GCSE MATHS 2026 EDEXCEL 2H PRACTICE PAPER

Added: 2026-05-19

[00:00:10]Hello and welcome to the video solutions to my 2026 EDI Excel practice paper 2.

[00:00:15]You can find a link to this paper on my website firstclassmaths.com and there's also a link in this video's description.

[00:00:21]I would also like to thank Casio Education for sponsoring the creation of these video solutions and papers.

[00:00:28]For this first question, we need to expand and simplify. Let's start by expanding. We first of all multiply 8 by C, which gives you 8 C and then 8 by -1, which gives you8.

[00:00:40]For the second bracket, we need to multiply by -3. So, -3 * 2 C is -6 C.

[00:00:47]And for this final one, you need to watch out. It's -3 * -5. And when you multiply two negatives, you get a positive. So, this is pos5.

[00:00:56]Now that we've expanded, we can simplify by collecting like terms. We have 8 C subtract 6 C, which is 2 C. And then we have 8 + 15, which is a positive 7. So + 7. So the answer is 2 C + 7.

[00:01:13]In this question, we're given a number line and we're told that n is an integer and we need to write down the greatest possible value of n. Looking at the number line, it goes between -4 and pos4. And since n is an integer, we just want to consider all of the integers between these two. So from -4 all the way to pos4. However, because this circle here is open rather than shaded, this means we do not include the number four itself. So let's remove that from the list. This means the inequality is showing all of the numbers from -4 to 3.

[00:01:42]So the greatest possible value is three.

[00:01:46]In the second part of the question, we need to solve this inequality. Here we would start by subtracting four from both sides. When you subtract 4 from the left hand side, this will cancel the positive 4. So, we're just left with 5x.

[00:01:58]On the right hand side, 20 subtract 4 is 16. It's really important here that you write this inequality symbol rather than an equal sign. The next step is to divide both sides by 5. 5x / 5 is just 1 x. And for 16 / 5, remember you have a calculator. That will give you 3.2. So, the solution is x is greater than or equal to 3.2.

[00:02:20]In this next question, we need to use a calculator to work out the answer to this calculation. So, let's take our calculator and we would begin by typing in a square root. Then, inside the square root, we have 9. And then, it says subtract. And then, we need a fraction. So, we'll press this button here. Then, we need to type in 2.1. And this needs to be to the power four. So, we click this button here to get under the powers. And then, type a four. Now, to get to the bottom of the fraction, we'll need to press the down arrow. And then we need to type in pi. On this calculator, pi is above the seven. So I'll press shift and then 7 to get the pi and then press equals. That'll give me this number here. Question says to write down all of the figures of your calculator display. So we would write down this entire number. In the second part of the question, we need to round the answer from the previous part to three significant figures. So if we take this answer, the first number one here is the first significant figure. The six is the second and the seven is the third. So the answer could well be 1.67.

[00:03:19]However, we do need to check the number that comes after this to see if we need to round. The number that comes after is a six, which means we would need to round up. So the number before this line here, the 7, will round up to an 8. So the answer is 1.68.

[00:03:34]For this question, we need to change a speed of 15 m/s into kmh.

[00:03:40]A speed of 15 m/s means the object moves 15 m every single second. There are 60 seconds in 1 minute. So if the object is moving 15 m in 1 second, we can multiply 15 by 60 to get 900. And this is now how many m it's moving every minute. So 900 m per minute. But there are also 60 minutes in 1 hour. So if we take this 900 and multiply it by 60, we get 54,000. And it's no longer me per minute, but m per hour. Now, we're getting very close to the units required in the question. It's km hour. You should know that 1 km is the same as 1,000 m. So, to go from m to km, you need to divide by 1,000. So, if we take this 54,000 and divide it by 1,000, we get 54. So, this is 54 km per hour. And that's the answer to the question.

[00:04:37]In the first part of this question, we need to write down the name of the country with the greatest population. To do this, we're going to convert all of the populations in the table from standard form to ordinary numbers. Let's start with the population of Estonia, which is 1.4 * 10 ^ 6. This means we take 1.4 and multiply it by 10 6 times.

[00:04:58]If we multiply it by 10 once, we get 14.

[00:05:02]A second time would be 140. A third time, 1,400.

[00:05:07]A fourth time, a fifth time, and a sixth time. So 1.4 4 * 10 ^ 6 is the same as 1,400,000.

[00:05:15]If you then repeat the same process for all of the other countries, the population of France is 68 million, Germany is 84 million, Hungary is 9,600,000, and San Marino is 34,000. We can now see that the largest population is 84 million, which was for Germany. So the answer is Germany. In the second part of the question, we need to find the name of the country that has the median population. The median is the value that's in the middle when they are in order. So let's label up these countries and we'll rewrite them in order from smallest to largest. We can now see the population in the middle is this one here which corresponds to Hungary. So the answer is Hungary. For the final part of the question, we need to work out how many times greater the population of France is compared to the population of San Marino. So we only need the populations of those two countries. To find how many times greater it is, we would simply divide the populations. So using your calculator, you can do 68 million divide by 34,000. This will give you 2,000, which is the answer to this question.

[00:06:21]For the first part of this question, we need to complete the table of values.

[00:06:25]Notice we've been given all of the x values, and we just need to fill in some of the y values. We can do this by substituting the x values into this equation here. So, what we're going to do is write out x^2 + 2x - 4, but substitute the x for each of our missing coordinates. We'll start with x= -4. So, we have x^2, which is now -4^ 2. And notice I've put this4 in brackets.

[00:06:50]That's particularly important when you substitute in negative numbers. Then, it's + 2x. So, plus two lots of -4. Once again, the -4 in brackets, then subtract four. If you type this into your calculator, you will get the answer four. So that's the y-coordinate and we place that into our table. Now they've already done -3 for us. So let's move on to x= -2. So we have x^2, so -2^2 plus 2 lots of x, so plus 2 lots of -2 subtract 4. Typing this into your calculator will give you -4. So let's add this into the table. And finally, we need to do x= 0. So we have 0^ 2 + 2 lots of 0 subtract 4 which will give you -4 again. So let's place -4 into the table. For the second part of the question, we need to complete the graph of this equation. To do that, we're going to need to plot each of these pairs of coordinates onto the graph.

[00:07:48]We'll start with when x is -4 and y is pos4. That's the coordinates -4,4 which goes here. Then if we move on to -31 that goes here -2 -4 goes here -1 -5 0 -4 1 and 2 4. Once you've plotted all of the points, you can connect them together with a nice smooth curve. Something that looks like this. For the final part of the question, we need to use this graph to estimate the roots of the equation x^2 + 2x - 4 = 0. Since this equation matches the equation of the graph, when it equals 0 is when it crosses the x-axis. So at this point here and this point here. So we just need to read off these two x values. They are about x= -3.2 and x= positive 1.2.

[00:08:42]In the first part of this question, we need to work out an estimate for the mean mass of the 20 bolts. To start this question, we need to identify all of the midpoints of each of the groups in the table. The first group is from 20 to 30.

[00:08:53]So the midpoint is 25. The next group from 30 to 40, the midpoint is 35. Then it's 45 and 55.

[00:09:02]Next we want to multiply these midpoints by the frequencies. So 25 * 4 is 100, 35 * 5 is 175, 45 * 9 is 405, and 55 * 2 is 110. Then if we add up all of these numbers, we get 790, which is the estimate for the total mass of all of the bolts. To find the mean, we then need to divide this by how many bolts there are. You can get this by adding up the frequency column, which will give you 20. But you may have also noticed that this number was given in the question. The question told us there were 20 metal bolts. So to finish the question, we do 790 / 20, which is 39.5.

[00:09:42]and the answer to the question. In the second part of the question, we are told that each of the bolts is made from a metal with a density of 8 g per cm cubed. And we need to work out the volume of a bolt that has a mass of 44 g. To do this, we're going to use the formula linking density, mass, and volume, which is density equals mass / volume.

