in this video I'll be taking you through an AQA GCSE higher paper 2 maths exam you are allowed to use a calculator in this one so feel free to I recommend trying each question first before I do it and then I will provide some model solutions that you can go through once you've attempted it thank you for watching and I hope you enjoy question number one write 30 to 12 in the form n to1 now as this is a calculator paper straight away we know that the right hand side has to be one so let's do 30 / 12 and that is going to give us 2.5 and that is going to be our final answer for the first one four consecutive triangular numbers are six 10 15 21 what is the next triangular number you can see add four add five add six next one will be add 7 so 28 write down the reciprocal of 4 over 7 reciprocal just means flip it so 7 over4 the price of a toy increases by 12.5% to £1 1953 work out the original price of the toy now important with this originally it was 100% so now that it's had the increase it's 112.5 and that is now equal to 19953 so if we want 100% we could first of all work out 1% and do 19.53 divided by 112.5 and that is going to give us 0.1 736 and then if we multiply that by 100 we're going to get £736 and that is our final answer Jess saves 2p 5p and 10p coins she has 45 10p coins eight times as many 2p coins as 10p coins £770 in total work out the total value of the two P coins to the total value of the 5 P coins okay so let's just work down the bullet points 45 * by 10p is going to be £450 bullet point number two8 times as many 2p coins as 10p coins so if she has 45 10p coins multiply that by eight and you're going to get 362 P coins and we can see that that is going to be £720 again put that in your calculator if you're not sure and the third bullet point we're told is £770 in total so let's write down our total £770 and we're going to to take away the total of these two which is £170 leaving us with £6 and that must be for the five PS so the total value of the 2p coins £720 the total value of the 5 P coins is £6 so we can multiply these two numbers by 10 giving us 72 to 60 and dividing by 12 we're going to get 6 to 5 because it does say simplest form always look out for that part of a regular polygon is shown assume that the polygon is an octagon work out the size of an exterior angle something to remember the external angles of a regular polygon will always add up to 360 so we know this is an octagon so there's eight of them so 8 * some angle is equal to 360 so 360 / 8 is going to give us that angle and that angle is 45° in fact the polygon has more sides than an octagon what does this mean about the size of an exterior angle so because there's more sides when we divide 360 by a number that's bigger than eight it's going to give a smaller angle so it will be less than the answer to part A in a game an ordinary Fair six-sided dice is rolled the fair spinner shown is spun the scor is the dice number substituted into the spinner expression complete the table to show all of the possible scores so if you get one on the dice and 2 X on the triangle that's going to give us two likewise for two three and five and six they are all just going to be doubled so 10 and 12 in there 3x is going to be a very similar idea 3 6 9 12 15 and 18 and then for the squared one we're going to have 1 4 9 16 25 and 36 if you want to do any workings feel free to use this space a player wins the game if their score is 10 or more work out the probability so going back to this you can see that 10 or more is going to be those areas there so that is eight out of a possible 18 and we could simplify that if you wanted to which is 4 over9 the game is played 711 times estimate the number of games that are won so using our probability from before 700 Times by 4 9ths because they win 4 9ths of the time is going to equal 316 now in the past I've said if the question says estimate in it typically you'd want to round something like this down to 700 but in this case because it's a calculated paper none of this stuff has to be done in your head so you can give a more accurate result the reason it is estimate is because based off the sample of games that we did there was a 49th chance of winning does not mean there is always going to be a 49th chance of winning so another sample could happen and this fraction could be completely different which is obviously going to give a different score than 316 here so that is why the word estimate is used there and I don't want you to round the number at all question number eight this is going to be about comparing coefficients work out the values of A and B so what I would recommend with these is expand it out first we've got ax s if we use the kind of claw method - 3x^2 + 2 B = 5x^2 + 12 now straight away the easiest thing to do here is ignore the X values and just focus straight on the numbers so we've got 12 on one side and 2 B on the other side that 12 cannot be made up of any other elements on the left hand side because these both contain the x s values so because of that b must equal six because 2 * 6 makes that 12 so now if we just look at the X2 terms a - 3 must equal this five so by rearranging we can add the three onto the other side and a is equal to eight so that is your final two values there 1 3 and 29 