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1. Question 1.
Find the squares of the following numbers using column method. Verify the
result by finding the square using the usual multiplication :
(i) 25
(ii) 37
(iii) 54
(iv) 71
(v) 96
Solution:
(i) (25)2

3. Question 2.
Find the squares of the following numbers using diagonal method :

4. (i) 98
(ii) 273
(iii) 348
(iv) 295
(v) 171
Solution:

5. Question 3.
Find the squares of the following numbers :
(i) 127
(ii) 503
(iii) 451
(iv) 862
(v) 265
Solution:
(i) (127)2
= (120 + 7)2
{(a + b)2
= a2
+ lab + b2
}

6. = (120)2
+ 2 x 120 x 7 + (7)2
= 14400+ 1680 + 49 = 16129
(ii) (503)2
= (500 + 3)2
{(a + b)2
= a2
+ lab + b1
}
= (500)2
+ 2 x 500 x 3 + (3)2
= 250000 + 3000 + 9 = 353009
(iii) (451)2
= (400 + 51)2
{(a + b)2
= a2
+ lab + b2
}
= (400)2
+ 2 x 400 x 51 + (5l)2
= 160000 + 40800 + 2601 = 203401
(iv) (451)2
= (800 + 62)2
{(a + b)2
= a2
+ lab + b2
}
= (800)2
+ 2 x 800 x 62 + (62)2
= 640000 + 99200 + 3844 = 743044
(v) (265)2
{(a + b)2
= a2
+ 2ab + b2
}
(200 + 65)2
= (200)2
+ 2 x 200 x 65 + (65)2
= 40000 + 26000 + 4225 = 70225
Question 4.
Find the squares of the following numbers
(i) 425
(ii) 575
(iii) 405
(iv) 205
(v) 95
(vi) 745
(vii) 512
(viii) 995
Solution:
(i) (425)2
Here n = 42
∴ n (n + 1) = 42 (42 + 1) = 42 x 43 = 1806
∴ (425)2
= 180625
(ii) (575)2
Here n = 57
∴ n (n + 1) = 57 (57 + 1) = 57 x 58 = 3306
∴ (575)2
= 330625
(iii) (405)2
Here n = 40

7. ∴ n (n + 1) = 40 (40 + 1) -40 x 41 = 1640
∴ (405)2
= 164025
(iv) (205)2
Here n = 20
∴ n (n + 1) = 20 (20 + 1) = 20 x 21 = 420
∴ (205)2
= 42025
(v) (95)2
Here n = 9
∴ n (n + 1) = 9 (9 + 1) = 9 x 10 = 90
∴ (95)2
= 9025
(vi) (745)2
Here n = 74
∴ n (n + 1) = 74 (74 + 1) = 74 x 75 = 5550
∴ (745)2
= 555025
(vii) (512)2
Here a = 1, b = 2
∴ (5ab)2
= (250 + ab) x 1000 + (ab)2
∴ (512)2
= (250 + 12) x 1000 + (12)2
= 262 x 1000 + 144
= 262000 + 144 = 262144
(viii) (995)2
Here n = 99
∴ n (n + 1) = 99 (99 + 1) = 99 x 100 = 9900
∴ (995)2
= 990025
Question 5.
Find the squares of the following numbers using the identity (a + b)1 = a2 + lab
+ b1
(i) 405
(ii) 510
(iii) 1001
(iv) 209
(v) 605
Solution:
a + b)2
= a2
+ lab + b2
(i) (405)2
= (400 + 5)2
= (400)2
+ 2 x 400 x 5 + (5)2
= 160000 + 4000 + 25 = 164025
(ii) (510)2
= (500 + 10)2
= (500)2
+ 2 x 500 x 10 x (10)2

8. = 250000 + 10000 + 100
= 260100
(iii) (1001)2
= (1000+1)2
= (1000)2
+ 2 X 1000 x 1 + (1)
= 1000000 + 2000 + 1
=1002001
(iv) (209)2
= (200 + 9)2
= (200)2
+ 2 x 200 x 9 x (9)2
= 40000 + 3600 +81
= 43681
(v) (605)2
= (600 + 5)2
= (600)2
+ 2 x 600 x 5 +(5)2
= 360000 + 6000 25
=366025
Question 6.
Find the squares of the following numbers using the identity (a – b)2
= a2
– 2ab
+ b2
:
(i) 395
(ii) 995
(iii) 495
(iv) 498
(v) 99
(vi) 999
(vii) 599
Solution:
a – b)2
= a2
– lab + b2
(i) (395)2
= (400 – 5)2
= (400)2
– 2 x 400 x 5 + (5)2
= 160000-4000 + 25
= 160025-4000
= 156025
(ii) (995)2
= (1000 – 5)2
= (1000)2
– 2 x 1000 x 5 + (5)2
= 1000000- 10000 + 25
= 1000025- 10000
= 990025
(iii) (495)2
= (500 – 5)2
= (500)2
– 2 x 500 x 5 + (5)2
= 250000 – 5000 + 25

9. = 250025 – 5000
= 245025
(iv) (498)2
= (500 – 2)2
= (500)2
– 2 x 500 x 2 + (2)2
= 250000 – 2000 + 4
= 250004 – 2000
= 248004
(v) (99)2
= (100 – l)2
= (100)2
– 2 x 100 x 1 + (1)2
= 10000 – 200 + 1
= 10001 – 200
= 9801
(vi) (999)2
= (1000- l)2
= (1000)2
– 2 x 1000 x 1+ (1)2
= 1000000-2000+1
= 10000001-2000=998001
(vii) (599)2
= (600 – 1)2
= (600)2
-2 x 600 X 1+ (1)2
= 360000 -1200+1
= 360001 – 1200 = 358801
Question 7.
Find the squares of the following numbers by visual method :
(i) 52
(ii) 95
(iii) 505
(iv) 702
(v) 99
Solution:
(a + b)2
= a2
– ab + ab + b2
(i) (52)2
= (50 + 2)2
= 2500 + 100 + 100 + 4
= 2704
(ii) (95)2
= (90 + 5)2
= 8100 + 450 + 450 + 25
= 9025

10. (iii) (505)2
= (500 + 5)2
= 250000 + 2500 + 2500 + 25
= 255025
(iv) (702)2
= (700 + 2)2
= 490000 + 1400+ 1400 + 4
= 492804
(v) (99)2
= (90 + 9)2
= 8100 + 810 + 810 + 81
= 9801