Exercise 3A
Question 1
Find the following squares:
(a) 122
Sol :
=12×12=144
(b) 92
Sol :
=9×9=81
(c) 282
Sol :
=28×28=784
(d) 392
Sol :
=39×39=1521
(e) 2152
Sol :
=215×215=46,225
Question 2
Factories and find the which of the following numbers are not perfect squares.
(a) 784
Sol :
=2×2×2×2×7×7 (perfect square)
(b) 1296
Sol :
=2×2×2×2×9×9 (perfect square)
(c) 7500
Sol :
=5×5×5×5×5×5×3 (not perfect square)
(d) 5184
Sol :
=2×2×2×2×2×2×9×9 (perfect square)
(e) 980
Sol :
=2×2×7×7×5 (not perfect square)
(f) 4050
Sol :
=2×5×5×9×9 (not perfect square)
Question 3
Find the smallest number by which each of the given numbers must be multiplied so that the product is a perfect square.
(a) 240
Sol :
=2×2×2×2×[3×5]
∴multiplied by 15
(b) 432
Sol :
=2×2×2×2×3×3×[3]
∴multiplied by 3
(c) 2592
Sol :
=2×2×2×2×[2]×9×9
∴multiplied by 2
(d) 18000
Sol :
=2×2×2×2×3×3×5×5×[5]
∴multiplied by 5
(e) 21952
Sol :
=2×2×2×2×2×2×7×7×[7]
∴multiplied by 7
Question 4
Find the smallest number by which each of the following numbers should be divided so that question may be perfect square.
(a) 98
Sol :
=[2]×7×7
∴divided by 2
(b) 363
Sol :
=[3]×11×11
∴divided by 3
(c) 700
Sol :
=[7]×2×2×5×5
∴divided by 7
(d) 4400
Sol :
=[11]×2×2×2×2×5×5
∴divided by 11
(e) 4374
Sol :
=[2×3]×3×3×9×9
∴divided by 6
Question 5
Just by looking at the following numbers, decide which of them
(i) may or may not be perfect squares
(ii) cannot be perfect squares. Give reasons.
[Note: If Unit digit is 2,3,7,8 or number of zeros is odd then its not a perfect square]
(a) 537
Sol :
=3×179 (not perfect square)
Reasons: pairs not found
(b) 1042
Sol :
=2×521 (not perfect square)
Reasons: pairs not found
(c) 800
Sol :
=2×2×2×2×[2]×5×5 (not perfect square)
Reasons: pairs not found
(d) 384
Sol :
=2×2×2×2×2×2×[2×3] (not perfect square)
Reasons: pairs not found
(e) 625
Sol :
=25×25 (perfect square)
Reasons: pair is found
(f) 6398
Sol :
=2×7×457 (not perfect square)
Reasons: pair not found
(g) 33493
Sol :
=3 is unit digit (not perfect square)
Reasons: pair not found
(h) 960
Sol :
=2×2×2×2×2×2×[3×5] (not perfect square)
Reasons: pair not found
(i) 72000
Sol :
=2×2×2×2×2×2×3×3×5×5×[5] (not perfect square)
Reasons: pair not found (Number of zeros are odd)
(j) 1571
Sol :
1571 is a prime number.(not perfect square)
Reasons: pair not found
Question 6
Which of the following numbers would end with digit 1?
(a) 6092
(b) 3272
(c) 3252
(d) 3412
(e) 5462
Sol :
There is no need to find square of whole number , just find square of their unit digit .
(a) 92=81 and (d) 12=1
So, as you can see (a) and (d) end with unit digit 1.
Question 7
Which of the following numbers would have digit 6 at units place?
(a) 732
(b) 3242
(c) 2762
(d) 7322
(e) 2942
Sol :
There is no need to find square of whole number , just find square of their unit digit .
here,(b) 42=16 , (c) 62=36 and (e) 42=16
So,(b),(c) and (e) end with unit digit 6.
Question 8
What will be the units digit in the squares of the following numbers?
(a) 642
Sol : =42=16
unit digit=6
(b) 932
Sol : =32=9
unit digit=9
(c) 2062
Sol : =62=36
unit digit=6
(d) 1352
Sol : =52=25
unit digit=5
(e) 4992
Sol : =92=81
unit digit=1
(f) 2382
Sol :
Sol : =82=64
unit digit=4
(g) 6072
Sol : =72=49
unit digit=9
(h) 6522
Sol : =22=4
unit digit=4
(i) 6502
Sol : =02=0
unit digit=0
(j) 9712
Sol :
Sol : =12=1
unit digit=1
Question 9
The square of which of the following numbers would be an odd number/ even number? Why?
