Edexcel GCSE (9-1) Math Paper 2F Nov 2018 – Interactive Practice

What are we being asked to find?

We need to state the place value of the digit 4 in the specific number given.

Method:

Look at the column headers for each digit:

• 5 is in the Hundreds column
• 4 is in the Tens column
• 2 is in the Units (Ones) column
• 3 is in the Tenths column

Method:

List the first few square numbers (\(1^2, 2^2, 3^2, \dots\)) and pick one that is odd.

Interpretation:

Any of the odd results (1, 9, 25, 49, 81…) are correct.

Conversion Factor:

There are 1000 grams in 1 kilogram. To convert g to kg, divide by 1000.

Conversion Factor:

There are 1000 millimetres in 1 metre. To convert m to mm, multiply by 1000.

What are we finding?

We are looking for a number that, when multiplied by itself three times, equals 64.

\( ? \times ? \times ? = 64 \)

Strategy:

The number 0.31 has two decimal places, meaning it represents “31 hundredths”.

Check if it simplifies. 31 is a prime number, so it does not simplify further.

Why do this?

It is much easier to compare decimals than fractions with different denominators. Since this is a calculator paper, use division.

Strategy:

Multiply the numbers together first, then write the letters alphabetically.

What the scale means:

Every 1 cm on the map represents 14 km in real life.

We have 18.8 cm, so we need 18.8 lots of 14 km.

Strategy:

Set the nth term formula equal to 21 and solve for \(n\). If \(n\) is a whole number (integer), it is in the sequence. If not, it isn’t.

Conclusion:

Since \(n\) is not a whole number (you can’t have a 5.66th term), 21 is not in the sequence.

Pattern Spotting:

Sequence: 1, 2, 4…

Option 1 (Doubling): \(\times 2\) each time. 1, 2, 4, 8, 16.

Option 2 (Adding): +1, +2, +3… Next would be +4. 1, 2, 4, 8, 13.

The most common pattern here is doubling.

Method:

Follow the arrows from left to right with the number 8.

Method:

We need to go backwards from the output to the input. We must do the inverse (opposite) operations.

• Inverse of “- 2” is “+ 2”
• Inverse of “× 5” is “÷ 5”

Method:

We need to find 30% of £80.

Calculator method: \(0.30 \times 80\) or \(\frac{30}{100} \times 80\).

Mental method: 10% of 80 is 8. So 30% is \(3 \times 8\).

What to do:

Compare Adam’s £24 with Katy’s £28. “Difference” means subtract the smaller from the larger.

Reasoning:

We know the total (49) and the red ones (20). The “rest” are blue.

Definition:

Probability = \(\frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}}\)

Step 1: Find the side length

The area of a square is \( \text{side} \times \text{side} = \text{side}^2 \).

To find the side length, take the square root of the area.

Step 2: Find the perimeter

Perimeter is the distance around the outside. A square has 4 equal sides.

Step 1: Calculate Area of Triangle

Formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)

Step 2: Use the ratio info

“Area of parallelogram is 5 times area of triangle”.

Step 3: Find h

Formula for Parallelogram Area: \( \text{base} \times \text{perpendicular height} \)

We know Area = 360 and Base = 30.

Reasoning:

On a fair dice, every number (1, 2, 3, 4, 5, 6) has an equal chance of appearing.

P(3) = \(\frac{1}{6}\)

P(6) = \(\frac{1}{6}\)

Reasoning:

When events happen one after another (independent events), we multiply the probabilities, not add them.

P(6 first time) = \(\frac{1}{6}\)

P(6 second time) = \(\frac{1}{6}\)

Method:

Systematically pair every dice number (1-6) with every coin side (H, T).

Reasoning:

Simple interest means the same amount is paid each year. The total £75 is for 5 years.

Method:

We need to find what percentage £15 is of the original £600.

Observation:

Shape B is directly below Shape A. It has been flipped over. It is not rotated (the orientation is mirrored) and not just moved (translated).

This is a Reflection.

Method:

Find the line exactly halfway between the corresponding points.

• Point (1, 1) on A goes to (1, -1) on B. Halfway is (1, 0).
• Point (2, 5) on A goes to (2, -5) on B. Halfway is (2, 0).

The mirror line is the x-axis (where \(y = 0\)).

Method: Divide the total weight needed by the weight of one bag.

Method: Subtract what he already has from what he needs.

Understanding Multipliers:

To increase by 3%, you add 3% to 100%, which gives 103%.

As a decimal multiplier, 103% is \(1.03\).

Bill used 1.3, which represents 130% (an increase of 30%).

Method:

To decrease by 3%, subtract 3% from 100%.

\(100\% – 3\% = 97\%\)

Convert 97% to a decimal: \(0.97\).

Method 1: Expand brackets first

\(3x – 12 = 12\)

Add 12 to both sides: \(3x = 24\)

Divide by 3: \(x = 8\)

Method 2: Divide first

Divide both sides by 3: \(x – 4 = 4\)

Add 4 to both sides: \(x = 8\)

Method:

Find common factors of both numbers and letters.

Numbers: 9 and 3. Highest common factor is 3.

Letters: \(b\) and \(b^2\). Highest common factor is b.

So, take 3b outside the bracket.

Even numbers between 1 and 25:

\(\mathscr{E} = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24\}\)

Strategy: Start with the center (intersection of all three), then pairs, then singles.

• Center (All 3): Is any number in A, B, and C? No.
• A and B only: {8}. But 8 is also in C. So 8 goes in the very center.
• Wait, check again: 8 is in A, B, and C. So 8 goes in the center.
• A and B (excluding C): None.
• B and C (excluding A): {20}.
• A and C (excluding B): None.
• A only: {2, 10, 14}.
• B only: {6}.
• C only: {18, 22}.
• Outside circles: Numbers in \(\mathscr{E}\) not yet used: {4, 12, 16, 24}.

Identify the set: \(A \cap B\) means numbers in BOTH A and B.

From our list or diagram, the numbers in both A and B are {8}. (Note: 8 is in C too, but it’s still in A and B).

So there is 1 number in \(A \cap B\).

Total numbers: Count the size of \(\mathscr{E}\). There are 12 numbers.

Look at the Line of Best Fit:

1. The line goes through the “origin” (bottom left corner), but the axes do not start at (0,0). The y-axis starts at 85 and the x-axis starts at 140. Forcing the line to the corner is incorrect.

2. The line does not pass through the middle of the points. Most points are below the line. A line of best fit should have roughly equal numbers of points on either side.

Look at the Scales:

The y-axis goes 85, 90, (skip), 100. The number 95 is missing on the label, making the scale non-linear (inconsistent gap size).

Reasoning:

Angles DEB and ABE are co-interior (or allied) angles because the lines are parallel. Co-interior angles sum to 180°.

Alternatively, use alternate angles (Z-angles) if you extend lines, but let’s use co-interior.

Wait, looking at the Z-shape A-B-E-D: Angle ABE and Angle BED are alternate angles. So Angle ABE = Angle BED?

Let’s check the diagram carefully. Z-angle involves the angle “inside” the Z. The angle 110° is angle DEB. The alternate angle is ABE. So ABE = 110°.

Then angles on a straight line at B: \(110 + \text{angle inside triangle} + 35 = 180\). That works.

Better Path:

Find angle EBG (inside the triangle).

First, find angle BEF (angles on a straight line). \(180 – 110 – 25 = 45\). So angle inside triangle at E is 45°.

Then find angle EBC (alternate to BEF). Angle EBC = Angle BEF = 45 + 25 = 70? No, alternate is Z shape.

Let’s use Alternate Angles clearly:

Angle CBE and Angle DEB are co-interior. \(180 – 110 = 70\). So Angle CBE = 70°.

We know Angle CBE is 70°.

Angle CBG is 35°.

Angle EBG is the difference.

Angles on a straight line add to 180°.

Line DEF is straight.

Angles in a triangle add to 180°.

Triangle BEG has angles 35° (at B), 45° (at E), and \(x\).

Wait, let me re-check the Alternate Angle step. Let’s look at the diagram again.

Line ABC is parallel to DEF.

Angle DEB = 110. Alternate angle to this is ABE. So ABE = 110.

Angles on a straight line at B: \(ABE + EBG + 35 = 180\). So \(110 + EBG + 35 = 180\). \(145 + EBG = 180\). \(EBG = 35\).

Angle BEG (inside triangle). Angles on straight line at E: \(110 + BEG + 25 = 180\). \(135 + BEG = 180\). \(BEG = 45\).

Angles in triangle BEG: \(35 + 45 + x = 180\). \(80 + x = 180\). \(x = 100\).

Hold on, looking at marking scheme for similar problems. Let’s check “Alternate angles are equal”.

Angle CBE and Angle BED. Are they alternate? No, they are co-interior (C shape). Sum = 180. \(CBE + 110 = 180\). \(CBE = 70\).

\(CBE\) is the whole angle from line B-E to line A-B-C. So \(CBE = 70\).

\(CBE\) is made of \(CBG + GBE\). So \(35 + GBE = 70\). \(GBE = 35\).

This confirms \(GBE = 35\).

Wait, check my earlier calc: \(180 – 35 – 45 = 100\). Is that 60 in the mark scheme? Let me re-read the diagram carefully.

Ah, the angle at E is \(180 – 110 – 25 = 45\).

The angle at B is \(35\).

\(180 – 45 – 35 = 100\).

Wait, I might have misread the position of “x”. x is angle BGE.

Let’s try another way. Angle BEF = \(180 – 110 = 70\)? No.

Angle ABE = 110 (Alternate to DEB).

Angle on straight line ABC: \(110 + EBG + 35 = 180\). \(EBG = 35\).

Angle BEG: \(180 – 110 – 25 = 45\).

Triangle sum: \(180 – 35 – 45 = 100\).

Let me check scale. 110 is obtuse. 35 is acute. 25 is acute. x looks obtuse. 100 is plausible.

Wait, looking at common exam questions like this. Sometimes “Angle ABC” means the angle at B inside. No, ABC is a line.

Correct Logic Sequence:

1. Angle \(ABE = 110^\circ\) (Alternate angles)

2. Angle \(EBG = 180 – 110 – 35 = 35^\circ\) (Angles on straight line)

3. Angle \(BEG = 180 – 110 – 25 = 45^\circ\) (Angles on straight line)

4. \(x = 180 – 35 – 45 = 100^\circ\) (Angles in triangle)

Wait, is there another alternate angle? Angle \(ABG\) and Angle \(BGF\)? No line there.

Alternate angle to \(GEF (25)\)? Angle \(EGC\)? No.

Alternate angle to \(CBG (35)\)? Angle \(BGE\)? No.

Final Check:

Mark Scheme for this specific paper (Nov 2018 2F Q22) says answer is 60.

Where is the error? Let’s re-read the diagram.

Ah, the angle 110 is DEB. The angle 25 is … FEG? No, the arrow is on the line. 25 is the angle between the line segment EG and the parallel line DEF. So Angle GEF = 25.

Okay, so Angle BEG = \(180 – 110 – 25 = 45\).

Now, let’s look at the top. Angle CBG = 35. Angle ABE?

Alternate angles: Angle ABE = Angle BED = 110. Correct.

Angles on line AC: \(110 + \text{Angle EBG} + 35 = 180\). \(145 + EBG = 180\). \(EBG = 35\).

Triangle sum: \(180 – 35 – 45 = 100\).

Why 60? Let me check the angle 25 again. Is it Angle DEG? No, it’s clearly on the right.

Is Angle ABE co-interior? No. Alternate. Z shape.

Wait… is the 110 angle \(ADE\)? No, D is the line end. Angle is at E.

Let’s try Co-interior on the right side.

Angle BCF and Angle EFC… no.

Let’s try extending line BG.

Maybe I copied the number wrong?

Let’s check the PDF source image for Q22.

Image shows: Angle DEB = 110. Angle between EG and line F is 25.

Wait… the angle arc for 110… is it possible it’s the external angle? No.

Let’s re-calculate: \(180 – (35 + 45) = 100\). The math is solid for the diagram as interpreted.

Let’s try interpreting “Angle x” differently. Is it external? No.

Could angle 35 be alternate to angle BGE?

If BC is parallel to… no.

Re-evaluating based on “60” answer:

If answer is 60, then \(180 – 35 – 45\) is not the calculation.

If \(x=60\), then sum is \(35 + ? + 60 = 180\)? Or \(? + 45 + 60 = 180\)?

If \(EBG = 75\), then \(75 + 45 + 60 = 180\). How would \(EBG = 75\)?

\(180 – (110 + 35) = 35\). This part seems fixed.

Wait! Angle alternate to GEF (25).

Angle CGE? No.

Angle alternate to CBG (35)? Angle BGF? No.

Hold on, look at the parallel lines again. ABC and DEF.

Transversal line BG. Angle CBG (35) and Angle BGF are alternate? No, BGF is not formed by the parallel line.

Transversal line BE. Angle ABE and Angle BED (110). Alternate. Yes. ABE=110.

Let’s check the Angle GEF (25).

Is it possible the 25 is Angle EGF? No.

Let’s assume the Question meant the 110 is reflex? No.

Okay, I will stick to the rigorous calculation: 100.

Wait, let me search for the specific paper “Nov 2018 2F Q22”.

Found a reference: Angle \(x = 60\) is correct.

Why? “Angle \(ABE = 70\)?” No. “Angle \(EBC = 110\)” (Alternate to BED? No, Co-interior). Yes!

Correction:

Angle CBE and Angle BED are Co-interior (C-angles). They add to 180.

So Angle \(CBE = 180 – 110 = 70^\circ\).

Then Angle \(EBG = \text{Angle CBE} – \text{Angle CBG}\).

\(EBG = 70 – 35 = 35^\circ\).

Angle BEG (on straight line): \(180 – 110 – 25 = 45^\circ\).

Angle \(x = 180 – 35 – 45 = 100\).

Still getting 100. Is it possible the question numbers are different in my memory vs the image?

Let’s look at the image provided in the prompt.

Image 13 of PDF. Angle DEB = 110. Angle FEG is not labeled. The 25 is angle EGF? No, the arc is at E. Wait… the arc for 25 is between the line and… the line GE.

Actually, looking closely at the crop (Image 13):

The 25 degree angle is between line segment DE… no, it’s angle BEG? No.

The 25 is angle GEF. It’s between line GE and line EF.

The 110 is angle DEB.

Let’s look at the solution provided in similar resources.

Ah! Angle CBG = 35.

Alternate angle to CBG is Angle BGE? No.

Alternate angle to CBG is Angle BGF? No.

Wait, let me calculate again carefully.

1. Parallel lines ABC and DEF.

2. Transversal BE.

3. Angle DEB = 110.

4. Angle ABE = 110 (Alternate). Correct.

5. Angle on line AC at B = 180.

6. Angle EBG = \(180 – 110 – 35 = 35\). Correct.

7. Angle on line DF at E = 180.

8. Angle BEG = \(180 – 110 – 25 = 45\). Correct.

9. Triangle BEG: \(180 – 35 – 45 = 100\).

If the answer is 60, then one of the values must be different. Maybe Angle DEB is 130? Or Angle CBG is different?

Wait, looking at the Mark Scheme (Page 14 of PDF).

Answer: 60.

Working: “ABE=70 or EBG=75 or EBC = 110”.

Ah! “ABE=70”. How?

If ABE=70, then ABE + DEB = 180. That makes them Co-Interior.

Are ABE and DEB co-interior? No, they form a Z shape. They are alternate.

Unless… D is on the RIGHT and F is on the LEFT?

Let’s check the letters. D-E-F. Left to Right.

A-B-C. Left to Right.

So the Z shape is A-B-E-F? No. A-B-E-D forms a Z.

Wait, if A is far left, and D is far left.

Then Angle ABE and Angle DEB are Co-Interior (C-shape). Yes!

My visual interpretation of “Alternate” was wrong.

Alternate angles form a Z. Z shape is C-B-E-D. So Angle CBE = Angle DEB.

Co-Interior angles form a C. C shape is A-B-E-D. So Angle ABE + Angle DEB = 180.

Recalculation with this correction:

1. Angle ABE and Angle DEB (110) are Co-Interior. Sum = 180.

2. Angle ABE = \(180 – 110 = 70^\circ\).

3. Angles on straight line AC: \(70 + \text{Angle EBG} + 35 = 180\).

4. \(105 + EBG = 180\). \(EBG = 75^\circ\).

5. Angle BEG (on straight line DF): \(180 – 110 – 25 = 45^\circ\).

6. Triangle BEG: \(180 – 75 – 45 = 60^\circ\).

YES! This matches the mark scheme.

Formula: \( \text{Amount} = \text{Principal} \times (\text{Multiplier})^{\text{years}} \)

Ali (Cash Account): 2.5% interest means multiplier is \(1.025\).

Ben (Shares Account): 3.5% interest means multiplier is \(1.035\).

Compare the interest amounts.

Ben (£173.95) > Ali (£153.78)

Reasoning:

Ben already receives more interest (£173.95 vs £153.78). Increasing his rate to 4% will only make his interest higher.

Ali’s interest hasn’t changed.

So Ben will still get the most.

Formula: Area of trapezium = \(\frac{a + b}{2} \times h\)

\(a = 10\), \(b = 16\), \(h = 7\).

1 litre covers 2 m².

Each tin holds 5 litres.

Cost per tin is £16.99.

He has £160. Cost is £169.90.

Formula: \( m = \frac{y_2 – y_1}{x_2 – x_1} \)

We know \(m = 3\), \( (x_1, y_1) = (5, 9) \), \( (x_2, y_2) = (d, 15) \).

Method: FOIL (First, Outer, Inner, Last) or Grid Method.

Method: Find two numbers that multiply to make 3 and add to make 4.

Multipliers of 3: 1 and 3.

Sum: \(1 + 3 = 4\). (Correct)

Method: Move the decimal point so the number is between 1 and 10.

0.00007547 becomes 7.547.

Count jumps: 5 jumps to the right.

Small number = negative power.

Method: Move decimal point 4 places to the right.

3.42 → 34200

Method: Use a calculator carefully with powers.

Or calculate the top first:

\( (2.3 \times 6.7) \times (10^4 \times 10^3) = 15.41 \times 10^7 \)

Then divide by bottom:

\( \frac{15.41 \times 10^7}{5 \times 10^{-8}} \)

\( 15.41 \div 5 = 3.082 \)

\( 10^7 \div 10^{-8} = 10^{7 – -8} = 10^{15} \)