Question 1

1. Find the smallest natural number by which 1008 must be multiplied so that the product is a perfect square.

A) 2B) 7C) 3D) 14

Answer

✓ Correct Answer: Option B

Explanation: Solution: First, find the prime factorization of 1008: $1008 = 2 \times 504 = 2^2 \times 252 = 2^3 \times 126 = 2^4 \times 63 = 2^4 \times 3^2 \times 7$. For a number to be a perfect square, all prime factors must have even exponents. Here, the factor 7 has an exponent of 1 (odd). Therefore, we must multiply by 7 to make the exponent even ($7^2$).

Question 2

2. If one member of a Pythagorean triplet is 32, find the sum of all three members of the triplet (using the form $2m, m^2-1, m^2+1$).

A) 512

Question 3

3. Evaluate the square root: $\sqrt{0.000529}$

A) 0.023B) 0.23C) 0.0023D) 2.3

Answer

✓ Correct Answer: Option A

Explanation: Solution: We can write 0.000529 as $\frac{529}{1,000,000}$. $\sqrt{\frac{529}{1,000,000}} = \frac{\sqrt{529}}{\sqrt{1,000,000}}$ Since $23^2 = 529$ and $1000^2 = 1,000,000$: $\frac{23}{1000} = 0.023$.

Question 4

4. Calculate the value of: $\sqrt{41 - \sqrt{21 + \sqrt{19 - \sqrt{9}}}}$

A) 4B) 5C) 6D) 7

Answer

✓ Correct Answer: Option C

Explanation: Solution: Work from the innermost root outwards: 1. $\sqrt{9} = 3$ 2. $\sqrt{19 - 3} = \sqrt{16} = 4$ 3. $\sqrt{21 + 4} = \sqrt{25} = 5$ 4. $\sqrt{41 - 5} = \sqrt{36} = 6$.

Question 5

5. What is the smallest number by which 54000 must be divided to make it a perfect cube?

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

Answer

✓ Correct Answer: Option A

Explanation: Solution: Prime factorization of 54000: $54000 = 54 \times 1000 = (2 \times 3^3) \times (10^3) = 2 \times 3^3 \times (2^3 \times 5^3)$ $54000 = 2^4 \times 3^3 \times 5^3$ To be a perfect cube, all exponents must be multiples of 3. The factor $2^4$ needs to become $2^3$. So we divide by $2^1 = 2$.

Question 6

6. How many non-square natural numbers lie between $99^2$ and $100^2$?

A) 100B) 199C) 200D) 198

Answer

✓ Correct Answer: Option D

Explanation: Solution: The number of non-square numbers between the squares of two consecutive numbers n and n+1 is given by 2n. Here, n = 99. Number of non-squares $= 2 \times 99 = 198$.

Question 7

7. If $\sqrt[3]{x} = 1.2$, find the value of x.

A) 1.44B) 1.728C) 17.28D) 0.1728

Answer

✓ Correct Answer: Option B

Explanation: Solution: $\sqrt[3]{x} = 1.2 ⇒ x = (1.2)^3$ $x = 1.2 \times 1.2 \times 1.2 = 1.728$.

Question 8

8. A square garden has an area of $1444 m^2$. A path of width 2 m is built inside the garden along its boundary. Find the area of the path.

A) $280 m^2$B) $288 m^2$C) $300 m^2$D) $272 m^2$

Answer

✓ Correct Answer: Option B

Explanation: Solution: Side of square garden $= \sqrt{1444} = 38 m$. The path is inside, so the side of the inner square = 38 - (2 + 2) = 34 m. Area of path = Outer Area - Inner Area $= 1444 - (34)^2 = 1444 - 1156 = 288 m^2$.

Question 9

9. Which of the following numbers cannot be a perfect square?

A) 1024B) 1296C) 4096D) 1234567

Answer

✓ Correct Answer: Option D

Explanation: Solution: A property of perfect squares is that they never end in the digits 2, 3, 7, or 8 at the unit's place. The number 1234567 ends in 7, so it cannot be a perfect square.

Question 10

10. Evaluate: $\sqrt[3]{\frac{0.000064}{0.008}}$

A) 0.2B) 0.02C) 2D) 0.4

Answer

✓ Correct Answer: Option A

Explanation: Solution: First, simplify the fraction inside the root: $\frac{0.000064}{0.008} = \frac{64 \times 10^{-6}}{8 \times 10^{-3}} = 8 \times 10^{-3} = 0.008$. Now, find the cube root: $\sqrt[3]{0.008} = \sqrt[3]{\frac{8}{1000}} = \frac{2}{10} = 0.2$.

Accepted Answer

✓ Correct Answer: Option C

Explanation: Solution: Let 2m = 32 ⇒ m = 16. The other two members are: $m^2 - 1 = 16^2 - 1 = 256 - 1 = 255$ $m^2 + 1 = 16^2 + 1 = 256 + 1 = 257$ Sum = 32 + 255 + 257 = 544.

Grade 8 : Mathematics (Ganita Prakash) - Square and Cube (Set 2)

1. Find the smallest natural number by which 1008 must be multiplied so that the product is a perfect square.

2. If one member of a Pythagorean triplet is 32, find the sum of all three members of the triplet (using the form $2m, m^2-1, m^2+1$).

3. Evaluate the square root: $\sqrt{0.000529}$

4. Calculate the value of: $\sqrt{41 - \sqrt{21 + \sqrt{19 - \sqrt{9}}}}$

5. What is the smallest number by which 54000 must be divided to make it a perfect cube?

6. How many non-square natural numbers lie between $99^2$ and $100^2$?

7. If $\sqrt[3]{x} = 1.2$, find the value of x.

8. A square garden has an area of $1444 m^2$. A path of width 2 m is built inside the garden along its boundary. Find the area of the path.

9. Which of the following numbers cannot be a perfect square?

10. Evaluate: $\sqrt[3]{\frac{0.000064}{0.008}}$