[00:10:02]Since we're trying to work out volume, we would rearrange this to give volumeals mass / density. So, we just need to do the mass of the bolt / the density. The mass is 44 and the density is 8. So we do 44 / 8 which will give you 5.5 which is the answer to the question.

[00:10:21]In this question we've been asked to find the size of angle CA which is this angle here. In the question we're told that AB is equal to AC which means we have an isoclesles triangle. And that also means that these two angles here are the same size. We can therefore write down that the first angle 11x subtract 4 is equal to the second one 5x + 35. We can solve this equation to find the value of x. Let's start by subtracting 5x from both sides. On the left hand side if you do 11 x subtract 5x you get 6x and then we have subtract 4. On the right side the 5x will cancel so we're left with 35. Next we would add four to both sides. On the left this will cancel the subtract 4. So we just have 6x. And on the right 35 add 4 is 39. And finally we can divide both sides by 6 which will give you x equals and 39 / 6 on your calculator will give you 6.5. We can now take this value of x and substitute it into the expressions here to work out the size of the angles.

[00:11:20]Let's start with this angle here. We would do 11 * x but x is 6.5. So 11 * 6.5 subtract 4 on your calculator. This will give you 67.5. So this angle is 67.5°.

[00:11:34]We can also do the same for the other angle. It's 5 lots of 6.5 + 35. And this will also give you 67.5, which shouldn't be surprising since we said they were the same size. Now that we know two of the angles in the triangle, we can use the fact that the angles in a triangle add up to 180°. So if we do 180, subtract these two angles, we'll get the answer to the question, which is 45°.

[00:12:00]In the first part of this question, we need to find an estimate for the median distance run by the year 9 students. We can see the cumulative frequency goes up to 60. So the median distance will be found at a frequency which is half of this. Half of 60 is 30. So we go across from a cumulative frequency of 30 until we hit the curve and then we go down and read off this value which will be 2,100 m. For the next part of the question, we need to find an estimate for the interquartile range. To do this, we need to calculate both the upper and lower quartiles. Let's start with the lower quartile. The lower quartile will be found at one quarter of the cumulative frequency. 1/4 of 60 is 15. So, we go from 15 across to the curve and down and read off this value, which is at 1,600.

[00:12:48]For the upper quartile, it's 3/4 of the cumulative frequency, which is at 45. So we go from 45 across and down and read off this value which is 2250.

[00:13:00]To find the interquartile range, we do the difference between the upper and lower cortiles. So 2250 subtract 1,600, which is 650. So that's the answer to the question. In the final part of the question, we're given some information about the year 10 students.

[00:13:17]And we are asked to compare the distances run by year 9 and year 10 students. Let's remind ourselves of the information from year 9 that we worked out in parts A and B. In part A, we worked out the median for year 9 was 2,100. And in part B, we got the intercortile range at 650. So, we just need to make two comparisons. One about the median and one about the interquartile range. Looking at the median, we can see that year 9 have a lower value of 2,100 compared to the value of year 10, which was 2,300.

[00:13:46]So, let's first of all write that down.

[00:13:48]Year 9 had a lower median. We then need to interpret what this means in the context of the question. This question was about the distances that the students had run. If year 9 had a lower median, then they have run a lower distance on average. So we could say this means they ran a lower distance on average. And it's important that we say on average because not every runner will have necessarily run a lower distance.

[00:14:10]It's just the average that we're comparing here. For the next part of the question, we'll compare the interquartile ranges. We'll once again start by stating what we know. We can see the intercortile range for year 9 is 650 and for year 10 it's 800. So the year 9's intercortile range is lower. So we would say that year 9 had a lower interquartile range. Now we need to interpret this in the context of the question. If you have a lower intercortile range it means that the distances run were more consistent. So we would say that this means that the distances run were more consistent.

[00:14:41]Another way of saying this is that they are less varied which will also score you the marks in the exam. You could also give both of these statements the other way round from the perspective of year 10's. For example, you could say that year 10 had a higher median. This means they ran a greater distance on average. Year 10 had a higher interquartile range. This means the distances ran were less consistent.

[00:15:06]In this question, we need to work out the equation of the circle. When a circle is centered at the origin, its equation is always x^2 + y^2= r, where r is the radius. Since we know a point on this circle has coordinate 68, we can simply substitute those values in. X will be 6 and Y will be 8. So instead of X^2, it's 6^2 and instead of Y^2, it's 82. So we find that 62 + 82 must equal R2. If you work out 62 + 82, you get 100. So 100 = r 2. This will mean that our circle has a radius of 10. Since if you do the square root of 100, you get 10. To write out the equation though, all we need to do is substitute this r^ 2 here for 100. So it's x^2 + y^2 = 100.

[00:15:53]You may have also written the answer as x^2 + y^2 = 10^2.

[00:16:00]In this question, we have L1 and L2 that are perpendicular lines and these are their equations. We need to write down the value of K. The first equation y= 3x + 4 is in the form y mx + c. That means this value here is the gradient. So the gradient is three. The second line is also in this form. So the gradient is K.

[00:16:22]We need to find the value of K. And since these lines are perpendicular, we know the gradient of perpendicular lines are negative reciprocals of each other.

[00:16:30]So we just need to find the negative reciprocal of 3. The reciprocal of 3 is 1 / 3. So the negative reciprocal is - 1 3. So that's the answer to this question.

[00:16:44]In this question, we're told that triangle A is enlarged with center of enlargement 01 to give triangle B. And we need to draw triangle B onto the grid. Now, at first, you might think that we don't have enough information here since we're not given the scale factor of enlargement. But we are told that triangle B is also translated by a vector to give triangle C. When you translate a shape, it doesn't change size. So, we can actually use triangle C to work out the scale factor instead.

[00:17:09]This length on triangle A is four squares. And the same length on triangle C is two squares. So going from shape A to shape C, the triangle is half the size. So the scale factor would be 1/2 or 0.5. However, the shape is also flipped upside down. This indicates to us that it must have been a negative scale factor enlargement. So the scale factor will be0.5 or - 1/2. Now that we know the scale factor, we're ready to do the enlargement and draw triangle B. The center of enlargement is 01. So let's mark that on. And we'll do each of the points on shape A one at a time. We'll start with this one down here at the bottom. For a negative scale factor enlargement, we want to look at how we get from this point back to the center of enlargement. And then we'll repeat this in the negative direction, but only half of the distance since the scale factor is negative - 1/2. So to get from this point to the center of enlargement, we go 1 2 to the right and then 1 2 3 4 5 6 7 8 up. So we're going to repeat this but only half of the journey. So rather than two right, one right. And rather than eight up, four up. So it's one to the right and four up. So this point would map to here. Now let's do this point. So to get from here to the center enlargement, we go 1 2 3 4 5 six to the right and two up. So we'll repeat this but only half of that. So rather than six right, three right. And rather than two up, one up. So three right and one up, which is here. And for the final point, it is 1 2 3 4 5 6 right and 1 2 3 4 5 6 up. If we do half of this, it's three right and three up. So it ends up here. We can now connect these points together to form triangle B.

[00:19:05]For the second part of the question, we need to write down the values of P and Q which came from this vector here. This vector represented the translation from triangle B to triangle C. So let's see what that translation was. If we pick a point on triangle B and the corresponding point on triangle C, we can work out what the translation was.

[00:19:24]To get from this point to the other, we would go 1 2 3 right and 1 2 3 4 5 6 down. So to work out the vector, the top number is how much we move left or right. And we move three to the right, so it's a positive 3. And the bottom number is for up and down, and we move six down, so it's a - 6. This means that P is 3 and Q is -6.

[00:19:49]In this question, we're told that X and Y are positive integers. And the first thing we need to do is simplify x y ^0.

[00:19:57]x y ^0 means x * y ^0. Since y is a positive integer, we can be sure that y ^0 is equal to 1 since any positive integer to the power 0 is equal to 1. So we just have x * 1 which is just x. And that's the answer to the question. In the second part, we're going to start by looking at what's inside the bracket here. When you divide like this, you subtract the indices. So if we do 400 subtract 256, we get 144. So inside the bracket is x ^ 144. And this is raised to the power 3 /2. When we have a bracket raised to another power, we multiply the indices. So if we do 144 * 3 over2, and you can just do this on your calculator, and it'll give you 216.

[00:20:45]So the answer is x ^ 216.

[00:20:49]In this question, we have Isaac who creates a three-digit code where each digit is a number from 1 to 9. We need to work out how many different codes Isaac can make where at least one of those digits is a prime number. So, we have a three-digit code and each of those digits is a number from 1 to 9.

[00:21:06]This means there are nine ways of choosing the first digit and there are also nine ways of choosing the second digit and there are also nine ways of choosing the third. So if we multiply these together, we get 729 total codes that could be made. But we're only interested in codes where at least one of those is a prime number. So let's keep the 729 for a moment. We'll need that later. Let's sort these numbers into prime and not prime. 2 3 5 and 7 are prime and 1 4 6 8 and 9 are not prime. So there are four primes and five not primes. Now for this question, we want at least one of those digits to be prime. But it's actually easier to think of the opposite. The opposite of at least one of those being prime is that none of them are prime. Let's work out the number of ways of making one of these codes where none of the digits are prime. What if we bring up the three digits again? And none of them are prime. All of them must be one of the numbers 1 4 6 8 or 9. Since there are five of those numbers, this means there's five ways of choosing the first digit, five for the second, and five for the third. And if we multiply these, we get 125 ways of making this code with no primes at all. Now remember, there were 729 total codes and 125 that have no primes. This means the difference between these must be the number of codes with at least one prime. So we do 729 subtract 125, which is 604. And that's the answer to this question.

[00:22:34]In this question, we need to find the area of the shaded region. Since this region is the difference between two sectors, we're going to need to find the area of those sectors. To find the area of a sector, you need to know its radius and its angle. So, we're going to try and find those first. First of all, notice that the length DB here is 1 cm, which means this length here is also 1 cm. That means the diameter of the larger sector is 1 + 13, which is 14 cm.

[00:23:02]So the larger sector has a diameter of 14 and the radius is half of the diameter. So half of this is 7. Now for the radius of the smaller sector, the line O D here is the radius of the larger one. So we said that was 7. And this line from O to B is the radius we want of the smaller one. So if we do 7 subtract 1, we get six. So the radius of the smaller sector is six.

[00:23:28]Now for the angles, we can see the angle for the smaller sector, that's definitely 40. And the angle for the larger sector is this angle here. Since AOC is a straight line, this angle is 180°. So the entire angle for the whole sector is 180 + 40, which is 220°.

[00:23:46]We can now work out the area of these sectors. For the larger sector, it's the angle, which is 220 / 360. And then we multiply this by the formula for the area of a circle which is p<unk> r^ 2.

[00:23:59]So *<unk> and time the radius which is 7^ 2. If you put this into a calculator, you'll get 539 over 18<unk>. For the smaller sector, it's the angle 40 over 360 *<unk> r 2 again. So *<unk> * 6^ 2.

[00:24:16]This will give you 4 pi. So to find the area of the shaded region, we'll do the area of the larger sector subtract the area of the smaller sector. So 539 over 18 pi subtract 4 pi, which will give you this number here. The question says to give your answer to three significant figures. So it's 81.5.

[00:24:36]In this question, we're told that x is directly proportional to the roo<unk> of y. This means we can write an equation x= k where k is some constant time the roo<unk> of y. Later in the question, we're told that when x= 0.8, y= 0.16. We can substitute these values in to help us find the value of k. So x= 0.8. So 0.8 equals k, which we don't know times the<unk> of y. So the<unk> of 0.16.

[00:25:05]You can do the roo<unk> of 0.16 on your calculator at 0.4. So we have k * 0.4 or 0.4k.

