Squares and Square Roots — Multiple Choice Questions

How many natural numbers lie between 25² and 26²?

1. 49

2. 50

3. 51

4. 52

Answer

We know that there are 2n non-square numbers between the squares of two consecutive numbers n and (n + 1).

Here, n = 25

Natural numbers between 25² and 26² = 2 × 25

= 50

Hence, option 2 is the correct option.

Square of an even number is always

1. even

2. odd

3. even or odd

4. none of these

Answer

According to the property of square numbers, squares of even natural numbers are even and squares of odd natural numbers are odd.

Hence, option 1 is the correct option.

1 + 3 + 5 + 7 + ..... up to n terms is equal to

1. n² - 1

2. (n + 1)²

3. n² + 1

4. n²

Answer

As we know that, the sum of first n odd natural numbers = n².

Hence, option 4 is the correct option.

$\sqrt{208 + \sqrt{2304}}$ is equal to

1. 18

2. 16

3. 14

4. 22

Answer

Finding prime factors of 2304

$\begin{array}{l|r} 2 & 2304 \\ \hline 2 & 1152 \\ \hline 2 & 576 \\ \hline 2 & 288 \\ \hline 2 & 144 \\ \hline 2 & 72 \\ \hline 2 & 36 \\ \hline 2 & 18 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array}$

2304 = (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (3 × 3)

$\Rightarrow \sqrt{2304} = 2 \times 2 \times 2 \times 2 \times 3 = 48$

$\Rightarrow \sqrt{208 + \sqrt{2304}}\\[1em] \Rightarrow \sqrt{208 + 48}\\[1em] \Rightarrow \sqrt{256}\\[1em] \Rightarrow 16$

Hence, option 2 is the correct option.

$\sqrt{0.0016}$ is equal to

1. 0.04

2. 0.004

3. 0.4

4. none of these

Answer

Converting decimal into proper fraction:

$\Rightarrow \sqrt{0.0016}\\[1em] \Rightarrow \sqrt{\dfrac{16}{10000}}\\[1em] \Rightarrow \dfrac{\sqrt{16}}{\sqrt{10000}}\\[1em] \Rightarrow \dfrac{4}{100}\\[1em] \Rightarrow 0.04$

Hence, option 1 is the correct option.

The smallest number by which 75 should be divided to make it a perfect square is

1. 1

2. 2

3. 3

4. 4

Answer

Finding prime factors of 75

$\begin{array}{l|r} 3 & 75 \\ \hline 5 & 25 \\ \hline 5 & 5 \\ \hline & 1 \\ \end{array}$

75 = (5 × 5) × 3

Since the prime factor 3 is not in pair.

The given number should be divided by 3.

Hence, option 3 is the correct option.

$\sqrt{3\dfrac{6}{25}}$ is equal to

1. $\dfrac{5}{9}$

2. $\dfrac{4}{5}$

3. $\dfrac{9}{5}$

4. $\dfrac{5}{4}$

Answer

Converting mixed fraction into improper fraction:

$\Rightarrow \sqrt{3\dfrac{6}{25}}\\[1em] \Rightarrow \sqrt{\dfrac{81}{25}}\\[1em] \Rightarrow \dfrac{\sqrt{81}}{\sqrt{25}}\\[1em] \Rightarrow \dfrac{9}{5}$

Hence, option 3 is the correct option.

The smallest number by which 162 should be multiplied to make it a perfect square is

1. 4

2. 3

3. 2

4. 1

Answer

Finding prime factors of 162

$\begin{array}{l|r} 2 & 162 \\ \hline 3 & 81 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array}$

162 = (3 × 3) × (3 × 3) × 2

Since the prime factor 2 is not in pair.

The given number should be multiplied by 2.

Hence, option 3 is the correct option.

If the area of a square field is 961 unit², then the length of its side is

1. 29 units

2. 41 units

3. 31 units

4. 39 units

Answer

Finding prime factors of 961

$\begin{array}{l|r} 31 & 961 \\ \hline 31 & 31 \\ \hline & 1 \\ \end{array}$

Area of square field = (side)²

$\Rightarrow (\text{side})^2 = 961\\[1em] \Rightarrow \text{side} = \sqrt{961}\\[1em] \Rightarrow \text{side} = \sqrt{31 \times 31}\\[1em] \Rightarrow \text{side} = 31$

Hence, option 3 is the correct option.

The smallest number that should be subtracted from 300 to make it a perfect square is

1. 11

2. 12

3. 13

4. 14

Answer

Using division method for 300:

We get the remainder = 11.

It means (17)² is less than 300 by 11.

So, the perfect square just less than 300 is 289.

Required number = 300 - 289 = 11

Hence, option 1 is the correct option.

If one number of Pythagorean triplet is 6, then the triplet is

1. (4, 5, 6)

2. (5, 6, 7)

3. (6, 7, 8)

4. (6, 8, 10)

Answer

We know that for any natural number m > 1, (2m, m² - 1, m² + 1) is a Pythagorean triplet.

Let 2m = 6

⇒ m = 3

∴ m² - 1 = 9 - 1 = 8 and m² + 1 = 9 + 1 = 10

So, (6, 8, 10) is a Pythagorean triplet.

Hence, option 4 is the correct option.

Given that $\sqrt{1521}$ = 39, the value of $\sqrt{0.1521} + \sqrt{15.21}$ is

1. 42.9

2. 4.29

3. 3.51

4. 35.1

Answer

Solving,

$\Rightarrow \sqrt{0.1521} + \sqrt{15.21}\\[1em] \Rightarrow \sqrt{\dfrac{1521}{10000}} + \sqrt{\dfrac{1521}{100}}\\[1em] \Rightarrow \dfrac{\sqrt{1521}}{\sqrt{10000}} + \dfrac{\sqrt{1521}}{\sqrt{100}}\\[1em] \Rightarrow \dfrac{39}{100} + \dfrac{39}{10}\\[1em] \Rightarrow 0.39 + 3.9\\[1em] \Rightarrow 4.29$

Hence, option 2 is the correct option.