(a) 517
Sol : 267289➝odd
(b) 234
Sol : 54756➝even
(c) 300
Sol : 90000➝even
(d) 718
Sol : 515524➝even
(e) 945
Sol : 893025➝odd
(f) 719
Sol : 516961➝odd
Question 10
What will be the following number of zeros in the square of the numbers?
(a) 40
Sol : 1600
Number of zeros=(2)
(b) 400
Sol : 160000
Number of zeros=(4)
(c) 8000
Sol : 64000000
Number of zeros=(6)
(d) 60,000
Sol : 3600000000
Number of zeros=(8)
Question 11
How many natural numbers lie between
(a) 62 and 72
Sol : 6×2=12
(b) 192 and 202
Sol : 19×2=38
(c) 492 and 502
Sol : 49×2=98
(d) 752 and 762
Sol : 75×2=150
Question 12
How many non-square numbers lie between the following pairs of numbers?
(i) 1002 and 1012
Sol : 100×2=200
(ii) 2152 and 2162
Sol : 215×2=430
(iii) 5002 and 5012
Sol : 500×2=1000
Question 13
Express the following as the sum of two consecutive natural numbers.
(i) 232
Sol : $=\frac{23^2-1}{2}+\frac{23^2+1}{2}$
=264+265
(ii) 152
Sol : $=\frac{15^2-1}{2}+\frac{15^2+1}{2}$
=112+113
(iii) 192
Sol : $=\frac{19^2-1}{2}+\frac{19^2+1}{2}$
=180+181
(iv) 252
Sol : $=\frac{25^2-1}{2}+\frac{25^2+1}{2}$
=321+313
(v) 172
Sol : $=\frac{17^2-1}{2}+\frac{17^2+1}{2}$
=144+145
Question 14
Using property of square numbers, find
(i) 242 – 232
Sol : 47
(ii) 592 – 582
Sol : 117
(iii) 752 – 742
Sol : 149
(iv) 1022 – 1012
Sol : 203
Question 15
Using adding, find the sum of the following sets of odd numbers.
(a) 1 + 3 + 5 + 7 + 9
Sol : 52=25
(b) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19
Sol : 102=100
(c) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23
Sol : 122=144
Question 16
Express:
(a) 169 as the sum of 13 odd numbers.
Sol :
=1+3+5+7+9+11+13+15+17+19+21+23+25
(b) 64 as the sum of 8 odd numbers.
Sol :
=1+3+5+7+9+11+13+15
Question 17
Which of the following sets of three numbers form a Pythagorean triples?
(a) (6, 8, 10)
Sol :
2m⇒$m=\frac{6}{2}$=3
m2-1=32-1=4-1=8
m2+1=32+1=9+1=10
∴ 6,8 and 10 is in Pythagorean triples
(b) (14, 18, 50)
Sol :
2m⇒$m=\frac{14}{2}$=7
m2-1=72-1=49-1=48
m2+1=72+1=49+1=50
∴ It is not Pythagorean triples
(c) (7, 9 ,12)
Sol :
2m⇒$m=\frac{7}{2}$
m2-1
m2+1
∴ It is not Pythagorean triples
(d) (16, 63, 65)
Sol :
2m⇒$m=\frac{16}{2}$=8
m2-1=82-1=64-1=63
m2+1=82+1=64+1=65
∴ It is in Pythagorean triples
(e) (12 ,25 ,37)
Sol :
2m⇒$m=\frac{12}{2}$=6
m2-1=62-1=36-1=35
m2+1=62+1=36+1=37
∴ It is not Pythagorean triples
Question 18
Observe the following pattern and supply the missing digits
112 = 121
1012 = 10201
10012 = 1002001
100012 = 100020001
1000012 = 1___2___1
100000012 = 1___2___1
Sol :
112=121
1012=10201
10012=1002001
100012=100020001
1000012=10000200001
100000012=100000020000001
Question 19
Using the given pattern find the missing number
12 + 22 + 22 = 32
22 + 32 + 62 = 72
32 + 42 + 122 = 132
42 + 52 + ____2 = 212
52 + ____2 + 302 = 312
62 + 72 + ______2 = _____2
Sol :
12 + 22 + 32 = 32
22 + 32 + 62 = 72
32 + 42 + 122 = 132
42+52+202=212
52+62+302=312
62+72+422=432
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