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Page No 42:

Question 1:

Using the prime factorisation method, find which of the following numbers are perfect squares:
(i) 441
(ii) 576
(iii) 11025
(iv) 1176
(v) 5625
(vi) 9075
(vii) 4225
(viii) 1089

Answer:

A perfect square can always be expressed as a product of equal factors.

(i)
Resolving into prime factors:
 441=49×9=7×7×3×3=7×3×7×3=21×21=(21)2

Thus, 441 is a perfect square.

(ii)
Resolving into prime factors:
 576=64×9=8×8×3×3=2×2×2×2×2×2×3×3=24×24=(24)2

Thus, 576 is a perfect square.

(iii)
Resolving into prime factors: 
11025=441×25=49×9×5×5=7×7×3×3×5×5=7×5×3×7×5×3=105×105=(105)2 

Thus, 11025 is a perfect square.

(iv)
Resolving into prime factors:
1176=7×168=7×21×8=7×7×3×2×2×2

1176 cannot be expressed as a product of two equal numbers. Thus, 1176 is not a perfect square.

(v)
Resolving into prime factors:
 5625=225×25=9×25×25=3×3×5×5×5×5=3×5×5×3×5×5=75×75=(75)2

 Thus, 5625 is a perfect square.

(vi)
Resolving into prime factors:
 9075=25×363=5×5×3×11×11=55×55×3

9075 is not a product of two equal numbers. Thus, 9075 is not a perfect square.

(vii)
Resolving into prime factors:
 4225=25×169=5×5×13×13=5×13×5×13=65×65=(65)2 

Thus, 4225 is a perfect square.

(viii)
Resolving into prime factors: 
1089=9×121=3×3×11×11=3×11×3×11=33×33=(33)2 

Thus, 1089 is a perfect square.

Page No 42:

Question 2:

Show that each of the following numbers is a perfect square. In each case, find the number whose square is the given number:
(i) 1225
(ii) 2601
(iii) 5929
(iv) 7056
(v) 8281

Answer:

A perfect square is a product of two perfectly equal numbers.

(i)
Resolving into prime factors: 

1225=25×49=5×5×7×7=5×7×5×7=35×35=(35)2

Thus, 1225 is the perfect square of 35.

(ii)
Resolving into prime factors:
 2601=9×289=3×3×17×17=3×17×3×17=51×51=(51)2

Thus, 2601 is the perfect square of 51.

(iii)
Resolving into prime factors: 

5929=11×539=11×7×77=11×7×11×7=77×77=(77)2

Thus, 5929 is the perfect square of 77.

(iv)
Resolving into prime factors:
7056=12×588=12×7×84=12×7×12×7=(12×7)2=(84)2

Thus, 7056 is the perfect square of 84.

(v)
Resolving into prime factors:

 8281=49×169=7×7×13×13=7×13×7×13=(7×13)2=(91)2

Thus, 8281 is the perfect square of 91.

Page No 42:

Question 3:

By what least number should the given number be multiplied to get a perfect square number? In each case, find the number whose square is the new number.
(i) 3975
(ii) 2156
(iii) 3332
(iv) 2925
(v) 9075
(vi) 7623
(vii) 3380
(viii) 2475

Answer:

1. Resolving 3675 into prime factors:
 3675=3×5×5×7×7 

Thus, to get a perfect square, the given number should be multiplied by 3.

New number= (32×52×72)=(3×5×7)2=(105)2

Hence, the new number is the square of 105.

2. Resolving 2156 into prime factors:
2156=2×2×7×7×11=(22×72×11)

Thus to get a perfect square, the given number should be multiplied by 11.

New number =(22×72×112)=(2×7×11)2=(154)2

Hence, the new number is the square of 154.

3. Resolving 3332 into prime factors:
3332=2×2×7×7×17=22×72×17

Thus, to get a perfect square, the given number should be multiplied by 17.

New number =(22×72×172)=(2×7×17)2=(238)2

Hence, the new number is the square of 238.

4. Resolving 2925 into prime factors:
2925=3×3×5×5×13=32×52×13

Thus, to get a perfect square, the given number should be multiplied by 13.

New number =(32×52×132)=(3×5×13)2=(195)2

Hence, the number whose square is the new number is 195.

5. Resolving 9075 into prime factors:

 ​9075=3×5×5×11×11=3×52×112

Thus, to get a perfect square, the given number should be multiplied by 3.

New number =(32×52×112)=(3×5×11)2=(165)2

Hence, the new number is the square of 165.

6. Resolving 7623 into prime factors:
 ​7623=3×3×7×11×11=32×7×112

Thus, to get a perfect square, the given number should be multiplied by 7.

New number =(32×72×112)=(3×7×11)2=(231)2

Hence, the number whose square is the new number is 231.

7. Resolving 3380 into prime factors:
 ​3380=2×2×5×13×13=22×5×132

Thus, to get a perfect square, the given number should be multiplied by 5.

New number =(22×52×132)=(2×5×13)2=(130)2

Hence, the new number is the square of 130.

8. Resolving 2475 into prime factors:
 ​2475=3×3×5×5×11=32×52×11

Thus, to get a perfect square, the given number should be multiplied by 11.

New number =(32×52×112)=(3×5×11)2=(165)2

Hence, the new number is the square of 165.

Page No 42:

Question 4:

By what least number should the given number be divided to get a perfect square number? In each case, find the number whose square is the new number.
(i) 1575
(ii) 9075
(iii) 4851
(iv) 3380
(v) 4500
(vi) 7776
(vii) 8820
(viii) 4056

Answer:

(i) Resolving 1575 into prime factors:
 1575=3×3×5×5×7=32×52×7

Thus, to get a perfect square, the given number should be divided by 7

New number obtained=(32×52)=(3×5)2=(15)2

Hence, the new number is the square of 15

(ii) Resolving 9075 into prime factors: 

9075=3×5×5×11×11=3×52×112

Thus, to get a perfect square, the given number should be divided by 3

New number obtained=(52×112)=(5×11)2=(55)2

Hence, the new number is the square of  55

(iii) Resolving 4851 into prime factors:

 4851=3×3×7×7×11=32×72×11

Thus, to get a perfect square, the given number should be divided by 11

New number obtained=(32×72)=(3×7)2=(21)2

Hence, the new number is the square of 21

(iv) Resolving 3380 into prime factors: 

3380=2×2×5×13×13=22×5×132

Thus, to get a perfect square, the given number should be divided by 5

New number obtained=(22×132)=(2×13)2=(26)2

Hence, the new number is the square of 26

(v) Resolving 4500 into prime factors: 

4500=2×2×3×3×5×5×5=22×32×52×5

Thus, to get a perfect square, the given number should be divided by 5

New number obtained=(22×32×52)=(2×3×5)2=(30)2

Hence, the new number is the square of 30

(vi) Resolving 7776 into prime factors:

 7776=2×2×2×2×2×3×3×3×3×3=22×22×2×32×32×3

Thus, to get a perfect square, the given number should be divided by 6 whish is a product of 2 and 3

New number obtained=(22×22×32×32)=(2×2×3×3)2=(36)2

Hence, the new number is the square of 36

(vii) Resolving 8820 into prime factors:

 8820=2×2×3×3×5×7×7=22×32×5×72

Thus, to get a perfect square, the given number should be divided by 5

New number obtained=(22×32×72)=(2×3×7)2=(42)2

Hence, the new number is the square of 42

(viii) Resolving 4056 into prime factors: 

4056=2×2×2×3×13×13=22×2×3×132

Thus, to get a perfect square, the given number should be divided by 6, which is a product of 2 and 3

New number obtained=(22×132)=(2×13)2=(26)2

Hence, the new number is the square of 26

Page No 42:

Question 5:

Find the largest number of 2 digits which is a perfect square.

Answer:

The first three digit number (100) is a perfect square. Its square root is 10.
The number before 10 is 9.
Square of 9 = (9)2=81
Thus, the largest 2 digit number that is a perfect square is 81.

Page No 42:

Question 6:

Find the largest number of 3 digits which is a perfect square.

Answer:

The largest 3 digit number is 999.

The number whose square is 999 is 31.61.

Thus, the square of any number greater than 31.61 will be a 4 digit number.
Therefore, the square of 31 will be the greatest 3 digit perfect square.

312=31×31=961



Page No 45:

Question 1:

Give reason to show that none of the numbers given below is a perfect square:
(i) 5372
(ii) 5963
(iii) 8457
(iv) 9468
(v) 360
(vi) 64000
(viii) 2500000

Answer:

By observing the properties of square numbers, we can determine whether a given number is a square or not.

(i) 5372
A number that ends with 2 is not a perfect square.
Thus, the given number is not a perfect square.

(ii) 5963
A number that ends with 3 is not a perfect square.
Thus, the given number is not a perfect square.

(iii) 8457
A number that ends with 7 is not a perfect square.
Thus, the given number is not a perfect square.

(iv) 9468
A number ending with 8 is not a perfect square.
Thus, the given number is not a perfect square.

(v) 360
Any number ending with an odd number of zeroes is not a perfect square.
Hence, the given number is not a perfect square.

(vi) 64000
Any number ending with an odd number of zeroes is not a perfect square.
Hence, the given number is not a perfect square.

(vii) 2500000
Any number ending with an odd number of zeroes is not a perfect square.
Hence, the given number is not a perfect square.

Page No 45:

Question 2:

Which of the following are squares of even numbers?
(i) 196
(ii) 441
(iii) 900
(iv) 625
(v) 324

Answer:

The square of an even number is always even.
Thus, even numbers in the given list of squares will be squares of even numbers.

(i) 196
This is an even number. Thus, it must be a square of an even number.

(ii) 441
This is an odd number. Thus, it is not a square of an even number.

(iii) 900
This is an even number. Thus, it must be a square of an even number.

(iv) 625
This is an odd number. Thus, it is not a square of an even number.

(v) 324
This is an even number. Thus, it is a square of an even number.



Page No 46:

Question 3:

Which of the following are squares of odd numbers?
(i) 484
(ii) 961
(iii) 7396
(iv) 8649
(v) 4225

Answer:

According to the property of squares, the square of an odd number is also an odd number.
Using this property, we will determine which of the numbers in the given list of squares is a square of an odd number.

(i) 484.
This is an even number. Thus, it is not a square of an odd number.

(ii) 961
This is an odd number. Thus, it is a square of an odd number.


(iii) 7396
This is an even number. Thus, it is not a square of an odd number.

(iv) 8649
This is an odd number. Thus, it is a square of an odd number.

(v) 4225
This is an odd number. Thus, it is a square of an odd number.

Page No 46:

Question 4:

Without adding, find the sum:
(i) (1 + 3 + 5 + 7 + 9 + 11 + 13)
(ii) (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19)
(iii) (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23)

Answer:

Sum of first n odd numbers = n2

(i) (1+3+5+7+9+11+13) = 72=49

(ii) (1+3+5+7+9+11+13+15+17+19)=102=100

(iii) (1+3+5+7+9+11+13+15+17+19+21+23) = 122 = 144

Page No 46:

Question 5:

(i) Express 81 as the sum of 9 odd numbers.
(ii) Express 100 as the sum of 10 odd numbers.

Answer:

Sum of first n odd natural numbers = n2

(i) Expressing 81 as a sum of 9 odd numbers:
81=92n=981=1+3+5+7+9+11+13+15+17

(ii) Expressing 100 as a sum of 10 odd numbers:
100=102n=10100=1+3+5+7+9+11+13+15+17+19

Page No 46:

Question 6:

Write a pythagorean triplet whose smallest member is
(i) 6
(ii) 14
(iii) 16
(iv) 20

Answer:

For every number m > 1, the Pythagorean triplet is 2m, m2-1, m2+1.

Using the above result:

(i)
 2m=6m=3, m2=9m2-1=9-1=8m2+1=9+1=10

Thus, the Pythagorean triplet is 6,8,10.

(ii)
2m=14m=7, m2=49m2-1=49-1=48m2+1=49+1=50

Thus, the Pythagorean triplet is 14,48,50.

(iii)
2m=16m=8, m2=64m2-1=64-1=63m2+1=64+1=65

Thus, the Pythagorean triplet is: 16,63,65

(iv)
2m=20m=10, m2=100m2-1=100-1=99m2+1=100+1=101

Thus, the Pythagorean triplet is 20,99,101.

Page No 46:

Question 7:

Evaluate:
(i) (38)2 − (37)2
(ii) (75)2 − (74)2
(iii) (92)2 − (91)2
(iv) (105)2 − (104)2
(v) (141)2 − (140)2
(vi) (218)2 − (217)2

Answer:

Given: n+12-n2 = n+1+n

(i) 382-372=38+37=75

(ii) 752-742=75+74=149

(iii) 922-912=92+91=183

(iv) 1052-1042=105+104=209

(v) 1412-1402=141+140=281

(vi) 2182-2172=218+217=435

Page No 46:

Question 8:

Using the formula (a + b)2 = (a2 + 2ab + b2), evaluate:
(i) (310)2
(ii) (508)2
(iii) (630)2

Answer:

(i) 3102=300+102=3002+2300×10+102=90000+6000+100=96100

(ii) 5082=500+82=5002+2500×8+82=250000+8000+64=258064

(iii) 6302=600+302=6002+2600×30+302=360000+36000+900=396900

Page No 46:

Question 9:

Using the formula (ab)2 = (a2 − 2ab + b2), evaluate:
(i) (196)2
(ii) (689)2
(iii) (891)2

Answer:

(i) 1962=200-42=2002-2200×4+42=40000-1600+16=38416

(ii) 6892=700-112=7002-2700×11+112=490000-15400+121=474721

(iii) 8912=900-92=9002-2900×9+92=810000-16200+81=793881

Page No 46:

Question 10:

Evaluate:
(i) 69 × 71
(ii) 94 × 106

Answer:

(i) 69×71=70-1×70+1=702-12=4900-1=4899

(ii) 94×106=100-6×100+6=1002-62=10000-36=9964

Page No 46:

Question 11:

Evaluate:
(i) 88 × 92
(ii) 78 × 82

Answer:

(i) ​88×92=90-2×90+2=902-22=8100-4=8096

(ii) 78×82=80-2×80+2=802-22=6400-4=6396

Page No 46:

Question 12:

Fill in the blanks:
(i) The square of an even number is .........
(ii) The square of an odd number is .........
(iii) The square of a proper fraction is ......... than the given fraction.
(iv) n2 = the sum of first n ......... natural numbers.

