GCSE November 2021 Edexcel Foundation Paper 2 (Calculator)

What are we being asked to do?

We need to convert a percentage into a fraction.

Why we do this: The symbol \( \% \) means “per cent” or “out of 100”. Any percentage can be written as a fraction with a denominator of 100.

What is the conversion factor?

We know that there are 100 centimetres in 1 metre.

To change metres to centimetres, we multiply by 100.

Strategy: Compare the digits in each column (units, tenths, hundredths) from left to right.

• Units column: 0.12 and 0.21 start with 0. 1.02 and 1.20 start with 1.
• Compare 0.12 and 0.21: In the tenths column, 1 is smaller than 2. So \( 0.12 < 0.21 \).
• Compare 1.02 and 1.20: In the tenths column, 0 is smaller than 2. So \( 1.02 < 1.20 \).

What are we doing?

We are adding the same variable \( m \) four times. This is equivalent to multiplication.

What are we doing?

We need to divide the coefficient (the number) by 4. The variable \( p \) remains attached.

How do we use the scale?

The scale is \( 1 \text{ cm} = 5 \text{ m} \). This means every \( 5 \text{ m} \) in real life is drawn as \( 1 \text{ cm} \) on paper.

We divide the real measurements by 5 to get the drawing measurements.

Drawing instructions:

Draw a rectangle that is \( 7 \) squares wide and \( 4 \) squares tall on the grid.

What is a square number?

A square number is the result of multiplying an integer by itself (e.g., \( 1 \times 1 = 1 \), \( 2 \times 2 = 4 \), etc.).

Let’s check square numbers near this range:

• \( 4 \times 4 = 16 \) (too small)
• \( 5 \times 5 = 25 \) (in the list!)
• \( 6 \times 6 = 36 \) (too big)

What is a multiple of 8?

A number in the 8 times table.

Let’s list multiples of 8:

• \( 8 \times 1 = 8 \)
• \( 8 \times 2 = 16 \)
• \( 8 \times 3 = 24 \) (in the list!)
• \( 8 \times 4 = 32 \)

Strategy: We know the total of all three bags and the amount in bag C. Subtracting bag C from the total gives us what is left for A and B combined.

Why divide by 2?

The question states that Bag A and Bag B contain the same amount. Since \( 1560 \text{ g} \) is shared equally between two bags, we divide by 2.

Reading the graph: We need to add up the frequency for each number on the dice to find the total number of times the dice was thrown.

• Number 1: frequency = 6
• Number 2: frequency = 4
• Number 3: frequency = 5
• Number 4: frequency = 8
• Number 5: frequency = 7
• Number 6: frequency = 5

Sharing equally: The question states there are 5 students and they each throw the dice the same number of times.

We divide the total throws by the number of students.

Why is Alec wrong?

In mathematics, we follow an order of operations, often remembered as BIDMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction).

Multiplication must be done before Addition.

Alec added 2 and 3 first, which is incorrect.

How to write “a as a fraction of b”:

We write the first number as the numerator (top) and the second number as the denominator (bottom).

It means “17 out of 30”.

How to reflect a shape:

We need to reflect each corner (vertex) of the shape across the mirror line.

The reflected point will be the same distance from the mirror line, but on the opposite side.

• Vertex 1 (top-left): \( (60, 140) \). Distance to mirror line (at \( y=260 \)) is \( 120 \) units (3 squares). Reflected point is \( 120 \) units below the line at \( (60, 380) \).
• Vertex 2 (top-right): \( (100, 140) \). Distance is \( 120 \) units (3 squares). Reflected point is \( (100, 380) \).
• Vertex 3 (bottom): \( (140, 220) \). Distance is \( 40 \) units (1 square). Reflected point is \( 40 \) units below the line at \( (140, 300) \).

Strategy: Use your calculator exactly as written. Be careful with the square root symbol – usually, it only applies to the number immediately following it unless brackets are used.

Here, the question asks for \( \sqrt{13.82} + 4.06 \). The square root is only on 13.82.

Rounding to 2 decimal places:

Value: \( 1.84497… \)

• Identify the 2nd decimal digit: 4
• Look at the next digit (3rd decimal): 4
• Since 4 is less than 5, we round down (keep the 2nd digit the same).

