Question 1

1. Without performing long calculation, which of the following numbers cannot be a perfect square based on its unit digit?

A) 1024B) 5625C) 7923D) 1296

Answer

✓ Correct Answer: Option C

Explanation: A perfect square never ends with the digits 2, 3, 7, or 8. Since 7923 ends in 3, it cannot be a perfect square.

Question 2

2. A square garden has an area of $2025 m^2$. What is the length of the decorative fence required to cover one side of the garden?

A) 35 mB) 45 mC) 55 mD) 25 m

Question 3

3. In a Pythagorean triplet (6, x, y), if 6 is the smallest member, what are the values of x and y?

A) 8, 10B) 7, 9C) 10, 12D) 9, 15

Answer

✓ Correct Answer: Option A

Explanation: For an even number 2m = 6, we get m = 3. The other members are $m^2 - 1 = 3^2 - 1 = 8$ and $m^2 + 1 = 3^2 + 1 = 10$. The triplet is (6, 8, 10).

Question 4

4. Find the smallest natural number by which 180 must be multiplied so that the product becomes a perfect square.

A) 2B) 3C) 4D) 5

Answer

✓ Correct Answer: Option D

Explanation: Prime factorization of $180 = 2 \times 2 \times 3 \times 3 \times 5 = 2^2 \times 3^2 \times 5$. To make it a perfect square, all factors must be in pairs. The factor 5 is unpaired. So, we must multiply by 5.

Question 5

5. What is the sum of the first n odd natural numbers if n = 12?

A) 121B) 144C) 169D) 100

Answer

✓ Correct Answer: Option B

Explanation: The sum of the first n odd natural numbers is always $n^2$. Here n = 12, so the sum is $12^2 = 144$.

Question 6

6. The volume of a cubical box is $512 cm^3$. What is the surface area of one face of this box?

A) $8 cm^2$B) $16 cm^2$C) $64 cm^2$D) $48 cm^2$

Answer

✓ Correct Answer: Option C

Explanation: Volume $V = a^3 = 512 cm^3$. $a = \sqrt[3]{512} = 8 cm$. Area of one face $= a^2 = 8^2 = 64 cm^2$.

Question 7

7. Which of the following is the cube root of -216?

A) -6B) 6C) -16D) Does not exist

Answer

✓ Correct Answer: Option A

Explanation: Cube roots of negative numbers are negative. Since $6 \times 6 \times 6 = 216$, then $(-6) \times (-6) \times (-6) = -216$. Therefore, $\sqrt[3]{-216} = -6$.

Question 8

8. Find the smallest number by which 250 must be divided to obtain a perfect cube.

A) 5B) 10C) 4D) 2

Answer

✓ Correct Answer: Option D

Explanation: Prime factorization of $250 = 2 \times 5 \times 5 \times 5 = 2 \times 5^3$. To make it a perfect cube, we need to remove the extra factor 2. $250 \div 2 = 125$, which is $5^3$.

Question 9

9. If the digit in the units place of a number is 7, what will be the digit in the units place of its cube?

A) 7B) 3C) 9D) 1

Answer

✓ Correct Answer: Option B

Explanation: To find the unit digit of the cube, we cube the unit digit of the number: $7^3 = 7 \times 7 \times 7 = 343$. The unit digit is 3.

Question 10

10. The value of $\sqrt[3]{0.027}$ is:

A) 0.03B) 3.0C) 0.3D) 0.003

Answer

✓ Correct Answer: Option C

Explanation: $\sqrt[3]{0.027} = \sqrt[3]{\frac{27}{1000}} = \frac{\sqrt[3]{27}}{\sqrt[3]{1000}} = \frac{3}{10} = 0.3$.

Grade 8 Mathematics Quiz: Squares and Cubes

1. Without performing long calculation, which of the following numbers cannot be a perfect square based on its unit digit?

2. A square garden has an area of $2025 m^2$. What is the length of the decorative fence required to cover one side of the garden?

3. In a Pythagorean triplet (6, x, y), if 6 is the smallest member, what are the values of x and y?

4. Find the smallest natural number by which 180 must be multiplied so that the product becomes a perfect square.

5. What is the sum of the first n odd natural numbers if n = 12?

6. The volume of a cubical box is $512 cm^3$. What is the surface area of one face of this box?

7. Which of the following is the cube root of -216?

8. Find the smallest number by which 250 must be divided to obtain a perfect cube.

9. If the digit in the units place of a number is 7, what will be the digit in the units place of its cube?

10. The value of $\sqrt[3]{0.027}$ is: