × 2) × (2 × 2) × (3 × 3) × (7 × 7)
Taking one factor from each pair, we get,
$\sqrt{7056}=2\times2\times3\times7=84$

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Question 693 Marks

Find the square root of the following correct to three places of decimal:
$\frac{7}{8}$

Answer

We 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

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Question 703 Marks

Find the square root of the following correct to three places of decimal:0.00064

Answer

We 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

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Question 713 Marks

Find the squares of the following numbers using diagonal method:
98

Answer

$(98)^2$

$\therefore\ (98)^2=9604$

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Question 723 Marks

Write five numbers which you cannot decide whether they are square just by looking at the unit's digit.

Answer

A 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

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Question 733 Marks

Write the prime factorization of the following numbers and hence find their square roots.
9604

Answer

The prime factorisation of 9604,
9604 = 2 × 2 × 7 × 7 × 7 × 7
Grouping them into pairs of equal factors, we get:
9604 = (2 × 2) × (7 × 7) × (7 × 7)
Taking one factor from each pair, we get,
$\sqrt{9604}=2\times7\times7=98$

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Question 743 Marks

Find the square root of the following by prime factorization.7056

Answer

Resolving 7056 into prime factors,
7056 = 2 × 2 × 2 × 2 × 3 × 3 × 7 × 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 × 2) × (2 × 2) × (3 × 3) × (7 × 7)
Taking one factor for each pair, we get the square root of 705
2 × 2 × 3 × 7 = 84

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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

$\text{a}^2$	$2\text{ab}$	$\text{b}^2$
$\ \ \ (7)^2\\=49\\+\ \ {1}$	$2\times7\times1\\=14$	$\ \ \ (1)^2\\=\ 1$
$50$

$(71)^2=71 \times 71=5041$
$\therefore(71)^2=5041$

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Question 763 Marks

Find the squares of the following numbers using the identity $(a-b)^2=a^2-2 a b+b^2$ :
505

Answer

$(a+b)^2=a^2-a b+a b+b^2$
$(505)^2=(500+5)^2$
$=250000+2500+2500+25$
$=255025$

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Question 773 Marks

Using square root table, find the square root:
21.97

Answer

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 × 0.97 = 0.104
$\therefore\sqrt{21.97}=4.583+0.104\approx4.687$

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Question 783 Marks

Find the square root of the following by prime factorization.1764

Answer

Resolving 1764 into prime factors,
1764 = 2 × 2 × 3 × 3 × 7 × 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 × 2) × (3 × 3) × (7 × 7)
Taking one factor for each pair, we get the square root of 1764,
2 × 3 × 7 = 42

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Question 793 Marks

Find the square root of the following correct to three places of decimal:0.1

Answer

We 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

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Question 803 Marks

Using square root table, find the square root:
13.21

Answer

From 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 × 0.21 = 0.02856
$\therefore\sqrt{13.21}=3.606+0.02856\approx3.635$

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Question 813 Marks

Find the square root of the following by prime factorization.586756

Answer

Resolving 586756 into prime factors,
586756 = 2 × 2 × 383 × 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 × 2) × (383 × 383)
Taking one factor for each pair, we get the square root of 586756
2 × 383 = 766

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Question 823 Marks

Find the square root of the following by prime factorization.4096

Answer

Resolving 4096 into prime factors,
4096 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 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 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2)
Taking one factor for each pair, we get the square root of 4096,
(2 × 2) × (2 × 2) × (2 × 2) = 64

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Question 833 Marks

Find the squares of the following numbers using the identity $(a-b)^2=a^2-2 a b+b^2$ :
52

Answer

$(a+b)^2=a^2-a b+a b+b^2$
$(52)^2=(50+2)^2$
$=2500+100+100+4$
$=2704$

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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.

Answer

The prime factorisation of 3645,
3645 = 3 × 3 × 3 × 3 × 3 × 3 × 5
Grouping the factors into pairs of equal factors, we get,
3645 = (3 × 3) × (3 × 3) × (3 × 3) × 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 × 3) × (3 × 3) × (3 × 3) = 729
Taking one factor from each pair on the L.H.S, the square root of the new number is 3 × 3 × 3, which is equal to 27.

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Question 853 Marks

Find the square root of the following by prime factorization.11664

Answer

Resolving 11664 into prime factors,11664 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 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 × 2) × (2 × 2) × (3 × 3) × (3 × 3) × (3 × 3)
Taking one factor for each pair, we get the square root of 11664
2 × 2 × 3 × 3 × 3 = 108

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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.

Answer

The prime factorisation of 1152,
1152 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3
Grouping the factors into pairs of equal factors, we get,
1152 = (2 × 2) × (2 × 2) × (2 × 2) × (3 × 3) × 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 × 2) × (2 × 2) × (2 × 2) × (3 × 3) = 576
Taking one factor from each pair on the LHS, the square root of the new number is 2 × 2 × 2 × 3, which is equal to 24

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Question 873 Marks

Find the square root of the following correct to three places of decimal:15.3215

Answer

We 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

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Question 883 Marks

Find the smallest number by which 28812 must be divided so that the quotient becomes a perfect square.

Answer

Prime factorisation of 28812
28812 = 2 × 2 × 3 × 7 × 7 × 7 × 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 × 2) × (7 × 7) × (7 × 7) × 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

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Question 893 Marks

Simplify:
$\frac{\sqrt{0.2304}\ +\sqrt{0.1764}}{\sqrt{0.2304}\ -\sqrt{0.1764}}$

Answer

We have,
$\sqrt{0.2304}=\sqrt{\frac{2304}{10000}}$
$=\frac{\sqrt{2\times2\times2\times2\times2\times2\times3\times3}}{\sqrt{10000}}$
$\frac{2\times2\times2\times2\times3}{100}$
$=0.48$
$\sqrt{0.1764}=\sqrt{\frac{1764}{10000}}$
$=\frac{\sqrt{2\times2\times3\times3\times7\times7}}{\sqrt{10000}}$
$=\frac{2\times3\times7}{100}$
$=0.42$
$\frac{\sqrt{0.2304}\ +\sqrt{0.1764}}{\sqrt{0.2304}\ -\sqrt{0.1764}}=\frac{0.48+0.42}{0.48-0.42}$
$=\frac{0.9}{0.06}$
$=15$

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Question 903 Marks

Find the square root of the following by prime factorization.24336

Answer

Resolving 24336 into prime factors,
24336 = 2 × 2 × 2 × 2 × 3 × 3 × 13 × 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 × 2) × (2 × 2) × (3 × 3) × (13 × 13)
Taking one factor for each pair, we get the square root of 24336
2 × 2 × 3 × 13 = 156

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Question 913 Marks

Find the square root of the following correct to three places of decimal:
427

Answer

We 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

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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.

Answer

Since 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 × 7 × 11 × 11
Grouping them into pairs of equal factors,
5929 = (7 × 7) × (11 × 11)
The square root of 5929
$=\sqrt{5929}=7\times11=77$
Hence, in the arrangement, there were 77 rows of students.

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3 Mark Question - Page 2 - Maths STD 8 Questions - Vidyadip