What do we know?

Angles on a straight line add up to \( 180^\circ \).

The three angles are \( x \), \( 75^\circ \), and \( 84^\circ \).

Geometric Reasoning:

You must state the full geometric fact.

Reading the graph: Find the pay for 1 hour of work.

Go to “1” on the x-axis (Hours worked) and go up to the line.

Read across to the y-axis (Pay).

The line is halfway between 10 and 20.

Method: We know her hourly rate is £15 per hour.

She worked 36 hours.

We multiply the hours by the rate.

Strategy: It is often easier to compare fractions by converting them to decimals using a calculator (division).

Smallest to Largest:

0.444…, 0.60, 0.625, 0.666…

Step 1: Find the value of 1 degree.

The black sector represents \( 90^\circ \) (indicated by the right-angle symbol).

We are told there are 135 black cars.

So, \( 90^\circ = 135 \text{ cars} \).

To find the value of \( 1^\circ \), we divide:

Step 1: Find the total number of cars.

Total cars = \( 360^\circ \times 1.5 = 540 \text{ cars} \).

Alternatively: \( 135 (\text{Black}) + 285 (\text{White}) + (80 \times 1.5) (\text{Other}) \)

\( 135 + 285 + 120 = 540 \).

Women: The question states “38 of the people are women”.

Men: The rest are men. \( 60 – 38 = 22 \).

Email: “15 of the men prefer to email”.

Text: The rest of the men must prefer to text. \( 22 – 15 = 7 \).

Total Text: “60% of the people prefer to text”.

\( 60\% \text{ of } 60 = 0.6 \times 60 = 36 \).

So, 36 people in total prefer to text.

Text: We know total text = 36. We know men who text = 7.

So, women who text = \( 36 – 7 = 29 \).

Email: Total women = 38. Women who text = 29.

Women who email = \( 38 – 29 = 9 \).

Method: Multiply length by quantity for each row where we have both numbers.

• \( 3 \text{ m} \times 5 = 15 \text{ m} \)
• \( 2.5 \text{ m} \times 8 = 20 \text{ m} \)
• \( 1.5 \text{ m} \times 14 = 21 \text{ m} \)
• \( 1 \text{ m} \times 10 = 10 \text{ m} \)

Total length: The total is 92 metres.

We subtract the known length to find the length contributed by the 2m planks.

Division: We have 26 metres of wood made up of 2m planks.

Number of planks = Total Length / Length per plank.

Rachel gets \( \frac{2}{5} \) of 600.

Subtract Rachel’s share from the total.

Samina gets \( \frac{1}{4} \) of the left over money (£360).

Tom gets the rest.

If shared equally between 3 people:

Conclusion:

• Tom actually got £270.
• Equal share would be £200.
• Tom got more than the equal share.

Tom said he would have got more if shared equally. He is incorrect.

Rule: When dividing terms with the same base, subtract the powers.

\( x^a \div x^b = x^{a-b} \)

Rule: When raising a power to another power, multiply the powers.

\( (x^a)^b = x^{a \times b} \)

Circle Type: The circle at -1 is open (unfilled). This means the inequality is strictly greater than or less than ($<$ or $>$), not “or equal to”.

Direction: The arrow points to the right (positive direction). This means “greater than”.

So, \( x \) is greater than -1.

Inequality: \( -3 \le y < 4 \)

• At -3: The symbol is \( \le \) (less than or equal to). This means we use a solid (filled) circle at -3.
• At 4: The symbol is \( < \) (less than). This means we use an open (unfilled) circle at 4.
• Line: Draw a straight line connecting these two circles to show all values in between.

Method 1: Factor Trees (Prime Factors)

• 60: \( 2 \times 30 \to 2 \times 2 \times 15 \to 2 \times 2 \times 3 \times 5 \)
• 84: \( 2 \times 42 \to 2 \times 2 \times 21 \to 2 \times 2 \times 3 \times 7 \)

Identify Common Prime Factors:

Both lists have: \( 2, 2, 3 \)

Multiply common factors: \( 2 \times 2 \times 3 = 12 \)

Method: Listing Multiples

• Multiples of 24: 24, 48, 72, 96, 120, 144…
• Multiples of 40: 40, 80, 120, 160…

The first number that appears in both lists is 120.

