Year 8 End of Year Mathematics Assessment Paper 1

Problem 1 Context: Evaluating expressions involving large numbers and powers of ten without a calculator.
Expression to Evaluate: $(2 \times 10^4) \times (3 \times 10^7)$ .
Procedural Steps:
- Multiply the coefficients: $2 \times 3 = 6$ .
- Multiply the powers using exponent laws: $10^4 \times 10^7 = 10^{4+7} = 10^{11}$ .
Target Format: The final answer must be expressed in standard form notation, i.e., $A \times 10^n$ where 1 \le A < 10.
Final Answer: $6 \times 10^{11}$ .
Point Allocation: This question is worth $2$ marks.

Problem 2 Context: A collection of pens categorized by three colors: Red, Blue, and Green.
Defined Ratio: The ratio of colors is Red:Blue:Green = $1:4:11$ .
Total Parts Calculation: Sum of parts = $1 + 4 + 11 = 16$ .
Sub-Problem (a): Work out the percentage of the pens that are green.
- Calculation: $\frac{\text{Green parts}}{\text{Total parts}} = \frac{11}{16}$ .
- Conversion to Percentage: $\frac{11}{16} \times 100 = 68.75\%$ .
- Point Allocation: This sub-question is worth $1$ mark.
Sub-Problem (b): Calculating the total inventory of pens.
- Given Condition: There are $14$ more green pens than blue pens (based on legible markers).
- Logic: The difference in parts between green and blue is $11 - 4 = 7$ parts.
- Equivalency: $7 \text{ parts} = 14 \text{ pens}$ .
- Value per Part: $1 \text{ part} = \frac{14}{7} = 2 \text{ pens}$ .
- Total Calculation: $16 \text{ total parts} \times 2 \text{ pens/part} = 32 \text{ pens}$ .
- Point Allocation: This sub-question is worth $3$ marks.

Problem 3 Context: A diagram depicting intersecting straight lines forming various angles.
Sub-Problem (a): Find the value of the angle denoted as $x$ .
- Mark: Worth $1$ mark.
Sub-Problem (b): Justification of the answer.
- Mathematical Reason: Often involves identifying "Angles on a straight line sum to $180^{\circ}$ " or "Vertically opposite angles are equal."
- Mark: Worth $1$ mark.

Problem 4 Context: Comparing the distance from Earth to Mars at two different points in time.
Requirements:
- Utilize the provided distance data for Earth and Mars.
- Work out the required calculation (often difference or combined distance).
- Answer must be expressed in standard form (e.g., $X.Y \times 10^Z$ ).
Point Allocation: This question is worth $4$ marks.

Problem 5 Context: A function machine that applies a sequence of operations to an input value $x$ to produce an output value $y$ .
Structure: Input $\rightarrow$ [Operation 1] $\rightarrow$ [Operation 2] $\rightarrow$ Output.
Sub-Problem (a)(i): Find the output when the input is $18$ .
- Marks: $2$ marks.
Sub-Problem (a)(ii): Find the input when the output is $17$ .
- Procedure: Apply inverse operations in reverse order (e.g., if the machine was $\times 2, - 5$ , the inverse is $+ 5, \div 2$ ).
Sub-Problem (b): The input is $x$ and the output is $y$ . Write an expression for $y$ in terms of $x$ .
- Format: $y = f(x)$ .
- Marks: $2$ marks.

Problem 6 Context: Alice (A), Beth (B), and Charlie (C) are comparing their ages in years.
Relationships:
- Beth is older than Alice by a specific factor or fixed number of years.
- Charlie's age is linked to either Alice or Beth.
Objective: Determine the actual ages or a collective ratio representing their relative ages.
Point Allocation: This question is worth $3$ marks.

Problem 7 Context: An experiment involving two spinners where the two scores are combined (summed) to get a total.
Sub-Problem (a): Complete the sample space table showing all possible totals.
- Table Structure: Rows represent Spinner 1, columns represent Spinner 2.
- Marks: $2$ marks.
Sub-Problem (b): Find the probability that the total is a square number.
- Identification of Squares: Identify which outcomes in the table are square numbers ( $1, 4, 9, 16, 25, \dots$ ).
- Calculation: $\text{Probability} = \frac{\text{Number of square outcomes}}{\text{Total number of outcomes}}$ .
- Point Allocation: This sub-question is worth $2$ marks.

Problem 8 Context: Basic understanding of integer powers.
Question: Write down the value of $5^0$ .
Fundamental Principle: For any non-zero real number $a$ , the zero exponent rule states that $a^0 = 1$ .
Final Answer: $1$ .
Point Allocation: This question is worth $1$ mark.

Problem 9 Context: A person is purchasing a car using a combination of a deposit and monthly payments.
Financial Data:
- Deposit: $15\%$ of the total price of the car.
- Remaining Balance: Paid in $24$ equal monthly payments.
- Monthly Installment: $£300$ .
Calculation Task: Work out the total price of the car.
Step-by-Step Logic:
- Step 1: Calculate total installment amount: $24 \text{ months} \times £300/\text{month} = £7,200$ .
- Step 2: Determine the percentage of the car price covered by installments: $100\% - 15\% = 85\%$ .
- Step 3: Set up the equation where $P$ is the total price: $0.85P = £7,200$ .
- Step 4: Solve for $P$ : $P = \frac{7200}{0.85}$ .
Point Allocation: This comprehensive multi-step problem is worth $4$ marks.