[00:25:13]Dividing both sides of this equation by 0.4 4 will give you 2 = k or k = 2. Now that we have the value of k, we can substitute that back into the original formula. So it's x= k, which we now know is 2 * the<unk> of y. We now have a formula linking x and y. We want a formula linking y and zed. We're told that y is inversely proportional to z^ 3, which means that y is equal to some constant, and we'll use a different letter this time. let's say c / z ^ 3 since it's inversely proportional rather than directly. We also have a corresponding value of zed which is 5 and this is when y is equal to 0.16. So we can substitute those values in to find c. y is equal to so 0.16 is equal to c the constant / z ^ 3 so 5 ^ 3.

[00:26:09]Multiplying both sides of this equation by 5 ^ 3 will give you 20 = c or c = 20.

[00:26:17]We can now substitute this value of c back into this formula. So it's y = 20 / z ^ 3. So we now have two formulas, one linking x and y and one linking y and zed. We're asked to find the value of x when zed equals 20. We can substitute zed for 20 into our second formula. So it's y = 20 over z ^ 3. But zed is 20.

[00:26:40]So 20 ^ 3. Typing this into your calculator will give you y = 0.25.

[00:26:47]We can then take this y value and substitute that into the first formula.

[00:26:51]So x = 2 * the<unk> of y. So the<unk> of 0.25.

[00:26:57]Typing this into the calculator gives you x= 0.1, which is the answer to this question.

[00:27:04]In this question, we've been given the graph of y= fx. and we need to draw the graph of y = fx -1 + 2. Let's take a closer look at this equation. The plus2 here is outside of the function. Any number outside of the function here will correspond to moving the graph upwards or downwards. Since it's positive, that's upwards. So, the graph will need to move 2 units up. This negative 1 though is inside the function which corresponds to moving left or right, but it's the opposite of what you might normally expect. A negative one actually means we need to move the graph one unit to the right. So it needs to go one right. So all we need to do is move this graph one right and two up. But you need to be extremely careful with the scale on the x-axis here. Let's take the graph and if we're going to move it one to the right that's actually two squares because every two squares corresponds to one unit. So if we go one right it looks like this. But on the vertical scale one square represents one unit. So when we move two up, we will just go two squares up like this.

[00:28:07]In this question, we're told the value of a house increases by x% each year. In 2023, the house was valued at 224,000, but then in 2026, it was valued at 268,000.

[00:28:19]And both of those values are given to the nearest,000. This is a big clue that this is a bounds question. But an even bigger clue is the next line that tells us we need to work out the upper bound for x. Let's call the value of the house in 2023 V1. Since it's 224,000 to the nearest thousand, it could be anywhere in between 223,500 and 224,500.

[00:28:44]Let's say the value 3 years later in 2026 is V2. Then this is any number in between 267,500 and 268,500.

[00:28:55]If you think about what's happened in this question, we had the original value in 2023 V1 multiplied by a multiplier for an X% increase which we're just going to call K in this question. And this would be to the power three to represent the 3 years between 2023 and 2026.

[00:29:13]And this would give you the final value which we called V2. If we were trying to find the value of K, we would divide both sides by V1. So we'd have k ^ 3 = v2 / v1 and then we would do the cube root of both sides. So k would equal the cube root of v2 over v1.

[00:29:32]Now this k is going to include our value of the rate x%. So we want to try and make this value as big as possible since we're looking for the upper bound. So how would we choose our values of v2 and v_sub_1 to make the value of k as big as possible? Well, k will equal the cube root of the number for v2 on the top. We want to make as large as possible. So, we'll take the upper bound 268,500.

[00:29:58]But then, we're going to divide this by the value of v1. Here, we want to choose the lower value 223,500 since when you divide by a smaller number, you get a greater answer. Typing this into your calculator will give you the largest possible value for k, which comes out at k= 1.06305. 06305464 which is a multiplier for a 6305464% increase.

[00:30:25]So this must be our value for x. The question says to give the answer to one decimal place though. So we would round this to 6.3%.