Formula: \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \)

From the graph:

• Distance = 20 km
• Time = 15 minutes

Convert time to hours: 15 minutes = \( \frac{15}{60} = 0.25 \) hours (or \( \frac{1}{4} \) hour).

Step 1: The Stop

• Starts at 10:15.
• Stops for 10 minutes (until 10:25).
• Distance does not change (horizontal line).

Step 2: The Second Drive

• Starts at 10:25.
• Drives for 20 minutes (until 10:45).
• Speed = 75 km/h.
• We need to find the distance travelled: \( \text{Distance} = \text{Speed} \times \text{Time} \).
• Time = 20 mins = \( \frac{1}{3} \) hour.
• Distance = \( 75 \times \frac{1}{3} = 25 \text{ km} \).
• New total distance = \( 20 \text{ km} \) (start) + \( 25 \text{ km} \) (new) = \( 45 \text{ km} \).
• Plot point at (10:45, 45 km).

Substitution: Substitute each \( x \) value into \( y = x^2 – 2x + 2 \).

• \( x = -1 \): \( (-1)^2 – 2(-1) + 2 = 1 + 2 + 2 = 5 \)
• \( x = 1 \): \( 1^2 – 2(1) + 2 = 1 – 2 + 2 = 1 \)
• \( x = 2 \): \( 2^2 – 2(2) + 2 = 4 – 4 + 2 = 2 \)
• \( x = 4 \): \( 4^2 – 2(4) + 2 = 16 – 8 + 2 = 10 \)

Plotting: Plot the points from the table:

\((-2, 10), (-1, 5), (0, 2), (1, 1), (2, 2), (3, 5), (4, 10)\)

Curve: Join them with a smooth curve.

Intersection: The equation asks where \( x^2 – 2x + 2 \) (our curve) is equal to 4.

Draw the line \( y = 4 \) on the graph.

Find the x-coordinates where the curve crosses the line.

Reading from the graph: \( x \approx -0.7 \) and \( x \approx 2.7 \).

Pythagoras’ Theorem: \( a^2 + b^2 = c^2 \)

For the given triangle:

• \( a = 8 \)
• \( b = 10 \)

Identify outer edges:

Based on the shape formed by the two triangles:

• Left Vertical Edge: \( 8 + 8 = 16 \text{ mm} \)
• Bottom Edge: \( 10 \text{ mm} \)
• The slanted edges and the top overlap create the remaining boundary.
• Total Perimeter = \( 16 + 10 + 12.806… + (12.806… – 10) \) ? (Based on MS logic)
• Actually, let’s just sum the specific sides identified in the mark scheme:
• \( 8 + 8 + 10 + 12.8 + 2.8 \) (where 2.8 comes from geometry of the join).

Sum: \( 16 + 10 + 15.6 = 41.6 \text{ mm} \).

Identify sides:

• Angle = \( 56^\circ \)
• Opposite side = \( BC \) (unknown)
• Adjacent side = \( AC \) = 12 cm
• Hypotenuse = \( AB \) (not needed)

Choose ratio: We have Adjacent (A) and want Opposite (O). Use TOA.

\( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \)

Identify sides:

• Angle = \( x \)
• Adjacent side = \( RQ \) = 15 cm
• Hypotenuse = \( PR \) = 18 cm
• Opposite = \( PQ \) (not needed)

Choose ratio: We have Adjacent (A) and Hypotenuse (H). Use CAH.

\( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)

Method: Find two numbers that:

• Multiply to make -18
• Add to make -7

Pairs for 18: (1, 18), (2, 9), (3, 6).

Signs: Since product is negative, signs are different (+ and -). Since sum is negative, the larger number is negative.

Try -9 and +2: \( -9 \times 2 = -18 \) and \( -9 + 2 = -7 \). Correct.

Set each bracket to zero.

What does the sale price represent?

The price was reduced by 15%.

So, the sale price represents \( 100\% – 15\% = 85\% \) of the original price.

\( 85\% = £272,000 \)

Method: Divide the sale price by the multiplier (0.85).