Answer:

(i) The square of an even number is even.

(ii) The square of an odd number is odd.

(iii) The square of a proper fraction is smaller  than the given fraction.

(iv) n2=the sum of first n odd natural numbers.

Page No 46:

Question 13:

Write (T) for true and (F) for false for each of the statements given below:
(i) The number of digits in a perfect square is even.
(ii) The square of a prime number is prime.
(iii) The sum of two perfect squares is a perfect square.
(iv) The difference of two perfect squares is a perfect square.
(v) The product of two perfect squares is a perfect square.

Answer:

(i) F
     The number of digits in a square can also be odd. For example: 121

(ii) F
      A prime number is one that is not divisible by any other number, except by itself and 1. Thus, square of any number cannot be a prime number.

(iii) F
      Example: 4+9=13
4 and 9 are perfect squares of 2 and 3, respectively. Their sum (13) is not a perfect square.

(iv) F
       Example: 36-25=11
36 and 25 are perfect squares. Their difference is 11, which is not a perfect square.

(v) T



Page No 48:

Question 1:

Find the value of using the column method:
(23)2

Answer:

Using the column method:
∴  a = 2
     b = 3
 

a2 2ab b2
04 + 1= 5 12+0 = 12 9

∴ 232=529

Page No 48:

Question 2:

Find the value of using the column method:
(35)2

Answer:

Using the column method:
Here, a = 3 and b = 5
 

a2 2ab b2
09
+3
= 12
30
+2
= 32
25

∴ 352 = 1225

Page No 48:

Question 3:

Find the value of using the column method:
(52)2

Answer:

Using the column method:

Here, a = 5
        b = 2
 

a2 2ab b2
25+2
=27

20
4

∴ 522=2704

Page No 48:

Question 4:

Find the value of using the column method:
(96)2

Answer:

Using column method:

Here, a =9b = 6
 

a2 2ab b2
81+11
=92
108 +3
=111
36

∴ 962=9216



Page No 49:

Question 5:

Find the value of using the diagonal method:
(67)2

Answer:


672=4489

Page No 49:

Question 6:

Find the value of using the diagonal method:
(86)2

Answer:



862=7396

Page No 49:

Question 7:

Find the value of using the diagonal method:
(137)2

Answer:


1372=18769

Page No 49:

Question 8:

Find the value of using the diagonal method:
(256)2

Answer:


2562=65536



Page No 50:

Question 1:

Find the square root of number by using the method of prime factorisation:
225

Answer:

By prime factorisation method:

225=3×3×5×5225=3×5=15

Page No 50:

Question 2:

Find the square root of number by using the method of prime factorisation:
441

Answer:

By prime factorisation:

      441=3×3×7×7  441=3×7=21

Page No 50:

Question 3:

Find the square root of number by using the method of prime factorisation:
729

Answer:

Resolving into prime factors:

729=3×3×3×3×3×3

729=3×3×3=27

Page No 50:

Question 4:

Find the square root of number by using the method of prime factorisation:
1296

Answer:

Resolving into prime factors:

1296=2×2×2×2×3×3×3×3

1296=2×2×3×3=36

Page No 50:

Question 5:

Find the square root of number by using the method of prime factorisation:
2025

Answer:

Resolving into prime factors:

2025=3×3×3×3×5×5

2025=3×3×5=45

Page No 50:

Question 6:

Find the square root of number by using the method of prime factorisation:
4096

Answer:

Resolving into prime factors:
4096=2×2×2×2×2×2×2×2×2×2×2×2

4096=2×2×2×2×2×2=64

Page No 50:

Question 7:

Find the square root of number by using the method of prime factorisation:
7056

Answer:

Resolving into prime factors:

7056=2×2×2×2×3×3×7×7

7056=2×2×3×7=84

Page No 50:

Question 8:

Find the square root of number by using the method of prime factorisation:
8100

Answer:

Resolving into prime factors:

8100=2×2×3×3×3×3×5×5

8100=2×3×3×5=90

Page No 50:

Question 9:

Find the square root of number by using the method of prime factorisation:
9216

Answer:

Resolving into prime factors:

9216=2×2×2×2×2×2×2×2×2×2×3×3

9216=2×2×2×2×2×3=96

Page No 50:

