NOTE
* Square number never ends with 2, 3, 7 or 8.
* Square number can end with 0, 1, 4, 5, 6 or 9.
* Square of even number is even and those are odd numbers is odd.
* The set of 3 numbers (x,y,z) is called Pythagorean triplet, if x²+ y²= z².
* For any natural number 'm' greater than 1, (2m, m²-1, m²+1) is a Pythagorean triplet.
* For perfect square x and y
a) √(x . y)= √x . √y
b) √(x/y = √x/√y (y≠ 0).
* The square of a proper fraction is smaller than the fraction.
* (a+ b)²= a²+ 2ab + b².
(a - b)²= a²- 2ab + b²
* For any natural number
Sum of n odd natural numbers= n²
EXERCISE - A
a) 16
b) 253
c) 431
d) 9284
e) 52698
f) 78949
g) 66866
h) 4876937
2) Which of the following number would have 6 digit at units place ?
a) 19²
b) 84²
c) 324²
d) 4368²
e) 9487²
f) 14278²
g) 84769²
h) 325846²
3) How many square numbers are there between 1 to 100 ?
4) Find the largest 3-digit number which is perfect square.
5) Find the smallest 4-digit number which is a perfect square.
6) By what least number should the following numbers be divided to get a perfect square.
a) 512
b) 648
c) 1275
d) 3380
e) 4056
f) 8820
7) By what least number should the following numbers be multiplied to get a perfect square.
a) 648
b) 2475
c) 2925
d) 4056
e) 4410
f) 4851
g) 7776
h) 9075
8) Is the following numbers a perfect square:
a) 196
b) 625
c) 7056
d) 9075
9) Show that 648 is not a perfect square?
10) By what least number should 648 be a multiplied to get a perfect square number ?
11) By what least number 3125 be divided to get a perfect square number?
12) Find the square of following:
a) 15
b) 134
c) 3/19
d) 14/5
e) 0.03
f) 1.3
g) 0.29
h) 4.5
EXERCISE - B
1) Without adding, find the sum.
a) 1+3+5+7+9+11+13.
b) 1+3+5+.......+31.
c) 1+3+5+......+71.
2) Express
a) 64 as the sum of 8 odd natural numbers.
b) 225 as the sum of 15 odd natural numbers.
3) Write a Pythagorean triplet where one member is
a) 6
b) 12
c) 16
d) 35
4) What will be the unit digit of the square of the following:
a) 272
b) 1057
c) 5554
d) 12796
e) 28245
c) 35842
5) The square of which of the following is odd.
a) 248
b) 431
c) 2826
d) 143876
6) Using the given pattern, find the missing numbers.
a) 11²= 121
101²= 10201
1001²=1002001
100001²= 1.........2..........1
1000000001²= 1.........2......1
b) 11²=121
101²= 10201
10101²= 102030201
( )²= 1020304030201
( )²= 10203040504030201
c) 9²= 81
99²= 9801
999²= 998001
9999²= 99980001
99999²= __________
999999²= _______
d) 4²+5²+20²=21²
5²+6²+30²=31²
6²+7²+___=___
7) Evaluate:
a) 38²- 37²
b) 42²- 41².
c) 218²- 217²
d) 428²- 427²
e) 767²- 766²
f) 982²- 981².
9) Without adding find the sum:
a) 1+3+5+7+......+23=?
b) 1+3+5+7+.....+29=?
c) 1+3+5+7+.....+39=?
10) Write the following numbers as the difference of squares of consecutive natural numbers.
a) 21
b) 43
c) 59.
d) 65
EXERCISE - C
1) Write the possible ones digit of the square root of each of the following numbers.
a) 6084
b) 5209
c) 6561
d) 9025
e) 17424
f) 217156
2) Find the square root of the following by prime factorization method.
a) 144
b) 252
c) 324
d) 441
e) 576
f) 676
g) 784
h) 1225
i) 1764
j) 2601
k) 2916
l) 3528
m) 3969
n) 5328
o) 5625
p) 5929
q) 6084
r) 6561
s) 7056
t) 7921
u) 8281
v) 9075
w)11025
x) 21904
y) 24336
3) Find the square root by method of repeated subtraction.
a) 81
b) 144
c) 169
4) Find the smallest number by which the following numbers be multiplied to get a perfect square. Also find this square root of the square numbers so obtained .
a) 396
b) 768
c) 588
d) 720
e) 847
f) 882
g) 1152
h) 2592
i) 2925
j) 3332
k) 3380
l) 3645
m) 4802
5) Find the smallest number by which the following numbers be divided to get a perfect square. Also find the square root of the square number so obtained.
a) 245
b) 396
c) 968
d) 1008
e) 1200
f) 1452
g) 1728
h) 4500
i) 5445
j) 6000
k) 7776
l) 8820
m) 28812
6) Find the length of the side of the square having area.
a) 441 cm²
b) 676cm²
c) 729 cm²
d) 2304 cm²
e) 3136 cm²
7) The student of class 8 of a school donoted Rs1521 to PM'srelief fund. Each student donated as many rupees as the number of students in the class. Find the number of students in the class.