are points on a centimeter grid a b CD is a parallelogram a d and BC are horizontal and each has a length 5 cm the diagonals of ab c d cross at Point e and we need to work out the two possible pairs of e so we could have C and D to the right hand side of these points so 1 2 3 4 five there so that could be c 1 2 3 4 and five there that would be D so if we connect up these with the blue parallelogram and then we can also create another parallelogram with c and d mirrored onto the other side so C and and D could also be in that position now if we draw those diagonals on you can see that we have our two points of e right there so our two points for E could be Min -1 and 6 or it could also be 4 and six now I believe there could be one more answer to this Solution please let me know down below what you think that could be to see if you're on the same wavelength as me question number 10 write down the translation Vector that mat a onto b a translation Vector will tell us how many movements to the side and how many up or down we have to go to get from one shape to the other in this case we pick a point we're going to pick the top right point of the triangle as you can see we've gone four to the right and three down so to the right is positive on the horizontal and up is positive on the vertical so because we're going down it's going to be a minus three on the bottom the volume of a sphere is 43k R cubed a bowl is a hemisphere with radius 12 water is poured into a bowl at a rate of 325 cm cubed per second for 8 seconds does the water fill More than 70% of the bow so first of all find the volume of the hemisphere because the radius is 12 we are simply doing 4 over3 * < * 12 cubed and that is equal to 2304 piun we want to half that because it's a hemisphere and that gives us 1152 Pi I would probably work out 70% of that now so we can multiply by 0.7 in our calculator and that gives us 23338 always useful to remember the decimals in case it is very fractionally over or under so if the water is poured at 325 cubic cm/s for 8 Seconds that's going to be 325 multiplied by 8 and we get 2,600 cm cubed of water does it fill More than 70% of the bowl the 2600 is bigger than the 70% so yes question number 12 show that these two rectangles are similar for something to be similar you want a consistent scale factor throughout all the dimensions on the shape so you can see here to get between 5 and 8 we multiply by 1.6 you can check that on your calculator by doing 8 ID 5 or 5 * 1.6 and you will get eight what we also want to check is what is 12 times by something to get to 19.2 so likewise we can do 19.2 divid 12 and that is also 1.6 so it'll probably be worth just adding a sentence there to say because both the scale factors are the same the rectangles are similar a factory packs X boxes of tea bags per hour each box contains 80 tea bags show that the factory packs 4X over 3 teabags per minute so one thing I will say these questions can be incredibly cryptic to try and work out let's say that that X is three so three boxes per hour and each box contains 80 te bags so if we do 3 * 80 that is 240 t bags per hour if we now say that it's five boxes per hour instead of three that's going to be five boxes per hour or 400 tags per hour the point I'm trying to say is whatever X is the number of tags per hour is 80 times whatever that X is so we can say 80x tags per hour and this can be applied to every single question that is like this because I have seen 90% of them and I know it works now the question is asking for teabags per minute so how do we get from per hour to per minute we divide by 60 so 80x over 60 tags per minute and finally to get from there to here we simply simplify the fraction you could even put this straight in your calculator 80 ID 60 and it will give you 4 over3 so you don't necessarily need to be good at simplifying the fractions it will help because you can see you can quickly knock off both the zeros 8 over 6 divide that by two becomes 4 over3 but yes that is how to answer a very difficult sounding question in a fairly straightforward way question number 14 a company has 123 employees information about their hourly rates of pay is shown in the table so here we just have a grouped frequency table the owner of the company uses the data to make two statements over 30% of employees have an hourly rate that is more than £17 the average hourly rate of pay is more than £20 show working that supports statement a we can say for sure that all of these people have an hourly rate more than 17 so that is 25 people out of 123 if we put that in our calculator that gives us 02032 5 which is the same as 20.3% now clearly that is not above 30% however what we could also do is estimate within our group to try and make an educated guess at how many of those 32 people in this band are on more than7 so 17 is slap bang in the middle of that range so we could guess again this is all an educated guess that 16 of those people could be on 17 or more because that is in the upper half of that band so if all of a sudden we now add the 16 and the 25 together we'd have 41 over 123 and that gives us 33.3% you will see if you put that in your calculator and that is working that support statement a it could do it might not be true but from our educated