[00:30:34]In this question we're asked to work out h inverse of x. We're told the function for h is equal to gf of x. So let's work out gf of x first. To do this we take the g function. So x over 5 - x and then substitute the f function into it. To do this, I like to replace all of the x's with brackets and then substitute the f function x -1 into these brackets. So like this. Now the brackets on the top aren't actually necessary. It's just x - one. But the ones on the bottom do matter. We need to expand out these brackets. Some people find it a bit confusing when expanding these out. It might be slightly easier if you imagine that there's a -1 in front of the bracket. So we would need to multiply -1 by x which isx and then -1 by -1 which is a positive 1. So we can replace this whole section here withx + 1. We can simplify on the bottom 5 + 1 which is 6.

[00:31:32]So this is what gf of x is actually equal to. Now g f ofx is the same as h.

[00:31:38]So we just need to find the inverse of this function. To do that, I'll replace h of x with y. And then we'll switch all of the y's to x's and all of the x's to y's. So instead of y equals, it's x equals. And instead of x - 1 / 6 - x, it's y - 1 / 6 - y. We now just want to try and make y the subject. We have 6 - y on the bottom here. So I'm going to multiply both sides by 6 - y. On the left, that will give us x * 6 - y. And we need to put this in a bracket. And on the right, the 6 - y will cancel. So it's y - 1. Now on the left hand side, we can expand out the bracket. x * 6 is 6 x and x * - y is -x y. And on the right, y - 1. Then we want to get all of the terms involving y on one side and all of the terms that don't involve a y on the other. I'll start by adding one to both sides. This will give us 6x - x y + 1 on the left and on the right, the one will cancel, so it's just y. Then we can add x y to both sides. This will cancel the negative x y on the left hand side. So it's just 6x + 1. And on the right we now have y + x y. If we take this and actually it's probably easier to switch it around at this point, we can factoriize the y out on the left hand side. So we have y bracket. Then we need to think what would we multiply y by to get y? Well, that's just one. And what would we multiply y by to get x y?

[00:33:04]That's x. And this equals 6x + 1. We can then finally divide both sides by this bracket 1 + x to give us y = 6 x + 1 / 1 + x. Now that we've got y as the subject, this must be the inverse function. So it's 6x + 1 / 1 + x.

[00:33:24]In this question, we're told about two bags that contain counters. Let's draw out the bags. Bag A has eight blue counters and four red counters. And bag B has six blue counters and X red counters. We're told that a counter is taken at random from bag A and then placed into bag B and then a counter is taken at random from bag B. The probability that the counter taken from bag B is red is 2/3. So let's think about the ways that this could happen.

[00:33:52]Well, the second counter must be red as we're told the counter selected from bag B is red, but the first one could be either. So we're looking at the outcomes red followed by red or blue followed by red. And we need to work out these probabilities. So we first of all want to work out the probability that we select a red counter from bag A. In bag A there are four red counters. So it's four out of the total number of counters which is 4 + 8 which is 12. Then we need to move this counter into bag B. Then we want to multiply this by the probability of selecting a red counter from bag B.

[00:34:29]So this is the total number of red counters which is x + 1 divide by the total number of counters which if we add up the x and the seven individual counters is x + 7. Now let's return to the original scenario and imagine we picked a blue counter first instead.

[00:34:46]What's the probability of a blue counter from bag A? Well, there are eight of them and 12 in total. So it's eight out of 12.

[00:34:54]Then we would move this blue counter into bag B. And we need the next counter selected from bag B to be red. So we'll multiply this by the probability of selecting a red. And there are X of those out of a total of X + 7. Now when working with these probabilities, we can simplify the first two fractions. 42 is the same as 1 / 3 and 82 is the same as 2 over 3. Now let's actually do the multiplications. We have 1 * x + 1 which is just x + 1 and 3 * x + 7 which is 3 bracket x + 7. For the next one, it's 2 * x which is 2x and again 3 * x + 7. So 3 bracket x + 7. So these are the two probabilities of the different outcomes of ending with a red counter. If we add up those probabilities, it must therefore equal the number given in the question 2/3. We now just need to solve this equation. Since both of those fractions have the same denominator, we can actually add them together easily.

