Let's go through the mathematical test questions one by one:

Multiple Choice Questions (10 marks)

1. Anna's weight relative to Simon

• Anna weighs 47.5 kg, 5 kg lighter than Simon. Simon’s weight is therefore $47.5 + 5 = 52.5 \, \text{kg}$.
• The fraction of Simon's weight that is Anna’s is: $\frac{47.5}{52.5} = \frac{95}{105} = \frac{19}{21}$ Answer: D. $\frac{19}{21}$

2. Divisibility by 36

• 36 = $2^2 \times 3^2$. A number divisible by 36 must have at least two 2’s and two 3’s in its prime factorization.
• Checking each option for divisibility:
  • Answer: C. 875748 (divisible by 36)

3. Least number of squares from the sheet

• Dimensions: 63 cm × 273 cm. We need to find the greatest common divisor (GCD) of 63 and 273 to determine the largest possible square. $\text{GCD}(63, 273) = 21$
  • The least number of squares: $\frac{63 \times 273}{21 \times 21} = \frac{17199}{441} = 39$ Answer: B. 39

4. Correct Equations

• I: $\frac{4}{7} = \frac{3}{5}$ is false.
• II: True as the arithmetic works out correctly.
• III: Also true. Answer: D. II and III only

5. Smallest Value

• We calculate each option:
  • $2 \times 10^4 = 20000$
  • $20.17^2 = 406.8361$
  • $200^2 = 40000$
  • $2017 \times 20.17 \approx 40685.89$ Answer: B. $20.17^2$ is the smallest value

Conventional Questions (20 marks)

1. (a) Express 1650 in index notation.

• 1650 = $2 \times 3 \times 5^2 \times 11$ Answer: $2 \times 3 \times 5^2 \times 11$

(b) Smallest integer $k$ such that 1650k is a perfect square

• $1650 = 2 \times 3 \times 5^2 \times 11$, so $k = 2 \times 3 \times 11 = 66$

2. Calculation

• Calculate the given expression: $4 \left( \frac{1}{10} \left[ 1.56 + 7.5 \times \left( \frac{12}{5} - 2.08 \right) \right] \right) + 0.2$ Let's break this down step by step to get the answer:
• $\frac{12}{5} - 2.08 = 0.32$
• $7.5 \times 0.32 = 2.4$
• $1.56 + 2.4 = 3.96$
• $\frac{1}{10} \times 3.96 = 0.396$
• $4 \times 0.396 = 1.584$
• Adding 0.2: $1.584 + 0.2 = 1.784$

Would you like further details on any specific problem or explanation?

Here are 5 related questions:

1. How do you find the greatest common divisor (GCD) of two numbers?
2. What are the steps to check if a number is divisible by 36?
3. How can you express a number in terms of its prime factors?
4. What is the process for calculating a fractional expression involving multiple terms?
5. How do you simplify a nested mathematical expression?

Tip: Always simplify complex expressions step by step to avoid calculation errors.