Question 10:

Find the square root of number by using the method of prime factorisation:
11025

Answer:

Resolving into prime factors:

11025=3×3×5×5×7×7

11025=3×5×7=105

Page No 50:

Question 11:

Find the square root of number by using the method of prime factorisation:
15876

Answer:

Resolving into prime factors:

15876=2×2×3×3×3×3×7×7

15876=2×3×3×7=126

Page No 50:

Question 12:

Find the square root of number by using the method of prime factorisation:
17424

Answer:

Resolving into prime factors:

17424=2×2×2×2×3×3×11×11

17424=2×2×3×11=132

Page No 50:

Question 13:

Find the smallest number by which 252 must be multiplied to get a perfect square. Also, find the square root of the perfect square so obtained.

Answer:

Resolving into prime factors:

252=2×2×3×3×7

Thus, the given number must be multiplied by 7 to get a perfect square.

New number = 252×7=1764

1764=2×3×7=42

Page No 50:

Question 14:

Find the smallest number by which 2925 must be divided to obtain a perfect square. Also, find the square root of the perfect square so obtained.

Answer:

Resolving into prime factors:

2925=3×3×5×5×13

13 is the smallest number by which the given number must be divided to make it a perfect square.

New number = 2925÷13=225

225=3×5=15

Page No 50:

Question 15:

1225 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.

Answer:

Let the number of rows be x.
Therefore, the number of plants in each row is also x.
Total number of plants =x × x=x2=1225
x2=1225=5×5×7×7x=1225=5×7=35

Thus, the number of rows is 35 and the number of plants in each row is 35.

Page No 50:

Question 16:

The students of a class arranged a picnic. Each student contributed as many rupees as the number of students in the class. If the total contribution is Rs 1156, find the strength of the class.

Answer:

Let the number of students be x.
Hence, the amount contributed by each student is Rs x.

Total amount contributed =x×x=x2=1156

1156=2×2×17×17x=1156=2×17=34

Thus, the strength of the class is 34.

Page No 50:

Question 17:

Find the least square number which is exactly divisible by each of the numbers 6, 9, 15 and 20.

Answer:

The smallest number divisible by each of these numbers is their L.C.M.
L.C.M. of 6, 9, 15, 20 = 180

Resolving into prime factors: 
180=2×2×3×3×5
To make it a perfect square, we multiply it with 5.

Required number = 180×5=900



Page No 51:

Question 18:

Find the least square number which is exactly divisible by each of the numbers 8, 12, 15 and 20.

Answer:

The smallest number divisible by each of these numbers is their L.C.M.
L.C.M. of 8, 12, 15, 20 = 120

Resolving into prime factors: 
120=2×2×2×3×5

To make this into a perfect square, we need to multiply the number with 2×3×5=30.

Required number = 120×30=3600



Page No 54:

Question 1:

Evaluate:
576

Answer:

Using the long division method:



∴ 576 = 24

Page No 54:

Question 2:

Evaluate:
1444

Answer:

Using the long division method:




1444 = 38

Page No 54:

Question 3:

Evaluate:
4489

Answer:

Using the long division method:


4489 = 67

Page No 54:

Question 4:

Evaluate:
6241

Answer:

Using the long division method:


∴ 6241 = 79

Page No 54:

Question 5:

Evaluate:
7056

Answer:

Using the long division method:




7056 = 84

Page No 54:

Question 6:

Evaluate:
9025

Answer:

Using the long division method:



9025 = 95

Page No 54:

Question 7:

Evaluate:
11449

Answer:

Using the long division method:



∴ 11449=107

Page No 54:

Question 8:

Evaluate:
14161

Answer:

Using the long division method:

 
∴ 14161=119

Page No 54:

Question 9:

Evaluate:
10404

Answer:

Using the long division method:



∴ 10404=102

Page No 54:

Question 10:

Evaluate:
17956

Answer:

Using the long division method:


17956 = 134

Page No 54:

Question 11:

Evaluate:
19600

Answer:

Using the long division method:




∴ 19600=140

Page No 54:

Question 12:

Evaluate:
92416

Answer:

Using the long division method:



∴ 92416=304

Page No 54:

Question 13:

Find the least number which must be subtracted from 2509 to make it a perfect square.

Answer:

Using the long division method:


Therefore, the number that should be subtracted from the given number to make it a perfect square is 9.

Page No 54:

Question 14:

Find the least number which must be subtracted from 7581 to obtain a perfect square. Find this perfect square and its square root.

Answer:

Using the long division method:


Therefore, the number that should be subtracted from the given number to make it a perfect square is 12.
Perfect square = 7581-12
                      = 7569

Its square root is 87.

Page No 54:

Question 15:

Find the least number which must be added to 6203 to obtain a perfect square. Find this perfect square and its square root.

Answer:

Using the long division method:


Thus, to get a perfect square greater than the given number, we take the square of the next natural number of the quotient, i.e. 78.
792=6241
Number that should be added to the given number to make it a perfect square =6241-6203=38

The perfect square thus obtained is 6241 and its square root is 79.

Page No 54:

Question 16:

Find the least number which must be added to 8400 to obtain a perfect square. Find this perfect square and its square root.

Answer:

Using the long division method:


The next natural number that is a perfect square can be obtained by squaring the next natural number of the obtained quotient, i.e. 91.
Therefore square of (91+1) = 922=8464
Number that should be added to the given number to make it a perfect square =8464-8400=64
The perfect square thus obtained is 8464 and its square root is 92.

Page No 54:

Question 17:

Find the least number of four digits which is a perfect square. Also find the square root of the number so obtained.

Answer:

Smallest number of four digits =1000

Using the long division method:


1000 is not a perfect square.
By the long division method, the obtained square root is between 31 and 32.
Squaring the next integer (32) will give us the next perfect square.

322=1024

Thus, 1024 is the smallest four digit perfect square.

Also, 1024=32

Page No 54:

Question 18:

Find the greatest number of five digits which is a perfect square. Also find the square root of the number so obtained.

Answer:

Greatest number of five digits =99999

Using the long division method:



99999 is not a perfect square. 
According to the long division method, the obtained square root is between 316 and 317.
Squaring the smaller number, i.e. 316, will give us the perfect square that would be less than 99999.

3162=99856

99856 is the required number. 
Its square root is 316.

Page No 54:

Question 19:

The area of a square field is 60025 m2. A man cycles along its boundary at 18 km/h. In how much time will he return to the starting point?

Answer:

Area of the square field = 60025 m2
Length of each side of the square field =60025=245m
Perimeter of the field =4×245=980 m

                                   = 9801000 km
The man is cycling at a speed of 18 km/h.

                          Time = Distance travelledSpeed          = 980100018          = 9801000 × 18 hr                     =980 × 60 × 6018000 sec          =98 × 2 sec          = 196 sec           = 3 min 16 sec



Page No 56:

Question 1:

Evaluate:
1.69

Answer:

Using long division method:




1.69=1.3

Page No 56:

Question 2:

Evaluate:
33.64

Answer:

Using long division method:



∴ 33.64=5.8

Page No 56:

Question 3:

Evaluate:
156.25

Answer:

Using long division method:


 ∴ 156.25=12.5

Page No 56:

Question 4:

Evaluate:
75.69

Answer:

Using long division method:


∴ 75.69=8.7

Page No 56:

Question 5:

Evaluate:
9.8596

Answer:

Using long division method:



​∴ 9.8596=3.14

Page No 56:

Question 6:

Evaluate:
10.0489

Answer:

Using long division method:



∴  10.0489=3.17

Page No 56:

Question 7:

Evaluate:
1.0816

Answer:

Using long division method:

1.0816=1.04

Page No 56:

Question 8:

Evaluate:
0.2916

Answer:

Using long division method:



0.2916=0.54

Page No 56:

Question 9:

Evaluate 3 up to two places of decimal.

Answer:

Using long division method:




     3=1.732 3 = 1.73        (correct up to two decimal places) 
                          

Page No 56:

Question 10:

Evaluate 2.8 correct up to two places of decimal.

Answer:

Using long division method:


     2.8=1.673 2.8 =1.67    (correct up to two decimal places) 
                             

Page No 56:

Question 11:

Evaluate 0.9 correct up to two places of decimal.

Answer:

Using long division method:




    0.9=0.948 0.9 =0.95    ( correct up to two decimal places) 
                           

Page No 56:

Question 12:

Find the length of each side of a square whose area is equal to the area of a rectangle of length 13.6 metres and breadth 3.4 metres.

Answer:

Area of the rectangle =(13.6 × 3.4) =46.24 sq m
​Thus, area of the square is 46.24 sq m.

Length of each side of the square = 46.24 m

Using long division method:



46.24=6.8
 
Thus, the length of a side of the square is 6.8 metres.



Page No 58:

Question 1:

Evaluate:
1681

Answer:

1681=1681

16 = 4 and 81 = 9

∴  1681=1681=49

Page No 58:

Question 2:

Evaluate:
64225

Answer:

64225=64225

Using long division method:
64=8




225=15

 

 64225=64225=815

Page No 58:

Question 3:

Evaluate:
121256

Answer:

121256=121256

Using division method:

121=11






121256=121256=1116

Page No 58:

Question 4:

Evaluate:
625729

Answer:

625729=625729

Using long division method:


625=25



729=27

625729=625729=2527

Page No 58:

Question 5:

Evaluate:
31336

Answer:

  31336=12136=12136=11×116×6=116=1511

Page No 58:

Question 6:

Evaluate:
473324

Answer:

473324=1369324=1369324

Using long division method:
1369=37



324 = 2×2×9×9 = 2×9 = 18



473324=3718=2118

Page No 58:

Question 7:

Evaluate:
333289

Answer:

333289=900289=900289

Using long division method:


289=17

And

900 = 2×2×5×5×3×3 = 2×5×3 = 30


333289=3017=11317

Page No 58:

Question 8:

Evaluate:
80405

Answer:

We have:

80405=80405=1681=1681=49

Page No 58:

Question 9:

Evaluate:
11832023

Answer:

We have:

11832023=11832023= 169289=169289=13×1317×17=1317

Page No 58:

Question 10:

Evaluate:
98×162

Answer:

We have:

   98×162=98×162=2×7×7×2×9×9=2×7×9=126

Page No 58:

Question 1:

Tick (✓) the correct answer
Which of the following numbers is not a perfect square?
(a) 7056
(b) 3969
(c) 5478
(d) 4624

Answer:

(c) 5478

According to the properties of squares, a number ending in 2, 3, 7 or 8 is not a perfect square.

Page No 58:

Question 2:

Tick (✓) the correct answer
Which of the following numbers is not a perfect square?
(a) 1444
(b) 3136
(c) 961
(d) 2222

Answer:

(d) 2222

According to the property of squares, a number ending in 2, 3, 7 or 8 is not a perfect square.

Page No 58:

Question 3:

Tick (✓) the correct answer
Which of the following numbers is not a perfect square?
(a) 1843
(b) 3721
(c) 1024
(d) 1296

Answer:

(a) 1843
According to the property of squares, a number ending in 2, 3, 7 and 8 is not a perfect square.

Page No 58:

Question 4:

Tick (✓) the correct answer
Which of the following numbers is not a perfect square?
(a) 1156
(b) 4787
(c) 2704
(d) 3969

Answer:

(b) 4787

By the property of squares, a number ending in 2, 3,7 or 8 is not a perfect square.

Page No 58:

Question 5:

Tick (✓) the correct answer
Which of the following numbers is not a perfect square?
(a) 3600
(b) 6400
(c) 81000
(d) 2500

Answer:

(c) 81000

According to the property of squares, a number ending in odd number of zeroes is not a perfect square.

Page No 58:

Question 6:

Tick (✓) the correct answer
Which of the following cannot be the unit digit of a perfect square number?
(a) 6
(b) 1
(c) 9
(d) 8

Answer:

(d) 8

According to the property of squares, a perfect square cannot have 2, 3, 7 or 8 as the unit digit.

Page No 58:

Question 7:

Tick (✓) the correct answer
The square of a proper fraction is
(a) larger than the fraction
(b) smaller than the fraction
(c) equal to the fraction
(d) none of these

Answer:

(b) smaller than the fraction

Page No 58:

Question 8:

Tick (✓) the correct answer
If n is odd, then (1 + 3 + 5 + 7 +... to n terms) is equal to
(a) (n2 + 1)
(b) (n2 − 1)
(c) n2
(d) (2n2 + 1)

Answer:

(c) n2

Page No 58:

Question 9:

Tick (✓) the correct answer
Which of the following is a pythagorean triplet?
(a) (2, 3, 5)
(b) (5, 7, 9)
(c) (6, 9, 11)
(d) (8, 15, 17)

Answer:

(d) (8,15,17)

This can be understood from the property of Pythagorean triplets. According to this property, for a natural number m, (2m, m2-1, m2+1) is a Pythagorean triplet.
Here,  m = 4
          2m = 8
          m2 - 1=15
and    m2 + 1 = 17

Page No 58:

Question 10:

Tick (✓) the correct answer
What least number must be subtracted from 176 to make it a perfect square?
(a) 16
(b) 10
(c) 7
(d) 4

Answer:

(c) 7

(176-7)=169169=13

Page No 58:

Question 11:

Tick (✓) the correct answer
What least number must be added to 526 to make it a perfect square?
(a) 3
(b) 2
(c) 1
(d) 6

Answer:

(a) 3

526+3=529529=232



Page No 59:

Question 12:

Tick (✓) the correct answer
What least number must be added to 15370 to make it a perfect square?
(a) 4
(b) 6
(c) 8
(d) 9

Answer:

(b) 6

15370+6=1537615376=124

Page No 59:

Question 13:

Tick (✓) the correct answer
0.9=?
(a) 0.3
(b) 0.03
(c) 0.33
(d) 0.94

Answer:

(d) 0.94



0.9=0.94

Page No 59:

Question 14:

Tick (✓) the correct answer
0.1=?
(a) 0.1
(b) 0.01
(c) 0.316
(d) none of these

Answer:

(c) 0.316

Using long division method:



0.1= 0.316

Page No 59:

Question 15:

Tick (✓) the correct answer
0.9×1.6=?
(a) 0.12
(b) 1.2
(c) 0.75
(d) 12

Answer:

(b) 1.2

 0.9×1.6=1.44=1.2

Page No 59:

Question 16:

Tick (✓) the correct answer
288128=?
(a) 32
(b) 32
(c) 32
(d) 1.49

Answer:

(c) 32

 288128=288128=2×2×2×2×2×3×32×2×2×2×2×2×2=3×32×2=3×32×2=32

Page No 59:

Question 17:

Tick (✓) the correct answer
214=?
(a) 212
(b) 112
(c) 114
(d) none of these

Answer:

(b) 112

 214=94=94=32=112

Page No 59:

Question 18:

Tick (✓) the correct answer
Which of the following is the square of an even number?
(a) 196
(b) 441
(c) 625
(d) 529

Answer:

(a) 196

Square of an even number is always an even number.

Page No 59:

Question 19:

Tick (✓) the correct answer
Which of the following is the square of an odd number?
(a) 2116
(b) 3844
(c) 1369
(d) 2500

Answer:

(c) 1369

Square of an odd number is always an odd number.



Page No 62:

Question 1:

Evaluate 11236.

Answer:

Using long division method:



∴  11236=106

Page No 62:

Question 2:

Find the greatest number of five digits which is a perfect square. What is the square root of this number?

Answer:

The greatest 5 digit number is 99999.


316 < 99999 < 317

3162=99856 
Thus, this is the greatest 5 digit number.

99856=316

Page No 62:

Question 3:

Find the least number of four digits which is a perfect square. What is the square root of this number?

Answer:

The least number of 4 digits is 1000.



31< 100< 32322= 1024

1024 is the least four digit perfect square and its square root is 32.

Page No 62:

Question 4:

Evaluate 0.2809.

Answer:


0.2809=0.53

Page No 62:

Question 5:

Evaluate 3 correct up to two places of decimal.

Answer:


3=1.732Therefore, the value of 3 up to two places of decimal is 1.73.

Page No 62:

Question 6:

Evaluate 48243.

Answer:

    48243=48243=2×2×2×2×33×3×3×3×3=2×2×2×23×3×3×3=2×23×3=49

Page No 62:

Question 7:

Mark (✓) against the correct answer
Which of the following numbers is not a perfect square?
(a) 529
(b) 961
(c) 1024
(d) 1222

Answer:

(d) 1222

A number ending in 2, 3, 7 or 8 is not a perfect square.

Page No 62:

Question 8:

Mark (✓) against the correct answer
214=?
(a) 212
(b) 114
(c) 112
(d) none of these

Answer:

(c) 112


214=94=94=3×32×2=32=112

Page No 62:

Question 9:

Mark (✓) against the correct answer
Which of the following is the square of an even number?
(a) 529
(b) 961
(c) 1764
(d) 2809

Answer:

(c) 1764

The square of an even number is always even.

Page No 62:

Question 10:

Mark (✓) against the correct answer
What least number must be added to 521 to make it a perfect square?
(a) 3
(b) 4
(c) 5
(d) 8

Answer:

(d) 8

521+8=529529=23  

Page No 62:

Question 11:

Mark (✓) against the correct answer
What least number must be subtracted from 178 to make it a perfect square?
(a) 6
(b) 8
(c) 9
(d) 7

Answer:

(c) 9

178-9=169169=13

Page No 62:

Question 12:

Mark (✓) against the correct answer
72×98=?
(a) 42
(b) 84
(c) 64
(d) 74

Answer:

(b) 84

72×98=2×2×2×3×3×2×7×7=2×2×2×3×3×2×7×7=2×2×3×7=84

Page No 62:

Question 13:

Fill in the blanks.
(i) 1 + 3 + 5 + 7 + 9 + 11 + 13 = (.........)2.
(ii) 1681=.........
(iii) The smallest square number exactly divisible by 2, 4, 6 is .........
(iv) A given number is a perfect square having n digits, where n is odd. Then, its square root will have ......... digits.

Answer:

(i) 1+3+5+7+9+11+13=(7)2 


(ii)

 1681=41

(iii) The smallest square number exactly divisible by 2, 4 and 6 is 36.

LCM of 2,4 and 6 is12.Prime factorisation of 12 = 2×2×3To make it a perfect square, we need to multiply it by 3. 12×3 = 36 

(iv) A given number is a perfect square having n digits, where n is odd. then, its square root will have n+12 digits.



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