Grade 8 Mathematics Quiz: Squares and Cubes | Concept of Maths

Question 1

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

A) $1024$

B) $5625$

C) $7923$

D) $1296$

Answer

Correct Answer: C

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 $2025 \text{ m}^2$. What is the length of the decorative fence required to cover one side of the garden?

A) $35 \text{ m}$

B) $45 \text{ m}$

C) $55 \text{ m}$

D) $25 \text{ m}$

Answer

Correct Answer: B

Area of square $= \text{side}^2$. $s^2 = 2025 \implies s = \sqrt{2025}$. Since $40^2 = 1600$ and $50^2 = 2500$, and the number ends in $25$, the root is $45$. Side $= 45 \text{ m}$.

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) $8, 10$

B) $7, 9$

C) $10, 12$

D) $9, 15$

Answer

Correct Answer: A

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) $2$

Question 5

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

A) $121$

B) $144$

C) $169$

D) $100$

Answer

Correct Answer: B

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 $512 \text{ cm}^3$. What is the surface area of one face of this box?

A) $8 \text{ cm}^2$

B) $16 \text{ cm}^2$

C) $64 \text{ cm}^2$

D) $48 \text{ cm}^2$

Answer

Correct Answer: C

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

Question 7

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

A) $-6$

B) $6$

C) $-16$

D) Does not exist

Answer

Correct Answer: A

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) $5$

B) $10$

C) $4$

D) $2$

Answer

Correct Answer: D

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) $7$

B) $3$

C) $9$

D) $1$

Answer

Correct Answer: B

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) $0.03$

B) $3.0$

C) $0.3$

D) $0.003$

Answer

Correct Answer: C

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

Accepted Answer

Correct Answer: D

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 $5$.

Grade 8 : Mathematics (Ganita Prakash) - Square and Cube (Set 1) | Concept of Maths

Grade 8 : Mathematics (Ganita Prakash) - Square and Cube (Set 1) | Concept of Maths

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

6. The volume of a cubical box is $512 \text{ cm}^3$. 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:

Grade 8 Mathematics Quiz: Squares and Cubes | Concept of Maths

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 $2025 \text{ m}^2$. 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.

Grade 8 : Mathematics (Ganita Prakash) - Square and Cube (Set 1) | Concept of Maths

Grade 8 : Mathematics (Ganita Prakash) - Square and Cube (Set 1) | Concept of Maths

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

6. The volume of a cubical box is $512 \text{ cm}^3$. 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: