PROPERTIES OF PERFECT SQUARE

Problem 1 :

The difference between the squares of two consecutive number is equal to their

(a) difference    (b) sum     (c) product    (d) quotient

Solution :

Let us take tow consecutive numbers, say 3 and 4.

3² = 9 and 4² = 16

Difference = 16 - 9 

= 7

Sum of two consecutive numbers = 3 + 4

= 7

So, the answer is option b.

Problem 2 :

What will be the digit in the thousands place of (1111)² ?

(i) 3    (ii) 4    (iii) 2    (iv) 1

Solution :

1 x 1 = 1

11 x 11 = 121

111 x 111 = 12321

Then the product of

1111 x 1111 = 1234321

So, the value in the thousands place is 4.

Problem 3 :

Perfect squares cannot have 2, 3, __ and __ in its ones place.

(i) 1, 7     (ii) 5, 6     (iii) 7, 8     (iv) 7, 9

Solution :

Considering some of the perfect squares, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

The unit places are 1, 4, 5, 6, 9. So, perfect squares cannot have 2, 3, 7 and 8 in its ones place.

Problem 4 :

The smallest number by which 72 must be divided to make it a perfect square is—

(i) 4    (ii) 5    (iii) 3    (iv) 2

Solution :

72 is not a perfect square, to express it as a perfect square we can write it as 2 x 36.

So, 72 must be divided by 2 to make it as perfect square.

Problem 5 :

The square root of 3.052009 has ____ decimal places.

(i) 3    (ii) 4    (iii) 5     (iv) 1

Solution :

√1.44 = 1.2

Since we have 2 digits after the decimal, we will have one digit after the decimal in the square root.

In √3.052009, since we have 6 digits after the decimal in the result we must have 3 digits.

Problem 6 :

How many non–square numbers are there between 13² and 14² ?

Solution :

13² = 169 and 14² = 196

170 to 179 = 10

180 to 189 = 10

190 to 195 = 6

Total of 10 + 10 + 7, which is 27 numbers.

Problem 7 :

Write the first four triangular numbers.

Solution :

There are 13 triangular numbers in between 1 to 100.

1, 3, 6, 10, 15,....

Problem 8 :

Is 5, 7, 9 a Pythagorean triplets? Why? Justify.

Solution :

Every Pythagorean triples will be in the form of 2m, m²–1 and m² + 1

5² = 25, 7² = 49, 9² = 81

81 ≠ 25 + 49

81 ≠ 74

So, the given numbers is not Pythagorean triplets.

Problem 9 :

Find √9 by repeated subtraction method.

Solution :

Keep subtracting consecutive odd numbers

9 - 1 = 8

8 - 3 = 5

5 - 5 = 0

By subtracting 3 consecutive odd numbers, we get 0. So, the value of √9 is 3.

Problem 10 :

Find the measure of the side of a square handkerchief of area 324 cm²

Solution :

Area of square = 324 cm²

a² = 324

a² = 18²

a = 18

Problem 11 :

Which of the following is the square of an odd number?

(a) 256     (b) 361     (c) 144     (d) 400

Solution : 

By observing,

2² = 4, 4² = 16

5² = 25 and 7² = 49

Square of even numbers are even, square of odd numbers is odd. By observing the options given 361 is the odd number, then it must be the square of odd number.

Problem 12 :

Which of the following will have 1 at its units place?

(a) 19²     (b) 17²    (c) 18²    (d) 16²

Solution :

• Unit place 9 multiplied by itself is 1.
• Unit place 7 multiplied by itself is 9.
• Unit place 8 multiplied by itself is 4.
• Unit place 6 multiplied by itself is 6.

So, option a is correct.

Problem 13 :

How many natural numbers lie between 18² and 19²?

(a) 30     (b) 37     (c) 35    (d) 36

Solution :

18 and 19 are consecutive numbers.

Here n = 18 and n + 1 =19. So, there must be 2n natural numbers in between them. That is 36 natural numbers are there.