Squares And Square Roots Test

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Squares And Square Roots Test - 3

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Question 1
1 / -0

Which of the following statements is true?

A
The square of numbers is always even.

B
There are 2n + 1 non perfect square numbers between the squares of the numbers n and (n + 1).

C
The sum of 1 + 3 + 5 + 7 + ....upto 15 terms is 225.

D
All of the above

Solution

(1) The square of numbers is always even.
This statement is false. Example: Square of 5 is 25, which is not an even number.

(2) There are 2n non perfect square numbers between the squares of the numbers n and (n + 1) and not 2n + 1.
False statement.

(3) The sum of 1 + 3 + 5 + 7 + ....upto 15 terms is 225.
This is a true statement.

The sum of 1 + 3 + 5 + 7 + ....upto n terms is n².
So, 1 + 3 + 5 + 7 + ...upto 15 terms = 15² = 225.
Question 2
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Study the statements carefully and choose the correct option.

Statement 1: If a natural number can be expressed as a sum of successive odd natural numbers starting with 1, then it is a perfect square.
Statement 2: Each and every square number has even number of factors.

A
Only statement 1 is true

B
Only statement 2 is true

C
Both statements are true

D
None of these is true

Solution

(Statement 1):
If a natural number can be expressed as a sum of successive odd natural numbers starting with 1, then it is a perfect square.
For example:

1 + 3 + 5 = 3²
1 + 3 + 5 + 7 = 4²
1 + 3 + 5 + 7 + 9 = 5²
So, statement (1) is true.

(Statement 2): Each and every square number has even number of factors.
For example:

2² has 3 factors(1,2 and 4)
3² has 3 factors(1,3 and 9)
2² × 5² has 9 factors(1,2 4,5 ,10, 20, 25, 50 and 100)

So, statement (2) is false.
Question 3
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State T for true or F for false.

I. The square root of 0.4 is 0.2.
II. There are 313 natural numbers between 156² - 157².
III. Negative numbers have positive square numbers.
IV. Sum of first n odd natural numbers is n².

A
I - F, II - F, III - T, IV - T

B
I - T, II - F, III - T, IV - T

C
I - F, II - T, III - T, IV - T

D
I - T, II - T, III - T, IV - F

Solution

I. The square root of 0.4 is 0.632.
So, statement (I) is false.

II. There are 312 natural numbers between 156² - 157²
So, statement (II) is false.

III. (-3)² = 9 is a positive number.
So, statement (III) is true.

IV. Sum of first n odd natural numbers is n².
So, statement (IV) is true.
Question 4
1 / -0

Which among the following can definitely not be a square number?

A
16384

B
13689

C
28561

D
48542

Solution

Square numbers always end with digits 0, 1, 4, 5, 6, 9.

So, from looking at the units digits in the given options, we can say that only 48542 can definitely not be a square number.