Edexcel GCSE Mathematics – Nov 2017 Higher Paper 2

What are we being asked to find?

We need to solve a linear equation. “Solving” means we need to find the specific value of \( x \) that makes both sides of the equals sign match. First, we need to expand any brackets to get all our \( x \) terms and numbers out in the open.

Why we do this:

We have an expression grouped in brackets on the right-hand side. By multiplying the \( 3 \) by everything inside the bracket, we can separate the terms and start moving them around.

Why we do this:

We want all terms with an \( x \) on one side of the equation, and all normal numbers on the other side. Let’s subtract \( 3x \) from both sides to group the \( x \)’s on the left, and add \( 6 \) to both sides to group the numbers on the right.

Why we do this:

We currently know the value of \( 2x \), but we want to find out what a single \( x \) equals. To reverse a multiplication by \( 2 \), we divide both sides by \( 2 \).

What this tells us:

The solution is \( x = 1.5 \). We can check this by substituting it back into the original equation: \( 5(1.5) – 6 = 7.5 – 6 = 1.5 \). The right side is \( 3(1.5 – 1) = 3(0.5) = 1.5 \). Since both sides equal \( 1.5 \), our answer is absolutely correct!

What are we being asked to find?

We need to find out the percentage profit Emily made. Profit is the amount of money she made after covering her initial costs. To find a percentage profit, we compare the total profit against the original amount she paid.

Why we do this:

Before we can work out how much profit she made, we need to know how much money she collected in total by selling all 12 bottles.

Why we do this:

Profit is the difference between the total money she received and what the pack originally cost her.

Why we do this:

To find the percentage profit, we divide the actual profit by the original cost and multiply by 100.

What this tells us:

We have our percentage, but the question asks for it to 1 decimal place. The first decimal digit is 3, and the next digit is 8. Since 8 is 5 or bigger, we round the 3 up to a 4.

Why we do this:

Before we can find the distance between individual points, we need to know the entire distance around the outside of the circle. This total distance is called the circumference. We find it using the formula \( C = \pi \times \text{diameter} \).

Why we do this:

There are 8 points marked on the circle, which divides the total circumference into exactly 8 equal sections or “gaps”. We just need to divide the total distance by 8.

What this tells us:

Rounding to 1 decimal place, the distance between any two adjacent points is 31.4 metres.

Why we do this:

The “mean distance” between points is calculated by adding up all the individual distances between points and dividing by the number of gaps.

Even though the points have been moved around, Hasmeet still walks around the *entire* circle exactly once. This means the Total Distance is still the same (the circumference). There are also still exactly 8 points (meaning 8 gaps).

What are we being asked to find?

We need to find the probability of picking a yellow cube. To find this probability, we need to know what fraction or “parts” of the total cubes are yellow. A great way to handle this is by creating a ratio of Yellow : Blue : Green or by making up a simple number for the smallest group.

Why we do this:

Let’s use algebra or simple numbers. It says there are “twice as many blue cubes as yellow cubes”. Yellow is the smallest amount, so let’s start by imagining there is exactly 1 yellow cube. We can build the rest of the numbers from there.

Why we do this:

Probability is the number of successful outcomes (yellow cubes) divided by the total number of possible outcomes (all cubes in the bag). We must add up all our parts to find the denominator.

What this tells us:

For every 11 cubes in the bag, exactly 1 will be yellow. The fraction is already in its simplest form.

Why we do this:

To rotate a shape \( 180^{\circ} \) about the origin \((0,0)\), we can use a simple coordinate rule. Every coordinate \((x, y)\) transforms into \((-x, -y)\). Alternatively, imagine turning your paper completely upside down while keeping the origin in the same place.

Why we do this:

A translation vector \( \binom{-1}{-3} \) gives us specific instructions on how to slide the shape. The top number represents movement on the x-axis (left or right). A negative top number means move 1 unit Left. The bottom number represents the y-axis (up or down). A negative bottom number means move 3 units Down.

