https://file.aiquickdraw.com/mathbot/file/1726898940710oyjbho6w.jpg

Let's go through and solve the questions from the image one by one.

### Q.1 Multiple choice questions:

#### (a) **Which of the following operations holds true for commutative property of rational numbers?**
- Commutative property states that changing the order of the numbers does not change the result.
  
  **Answer:**
  - (ii) Addition and Multiplication are commutative.

#### (b) **If the largest number of three consecutive numbers is (x + 1), then find the smallest number out of these numbers.**
- Three consecutive numbers are \(x - 1\), \(x\), and \(x + 1\). The smallest is \(x - 1\).

  **Answer:**
  - (ii) \(x - 1\)

#### (c) **Which of the following statements is false?**
  - (i) The smallest three-digit perfect square number is 100.
  - (ii) A perfect square number cannot end with digit 7.
  - (iii) There are 202 non-perfect square numbers between \(100^2\) and \(101^2\).
  - (iv) If a number has 1 or 9 in the unit place, then its square ends in 1.
  
  **Answer:**
  - (iii) There are **198** non-perfect square numbers between \(100^2\) and \(101^2\), so (iii) is false.

#### (d) **Which of the following rational numbers is in the standard form?**
  - (i) \(\frac{-16}{26}\)
  - (ii) \(\frac{78}{-18}\)
  - (iii) \(\frac{9}{-28}\)
  - (iv) \(\frac{7}{28}\)

  **Answer:**
  - (iv) \(\frac{7}{28}\) (because it can be simplified to \(\frac{1}{4}\)).

#### (e) **Which of the following statements is correct?**
  - (i) If a number has 0 at its units place, then its cube never ends with zero.
  - (ii) The value of \(\sqrt[3]{-125} = -5\).
  - (iii) When you multiply a number by itself three times, you get a cube root of the number.
  - (iv) The cube of a single-digit number is always a single-digit number.

  **Answer:**
  - (ii) The value of \(\sqrt[3]{-125}\) is \(-5\).

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### Q.2 Fill in the blanks:

#### (i) **______ is the digit at the units place in the cube of 37.**
- To find the units place of \(37^3\), we calculate \(37^3 = 50653\), so the units digit is 3.
  
  **Answer:**
  - 3

#### (ii) **If two quantities ‘a’ and ‘b’ are in direct proportion, then ______ is constant.**
- In direct proportion, the ratio \(\frac{a}{b}\) is constant.

  **Answer:**
  - \(\frac{a}{b}\)

#### (iii) **On subtracting \((3t^2q)\) from \((-5t^2q)\), the result will be ______.**
- \((-5t^2q) - (3t^2q) = -5t^2q - 3t^2q = -8t^2q\).
  
  **Answer:**
  - \(-8t^2q\)

#### (iv) **The square root of 0.36 is ______.**
- \(\sqrt{0.36} = 0.6\).

  **Answer:**
  - 0.6

#### (v) **When a die is thrown once, the probability of getting a prime number is ______.**
- Prime numbers on a die: 2, 3, 5. So, probability = \(\frac{3}{6} = \frac{1}{2}\).

  **Answer:**
  - \(\frac{1}{2}\)

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Would you like further details on any of these solutions, or explanations for any specific steps?

Here are 5 related questions to explore the concepts further:
1. What is the associative property, and how does it differ from the commutative property?
2. How can you calculate the cube of any given number without a calculator?
3. What are the general steps to determine whether a number is a perfect square?
4. What is the difference between direct and inverse proportion, and how do you identify each?
5. How can the probability of composite numbers on a die be determined?

**Tip:** When calculating cube roots, remember that negative numbers can have real cube roots, while negative numbers under square roots result in complex numbers!

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Explore the solved sample paper for Grade 8 mathematics, covering key topics such as the commutative property, perfect square numbers, cube roots, and probability. Understand important concepts with detailed step-by-step solutions.

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