guest we can say that that could be a possibility 14b why might statement a not be true the facts that I literally just said so we took an educated guess that half of those people in that group were paid more than £17 so that might not be true and they might all be paid at the lower end say 145 an hour instead that is exactly what you would write in that area work out an estimate to the mean to support statement B this is typically where a lot of your marks will come from now grouped frequency means are difficult a lot of people struggle with them so what I'm going to do is take a little screenshot of that table just and transfer it to the other page so we can do some workings on it so here we have our table and what we want to do here is create a new column called midpoint what we're effectively doing here with the midpoint is taking that educated guess at the average hourly rate within each of those bands so we're going to take the midpoint 12 17 as we said 30 and that one will be 70 what we now want to do is again take an estimate of the total sum of all the people in that hourly R range and we can do that by doing 66 * 12 and the same for each of these rows so frequency times the midpoint this is equivalent of adding up the numbers in a sequence when you'd normally work out the mean but because this sequence is 123 numbers long this is a much quicker way of doing it so I'm just going to quickly put those into the calculator work out what each of those numbers are and then if we add them all up we get 2 486 and then finally just like we do with normal mean once you've added all the numbers up you divide them by how many numbers there are and again as I said there are 123 numbers so divide those two and we will get 2.21 as an average now statement B stated the average hourly rate is more than2 we have done workings to show that that is true why is the mean not the best average to represent the data typically the reason for an answer like this is because a lot of the data is skewed towards one end and what I mean by that is that a lot of the people that work for this company are down the lower end and there are very few people that are on a much higher salary as you can see people are earning up to £1 an hour and some people are down at £10 an hour so a better representation typically is the median because it captures those things like where the data is skewed towards one end so again in an actual exam put something more descriptive but just say something along the lines of median would be better because the data is disproportionate and not based around the center really a skew to one side expand this double bracket so using our normal claw method something like this we can say 2x Cub is the first one plus 5 x^ 2 y - 18 x^2 Y and then - 9x y * 5 y first of all going to be a negative 9 * 5 45 x y * Y is going to be x y^ 2 and then if we simplify all that which I guess you don't necessarily have to do but it is two Mark so they might expect you to simplify anyway 2x cubed these two have the same letters so we can combine them positive 5 take away 18 is going to be -3 and then - 45x y^ 2 that's our final answer number 16 line a has equation Y = ax - 1 passes through the point 713 and line B we have 5 y - 3x = 4 show that line a has a greater gradient so the gradient of a is that a value that is missing currently however if we know the point that passes through it we can work out what a is and the reason we can do that is because remember with every single coordinate you're given that has officially given you an X and A Y value that you can use within the equation and because it lies on that line the equation must be true once those points have been subbed in so our y value is going to become 13 and that is equal to ATI 7 take away 1 so 13 = 7 a - 1 if we add that one to the other side we get 14 = 7 a and therefore a is going to equal 2 because we can divide the 14 by the 7 to get two therefore the gradient of a is two we just need to now work out officially what the gradient of B is so 5 y this is B 5 y - 3x = 4 remember the equation of a line should be in the form y = mx + C so we just need to rearrange this to get y as the subject again a very common skill to do especially in those higher papers so take a note of this we want to move the 3x to the other side and then divide by 5 so 3 over 5x + 4 over 5 now the gradient specifically the gradient is what we're after is the number in front of the X and in this case a has a gradient of two we decided B in this case has a gradient of three over five which you can check in your calculator is the same as 0.6 and because two is bigger we have then proven the statement question number 17 work out the size of angle X so first thing to note about this question is it is effectively two right angle triangles that have been put together and whenever you see right angle triangles that involve angles your first thought should always be trigonometry or SOA TOA people remember it in different ways first of all we look at triangle ACD and we want to work out this middle line here so labeling our points if the angle is here this would be the opposite because it's opposite the angle leaving us with h the hypotenuse and a the adjacent so to in this case we have C because we have the