8) Find the least square number divisible by each one of 8, 9 and 10.
9) Find the least perfect square which is divisible by 5, 6 and 8.
EXERCISE- D
1) Without performing any calculation, find the number of digits in square root of the following numbers :
a) 256
b) 4489
c) 6241
d) 7056
e) 15129
f) 17956
g) 18769
h) 92416
i) 193600
j) 403225
k) 999999
l) 1046529
2) Find square root of the following using division method
a) 2401
b) 3249
c) 5184
d) 9225
e) 15129
f) 18769
g) 65536
h) 893304
i) 7.3441
j) 24.01
k) 147.1369
3) Find the value of
a) √117 x √625 x √13.
b) √1.44 + √6.25 + √506.25 - √3.24.
4) Find the square root of the following character 3 places of decimal.
a) √3
b) √7
c) √13
d) √0.8
e) √0.9
f) √2.25
g) √2.8
h) √7.29
i) √15.21
5) Evaluate
a) √1_49/576
b) √1_56/169
c) √2_1/4
d) √(1/16 + 1/9).
e) 12_69/121.
f) 9_43/49.
6) What least number must be subtracted from 893304 to get a perfect square. Find the square root of this perfect square.
7) Find the greatest number of 4 digits which is a perfect square.
8) Find the smallest number which must be added to 4931 to make it a perfect square.
9) The area of a square field is 12100 m². Find its side.
10) A Gardner has 1000 plants. He wants to plant these in such a way that the number of rows and the number of columns remain same. Find the minimum number of plants he needs more for this.
11) The area of a square field 60025 m². A man cycles along its boundary at 18 kmph. In how much time will he return to the original point ?
12) An army commander wishing to arrange his soldiers, who were 8289 in number in the form of square, found that there were 8 soldiers left. How many soldiers were there in each row?
13) What is the number which when multiplied by itself gives √152.5225.
14) The area of a square is equals to the area of a rectangle whose length is 7.26m and breadth is 6m. Find each side of a square.
15) Find the smallest number that must be added to 1750 in order to make it a perfect square. Also find the square root of perfect square.
16) If √105.0625 = 10.25, find the value of √10506.25.
17) Simplify √[{√(59.29) - √(5.29)}/{59.29 + √(5.29)}].
MULTIPLE CHOICE QUESTIONS:
1) How many 2's are there in the prime factorization of 800 ?
a) 7 b) 5 c) 10 d) 15
2) Solve: 49²- 48²
a) 87 b) 97 c) 90 d) 80
3) Simplify √(63 x 28)
a) 60 b) 62 c) 42 d) 24
4) Find the √0.0081.
a) 0 b) 0.9 c) 0.09 d) 0.009
5) How many perfect square number lie between 16 and 81?
a) 4 b) 5 c) 8 d) 6
6) How many perfect square numbers lie between 100 and 500?
a) 13 b) 12 c) 10 d) 77
7) What is the sum of first ten odd numbers?
a) 100 b) 80 c) 200 d) 120
8) Find the sum of 5 + 7+ 9 +...+21.
a) 118 b) 117 c) 120 d) 119
9) Find the missing number. 40 x ____=60².
a) 90 b) 50 c) 60 d) 80
Miscellaneous Exercise - 1
1) Find the square of 24.
2) Can we have a square number that ends with 8?
3) How many 2's are there in the prime factorization of 800 ?
4) How many digits will be there in the square root of 12321.
5) What is the value of 48² - 47²?
6) Find the value of (59 + 41)²?
7) What is the missing digit in (37)²= 1∆∆9.
8) Simplify √(63 x 28 x16).
9) How many natural number lie between 63² and 64²?
10) Find the value of √0.0081.
11) What is the sum of the first twenty odd numbers?
12) Find the sum of 5 + 7+ 9 + 11 + 13+ 15+ 17+ 19+ 21.
13) What should be added to 45² to get 46²?
14) What should be substed from 37² to 35²?
15) What is the value of √(441/1369)=?
16) Give a Pythagorean triplet if one number is 12.
17) The length of a rectangular park is 80m and its breadth is 60m. Find the length of its diagonal.
18) find the length of diagonal of a square if one side is 47.
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