[00:35:55]We can simplify what's on the top. x + 2x is 3x. So, it's 3x + 1 on the top.

[00:36:01]And we end up with this. To solve this, we can cross multiply by multiplying both sides by 3 and 3 bracket x + 7.

[00:36:08]That will give us this. If we expand out these brackets on the left, 3 * 3x is 9 x and 3 * 1 is 3. on the right. If we multiply the two and three here, we get 6. So, this is 6 lots of x + 7.

[00:36:23]Expanding that bracket gives us 6 * x, which is 6 x and 6 * 7, which is 42.

[00:36:30]Now, if we subtract 6x from both sides, 9 x * 6 x is 3x. So, 3x + 3. And on the right, the 6x will cancel. So, just 42.

[00:36:40]Subtracting 3 from both sides gives us 3x = 39. And then if we take this and divide both sides by 3, we find that X is equal to 13, which is the answer to this question.

[00:36:53]In this question, we need to find the area of triangle BCD, which is this triangle here. We're actually going to start this question by looking at the other triangle ADC. Now, it's worth noting there are lots of different ways of solving this question. I'm just going to show you one of them. You may have done it in a slightly different way. I'm going to start by finding this angle here, ADC. I'll label that X since that's the one I'm trying to find. We can now add some labels to this red triangle and use the cosine rule to find the value of X. Let's label X as capital A. Then the side opposite this the<unk> 7 will be lowerase A. The other sides are lowerase B and C. And it doesn't matter which way around I label them. So let's call the 1 B and the 2 C. Now we need to use the rearranged version of the cosine rule which is cos of a = b^2 + c^2 - a^2 over 2 bc. This one is not given to you on the formula sheet. So you either need to learn it or learn how to rearrange the other one to get to it.

[00:37:49]Let's substitute in the values from the question. Capital a is x. So we have cos of x = b^ 2 which is 1 2 + c^ 2 which is 2 2 - a 2 which is<unk> 7^ 2. All of this is divided by 2 * b which is 1 * c which is 2. Typing this right hand side into your calculator will give you cos of x=0.5 which means that x must be equal to inverse cos of 0.5 which if you do on your calculator will give you 120°. So x must be 120°.

[00:38:26]Next, I'm going to find this angle here.

[00:38:28]And since it's a straight line, I can do that by doing 180 subtract 120, which is 60. So this angle is 60°.

[00:38:35]Now we're ready to work inside the blue triangle. We can find this angle here.

[00:38:39]Since the angles inside a triangle add up to 180, so if we add the 60 and 70 together, we get 130. And if we subtract this from 180, we get 50. So this angle down here is 50. I'm now going to use the sign rule to find the length of DB, which I will label as X. If we label the triangle, then X will be the lowerase A.

[00:38:59]The angle opposite this, the 50 is the capital A. The two will be a lowerase B and the 70 A capital B. Then if we write out the sign rule, A over sin A= B over sin B. And this one is on your formula sheet. We can substitute in the values.

[00:39:13]So a is x, capital a is 50, b is 2, and the capital b is 70. If we multiply both sides by sin 50, we get x = 2 / sin 70 * sin 50. Typing the right hand side into your calculator here will give you 1.63 and some other digits. So this is the length of x db is 1.63 and so on. We can now label the remaining side and angle in this triangle as lowerase C and capital C and then use half A sin C to find the area of the triangle. It's 1/2 * lowerase A which is this 1.63 and so on time lowerase B which is 2 * S of C which is s of 60. If you type all of this into your calculator and make sure you don't use a rounded value for 1.63, you'll end up with this number here. The question says to give the answer to three significant figures. So it's 1.41.

[00:40:10]And that's the paper complete. I'd like to remind you that if you are thinking about doing Alevel maths, why not check out the Casio FXCG 100, the perfect tool for mastering A-level maths. There's a link in the description where you can find out more. I want to thank you for watching this video. Please consider subscribing because it helps out my channel. And also if you could like, comment or share the