Why we do this:

When we multiply terms that have the same base (in this case, \( p \)), the law of indices states that we must add the powers. So, \( p^a \times p^b = p^{a+b} \). We can use this to set up a simple equation.

Why we do this:

When a power is raised to another power (separated by a bracket), the law of indices states that we must multiply the powers. So, \( (a^m)^n = a^{m \times n} \).

Why we do this:

We need the final answer to have a base of \( 10 \). Currently, our bases are \( 100 \) and \( 1000 \). To combine them using index laws, we first need to rewrite \( 100 \) and \( 1000 \) as powers of \( 10 \).

What this tells us:

The question states that the expression equals \( 10^w \). By simplifying the expression down to a single base of 10, we can see directly that \( 10^{2a + 3b} = 10^w \). Therefore, the power \( w \) must be exactly \( 2a + 3b \), which completes the proof.

What are we being asked to find?

We need to find the angle at corner D (angle CDA). To do this, we should split the complex trapezium into simpler shapes: a rectangle in the middle and right-angled triangles on the left and right.

Let’s label the point where the dashed line hits the base as point \( X \). So \( BX = 6\text{ cm} \).
If we drop another vertical line straight down from \( C \) to the base, let’s call that point \( Y \). The height \( CY \) is also \( 6\text{ cm} \).

Why we do this:

To use trigonometry to find angle D, we need the lengths of the sides of the right-hand triangle (triangle \( CYD \)). The total base \( AD = 24\text{ cm} \). It is made up of three parts: \( AX + XY + YD \). We know \( XY = 10\text{ cm} \) (because it’s directly below \( BC \)), but we need to calculate \( AX \) first to eventually find \( YD \).

Why we do this:

Now we have enough information to find the base of the right-hand triangle \( CYD \), which contains the angle we are looking for.

Why we do this:

Look at the right-angled triangle \( CYD \). We want to find the angle at \( D \). We know the Opposite side (\( CY = 6\text{ cm} \)) and the Adjacent side (\( YD = 9.5\text{ cm} \)). Using SOH CAH TOA, Opposite and Adjacent means we use the Tangent (\( \tan \)) ratio.

What this tells us:

The angle is roughly \( 32.27^{\circ} \). The question asks for the answer to 1 decimal place. The first decimal digit is 2, and the next digit is 7. Because 7 is 5 or more, we round the 2 up to a 3.

Why we do this:

This question tests your ability to input complex expressions into a scientific calculator accurately. Make sure your calculator is set to Degrees (D or DEG) mode, not Radians or Gradients!

Why we do this:

We need to round \( 2.75603957… \) to 2 decimal places. Look at the second decimal place (5). Then look at the next digit (6) to decide whether to round up or keep it the same.

Why we do this:

This is a problem involving inverse proportion. Having fewer cleaners means it will take more time. To figure out how long it takes 3 cleaners, we first need to know the total “size” of the cleaning job if 1 person were to do it all by themselves.

Why we do this:

Now we share those total 22.5 hours of work equally among the 3 cleaners working today.

Why we do this:

The question contains a vital condition: “paid £8.20 for each hour or part of an hour“.
They worked 7.5 hours. This means they worked 7 full hours, plus a half hour. Because they get paid for “part of an hour” as if it were a full hour, we must round the 7.5 hours up to 8 hours for payment purposes.

What this tells us:

Each cleaner receives £65.60 today. (Note: If you didn’t read carefully and multiplied 7.5 by £8.20 to get £61.50, you would only score partial marks!)

Why we do this:

On a distance-time graph, the speed is represented by the steepness (gradient) of the line. The steeper the line goes up, the more distance is covered in less time (which means higher speed). We just need to look for the steepest section of the graph visually.

Why we do this:

To calculate the exact speed, we need to find the gradient of that steepest line section. The formula for speed is \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \). We read the values from the graph for the 0 to 20 second section.

What this tells us:

The maximum speed reached during the journey is 18 metres per second.

What are we being asked to find?

We need to find a fraction (or decimal proportion). The top of our fraction is the number of 0-19 year olds in Adley. The bottom of our fraction is the total number of people in BOTH towns combined.

The question states the populations are in the ratio of the areas of the pie charts. By measuring the diagrams (or using the information provided in the examiner marks), we find the radius of the Adley chart is 5 units, and the Bridford chart is 4 units. Also, measuring the angle for the 0-19 section in Adley gives \( 70^{\circ} \).

Why we do this:

To find the total population, we need to add the areas of both pie charts together. Since they represent a ratio, we can just use their areas as the “number of people”.

Why we do this:

The 0-19 section in Adley is a slice of the Adley pie chart. The angle is \( 70^{\circ} \), which means it represents \( \frac{70}{360} \) of Adley’s population.

To find the final proportion, we divide the “area” of this specific slice by the “area” of both towns combined.

What this tells us:

The proportion is 0.11856… To 3 significant figures, we look at the first three non-zero digits (1, 1, 8). The next digit is 5, so we round the 8 up to a 9.

What are we being asked to find?

We need to find the angle \( \text{STR} \) inside the triangle \( \text{RST} \). Because the shape is a regular polygon, all of its sides are the exact same length. This means side \( \text{RS} \) is equal to side \( \text{ST} \). Therefore, triangle \( \text{RST} \) is an isosceles triangle.

Why we do this:

To use the isosceles triangle, we first need to know the angle at \( \text{S} \). This angle is an interior angle of the 12-sided polygon. The easiest way to find an interior angle is to first find the exterior angle, and then subtract it from \( 180^{\circ} \).

Why we do this:

The angles in any triangle add up to \( 180^{\circ} \). We know the main angle is \( 150^{\circ} \). Because it’s an isosceles triangle, the remaining two angles (angle \( \text{STR} \) and angle \( \text{SRT} \)) must be exactly equal to each other.

What this tells us:

The size of angle STR is \( 15^{\circ} \).

Why we do this:

Because the company increases by a percentage every year, this is compound growth. The formula is: \( \text{New Value} = \text{Original Value} \times (\text{multiplier})^{\text{years}} \).

A 2% increase means we keep 100% of the value and add 2%, giving 102%. As a decimal multiplier, this is 1.02. From 2009 to 2017 is exactly 8 years.

Why we do this:

We have the start value, the end value, and the number of years. We don’t know the percentage increase, so let’s call our unknown multiplier \( y \). We can build an equation just like we did in part (a).

Why we do this:

We need to isolate \( y \). First, divide both sides by the starting value to find the total 6-year multiplier. Then, take the 6th root to find the single-year multiplier.

Why we do this:

A multiplier of \( 1.044697… \) represents \( 104.4697…\% \). Since the original amount is \( 100\% \), the increase is whatever is left over.

What this tells us:

The value of \( x \) is 4.5.

Why we do this:

To find the region, we first draw the boundary lines. Replace the inequality signs with equals signs. \( y = 1 \) is a horizontal line cutting through 1 on the y-axis.

Since the inequality is \( y > 1 \), we will draw a dashed line (though solid is often accepted if the region is clear) and we want the area above this line.

Why we do this:

Let’s find coordinates to plot this line. It’s easiest to find where it crosses the axes.

• When \( x = 0 \), \( 0 + y = 5 \rightarrow y = 5 \). Point: \( (0, 5) \)
• When \( y = 0 \), \( x + 0 = 5 \rightarrow x = 5 \). Point: \( (5, 0) \)

Since the inequality is \( x + y < 5 \), we want the area below/left of this line.

Why we do this:

Let’s find coordinates for this line.

• When \( x = 0 \), \( y = 2(0) = 0 \). Point: \( (0, 0) \)
• When \( x = 2 \), \( y = 2(2) = 4 \). Point: \( (2, 4) \)

Since the inequality is \( y > 2x \), we want the area above/left of this line.

✓ (M1) For two of the lines correctly drawn.

✓ (M1) For three lines correctly drawn.

Why we do this:

We need the section of the grid that satisfies ALL THREE conditions at the same time: Above \( y=1 \), below \( x+y=5 \), and above \( y=2x \).

Why we do this:

When you have a certain number of choices for one item, and a certain number of choices for a second item, and you are choosing one of each, you must multiply the options together to find the total number of combinations. This is because for EVERY single main course (8 options), she can pair it with ALL 7 desserts.

Why we do this:

There are 12 teams. Every team will play exactly 11 matches (because they play everyone except themselves). If we multiply \( 12 \times 11 \), we get the total number of times a team plays.

However, a game involves two teams. If Team A plays Team B, that’s one game. But our \( 12 \times 11 \) calculation counts Team A playing Team B, AND it counts Team B playing Team A as a separate game. To fix this double-counting, we must divide our answer by 2.

What this tells us:

There will be exactly 66 games played in the competition. (Alternatively, you could add \( 11+10+9+8+7+6+5+4+3+2+1 = 66 \)).

What are we being asked to find?

We need to find the values of \( x \) that make the equation true. Because the equation contains a squared term, it’s a quadratic equation, which usually means there will be two possible solutions. The equation is already in a nice format (a perfect square on one side, a number on the other), so we don’t need to expand the brackets.

Why we do this:

To get to the \( x \) trapped inside the bracket, we must first “undo” the squared power. The opposite of squaring is square rooting. We must remember that taking a square root gives both a positive and a negative result (because a negative times a negative is a positive).

Why we do this:

Now we add 2 to both sides to leave \( x \) on its own. Then we split it into two separate calculations (one for the plus, one for the minus) and use the calculator to find the decimal values to 3 significant figures.

What this tells us:

The two values of \( x \) that satisfy the equation are 3.73 and 0.268.

Why we do this:

A histogram is different from a bar chart. In a histogram, the area of the bar represents the frequency, not the height. Because the class widths (the groups of heights) are different sizes, we must calculate the “Frequency Density” for the y-axis.

The formula is: \( \text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}} \).

Why we do this:

Now we plot these bars. The width of the bar matches the class width on the x-axis, and the height matches the Frequency Density we just calculated.

Why we do this:

We need to find how many students are between 150cm and 170cm, and write it as a fraction of the total 150 students.

We know exact numbers for the 150-155, 155-160, and 160-165 groups. But the 165-180 group goes too far! We only want up to 170. We must assume the students are spread out evenly in that last group, so we take a fraction of that group’s frequency.

What this tells us:

The estimated fraction is \( \frac{123}{150} \). You can simplify this by dividing top and bottom by 3 to get \( \frac{41}{50} \), but it’s completely acceptable to leave it as \( \frac{123}{150} \) in the exam.

What are we being asked to find?

We are looking for \( k \), which is the multiplier used to get from one hour’s volume to the next hour’s volume (\( V_{t+1} = k \times V_t \)).

Why we do this:

If there is “2% less water”, it means we start with 100% and subtract 2%. This leaves 98% of the water remaining at the end of every hour. To turn a percentage into a decimal multiplier, we divide by 100.

What this tells us:

To find the volume in the next hour, you multiply the current volume by 0.98. The 2000 litres is extra information that you don’t actually need to use to find \( k \).

Why we do this:

To prove anything about the line MN, we first need to know exactly where the points M and N are on the coordinate grid. We find the midpoint between two points by finding the average of their x-coordinates, and the average of their y-coordinates: \( \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \).

Why we do this:

Two lines are parallel if they have exactly the same gradient (steepness). We can find the gradient by using the formula \( m = \frac{y_2 – y_1}{x_2 – x_1} \) (change in y divided by change in x).

What this tells us:

Both lines have a gradient of \( \frac{1}{2} \). Because their gradients are mathematically identical, the two lines must be parallel.

What are we being asked to find?

We need to find the area of the grey shaded part. The best strategy is to find the area of the entire large kite shape (OABC) and then subtract the unshaded pie slice (the sector OAC).

A key circle theorem: A tangent (like BA) always meets a radius (like OA) at exactly \( 90^{\circ} \). So angle OAB is \( 90^{\circ} \), and angle OCB is also \( 90^{\circ} \).

Why we do this:

If we split the kite straight down the middle from O to B, we get two identical right-angled triangles: Triangle OAB and Triangle OCB. The angle at O (\( 120^{\circ} \)) gets split exactly in half, so angle AOB is \( 60^{\circ} \).

We know the base of this triangle (the radius OA) is 10 m. We can use trigonometry to find the height (AB), and then calculate the triangle’s area.

Why we do this:

Now we calculate the area of the white “pie slice” so we can subtract it from the total kite area. A sector’s area is just a fraction of a full circle’s area (\( \pi r^2 \)). The fraction is the angle divided by 360.

Why we do this:

Subtract the unshaded sector from the entire kite. Finally, round to 3 significant figures as requested.

What this tells us:

Rounding 68.485… to 3 significant figures means looking at the 6, 8, and 4. The next digit is 8, which is 5 or bigger, so the 4 rounds up to a 5.

What are we being asked to find?

We need to find the probability of picking a red counter, then another red, then a third red, without replacing them in between. First, we need to know exactly how many red counters are in the bag.

Why we do this:

Because the counters are “taken” (and not put back until part b), this is conditional probability. The total number of counters decreases by 1 each time we pick, and the number of red counters also decreases by 1 each time we successfully pick a red.

Why we do this:

We are told MORE counters are added, but they remain in equal amounts. The best way to prove what happens to the probability is to invent a new total that fits the rules, calculate the new probability, and compare it to our answer from part (a).

Why we do this:

To see if it is more or less likely, we need to compare \( \frac{1}{55} \) and \( \frac{2}{91} \). Turning them into decimals makes this very easy.

What this tells us:

Because \( 0.02197… \) is larger than \( 0.01818… \), the probability has gone up. It is now more likely to pick 3 red counters.

What are we being asked to find?

We need to find two missing constants, \( a \) and \( b \). This means we’ll likely need to create two simultaneous equations. We can get the first one from the clue \( g(3) = 20 \). This means “if you put 3 into the function g, the answer is 20”.

Why we do this:

To use the second clue (\( f^{-1}(33) = g(1) \)), we need to calculate what the left side is worth. The notation \( f^{-1}(33) \) means “the inverse function of f applied to 33”. In simple terms, it’s asking: “What number \( x \) do I have to put into \( f(x) \) to get an answer of 33?”

Why we do this:

Now we substitute \( 6 \) back into the second clue, which becomes \( 6 = g(1) \). This means “if you put 1 into the function g, the answer is 6”. We can use this to make our second simultaneous equation.

Why we do this:

We now have a system of two equations with two unknowns:
Eq 1: \( 3a + b = 20 \)
Eq 2: \( a + b = 6 \)
Because both equations have a single “\(+ b\)”, we can simply subtract Equation 2 from Equation 1 to eliminate \( b \).

What this tells us:

The constants are \( a = 7 \) and \( b = -1 \). You can check by putting them back into Eq 1: \( 3(7) + (-1) = 21 – 1 = 20 \). It works perfectly!

What are we being asked to find?

We need to find the specific value of \( x \). A geometric sequence is a number pattern where you multiply by the exact same number (the common ratio, \( r \)) to get from one term to the next.

Because the multiplier is constant, we know that:
(Term 2 \( \div \) Term 1) MUST equal (Term 3 \( \div \) Term 2).

Why we do this:

To solve for \( x \), we can cross-multiply to get rid of the fractions. This will leave us with a Difference of Two Squares expansion on the right side.

Why we do this:

Now that we know \( x = 2 \), we can find out what the actual terms and the multiplier are. Then we just need to keep multiplying to reach the 5th term to prove the question is correct.

What this tells us:

By substituting our value of \( x \) and applying the rules of geometric sequences, we have mathematically proven the 5th term is exactly \( 7 + 5\sqrt{2} \), as requested.