Squares And Square Roots Test

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Squares And Square Roots Test - 5

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Question 1
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In a school assembly, the students are arranged in rows to form a perfect square; but in doing so, 25 students are left out. If the total number students is 1181, find the number of students in each row.

A
30

B
34

C
36

D
38

Solution

Total students who form a square:
1181 - 25 = 1156
Let the number of rows be x.
So, x × x = 1156
x² = 1156
x = 34
Number of students in each row = 34
Question 2
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Which of the following statements is/are true?

A
A number having 2, 3, 7, or 8 at the ones place is never a perfect square; in other words, no square number ends in 2, 3, 7, or 8.

B
The square of a natural number other than 1 is either a multiple of 4 or exceeds a multiple of 4 by 1 .

C
Squares of even numbers are always even numbers and square of odd numbers are always odd.

D
All of these

Solution

(1) A number having 2, 3, 7, or 8 at the ones place is never a perfect square; in other words, no square number ends in 2, 3, 7, or 8.

Example: None of the numbers 152, 7693, 14357, 88888 and 798328 is a perfect square because the unit digit of each number is 2, 3, 7, or 8.

(2) The square of a natural number other than 1 is either a multiple of 4 or exceeds a multiple of 4 by 1.

Example: 67, 146 and 10003 are not perfect squares as they leave respectively the remainders of 3, 2 and 3 when divided by 4.

(3) Squares of even numbers are always even numbers and square of odd numbers are always odd.

Example: 12²= 12 × 12 = 144 (both are even numbers)
19²= 19 × 19 = 361 (both are odd numbers)
Question 3
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Consider the following statements and mark the correct option accordingly.

Statement I: The unit digit of the square of a natural number is the unit digit of the square of the digit at the ones place of the given natural number.
Statement II: The square of a number n is equal to the sum of first n odd natural numbers.

A
Only statement II is true.

B
Both statement are false.

C
Both statements are true.

D
Only statement I is true.

Solution

Statement I: The unit digit of the square of a natural number is the unit digit of the square of the digit at the ones place of the given natural number.

For example, let's find out the unit digit of the square of (i) 146 and (ii) 321.

Solution:
(i) Since the unit digit of 6²= 36 is 6, the unit digit of the square of 146 is 6.
(ii) Since the unit digit of 1²= 1 is 1, the unit digit of the square of 321 is 1.

Statement II: The square of a number n is equal to the sum of first n odd natural numbers.

1²= 1
2²= 1 + 3

3³ = 1 + 3 + 5
4² = 1 + 3 + 5 + 7, and so on.

So, both statements are true.
Question 4
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State true (T) or false (F) for the following statements.

i) If a number has 3 or 7 at the ones place, then its square ends in 9.
ii) 33.5² lies between 500 and 1000.
iii) If a number has 5 at the ones place, then its square ends in 5.
iv) The area of a square, whose side is 8 cm, is 64 cm².

A
i) - T, ii) - F, iii) - T, iv) - T

B
i) - T, ii) - F, iii) - T, iv) - F

C
i) - T, ii) - F, iii) - F, iv) - T

D
i) - F, ii) - F, iii) - T, iv) - T

Solution

i) If a number has 3 or 7 at the ones place, then its square ends in 9.
E.g. 33² = 1089, 17²= 289, 13² = 169

So, statement (i) is true.

ii) 33.5²= 1122.25; it does not lie between 500 and 1000. So, statement (ii) is false.

iii) If a number has 5 in the ones place, then its square ends in 5.
E.g. 5²=25, 15²=225, 25²= 625

So, statement (iii) is true.

iv) The area of a square, whose side is 8 cm, is 64 cm².
8 × 8 = 64 cm²

So, statement (iv) is true.