A and the H involved in the question that means we are using C so C of 37 is equal to the a over the H our a is our X here and our H is our four rearranging this equation multiplying both sides by the four to get the X just on its own on the right hand side x is going to be equal to 4 cos 37 you can put that in your calculator if you want to but I'm going to keep it as it is at the moment because that will give us the most accurate answer possible so this length here is 4 cos 37 so we now know this side and we also know the hypotenuse on triangle ABC now completely ignore that triangle now we're completely done with that one so if we just look at this new triangle you can now see that the side that was the adjacent before is now the opposite so we want to rename this one to be the opposite and that means we have sin because it's O and H so so so triangle B we'll call it because this one can be a so now for triangle B I'm actually going to call the angle y instead because X is the side that we've named so we're going to say s of Y is equal to the opposite over the hypotenuse which is X over 9.3 now to do the inverse of a sign to get rid of it effectively we do s the minus one on our calculator and that is going to leave us with s Theus one of X over 9.3 we know the X is 4 cos 37 so again if you wanted to write that as a decimal work it out first absolutely fine but I'm just going to put in my calculator 4 cos 37 divide it by 9.3 and then finally do sin Theus one of that and our y angle is 2.09 de now just to be picky make sure you rename it afterwards just so that they know so I am going to go back to calling it X just so we don't confuse anything with the examiners and that is our final so four marks for that just for in theory just some basic trig which is exactly the same as stuff you would learn in a classroom or online through other resources you might use rearrange Y = X + 8 /x to make X the subject this if you want to get those High grades this comes up a lot and there's one method to solve it every single time the distinctive thing we have here is an x on the top and the bottom of a fraction and watch closely with these steps I will try and explain it as well as I can first first of all get rid of the fraction and to do that we can multiply both sides by X so we're going to have YX = x + 8 now if there were more numbers involved in this this is a very typical looking equation where we've got x's on both sides whenever we have x's on both sides we just want to move both the X's to the same side and move all the numbers to the other side to do that we're just going to Simply take away X and that is going to move the X onto the left hand side so we're going to have YX - x equal 8 now this is the trick that a lot of people don't see to remove the 2 X's because we can't add or subtract these together because there's no number behind that y so we can factorize that is the key step to making two x's become one if we factorize that we're going to be left with X braet Y -1 = 8 and all of a sudden because these two things are being multiplied together with the brackets we divide them to leave us with x = 8 over y - one and that is the final answer to that question three marks again I promise you that kind of question does come up a lot and it is the same method every time multiply to get rid of the fraction take away or add to move them both onto the same side factorize and then redivide them here are the first four terms of a quadratic sequence work out an expression for the MTH term of the sequence straight away write it out on the top line work out the difference the first difference you can see 17 27 and 37 second difference 10 10 now with a quadratic we now half that number and that's going to be the coefficient of our N2 value what we then do is write out the 5 n 2 sequence so for the first term we Sub in one into there so 1 squ is 1 multipli by five is going to be five 2^ 2 four multiply that by five 20 3 S is 9 times that by 5 is 45 and then 4 S is 16 times that by 5 and you get 80 next we want to compare the difference between these two numbers on every single column I guess 3 takeway 5 is minus 2 20 takeway 20 0 2 and four another thing that people get wrong about quadratic MTH term is once you've done that first bit forget about the rest of the question and just do nth term again EX exactly the same as basic MTH term work out the difference they are going up by two therefore it is 2 N underneath that just like before we write out now the two * table 2 4 6 8 and we take them away from each other again - 2 - 2 - 4 0 - 4 - 4 and you can see you will get min -4 for the rest of them and that is our final term so 5 n^ 2 + 2 N - 4 is our final answer p q and R are points on a Circle S is a point inside Triangle pqr assume that s is the center of the circle work out the size of angle X this comes under one of our Circle theorems whatever the angle in the center is the angle at the circumference is going to be half of that so 130 / 2 will give us 65° in fact the center of the circle is on PS but not at s what does this mean about the size of the angle it would mean that angle X is bigger than part X because if the center is somewhere here the angle between there and there would therefore be bigger than 130 because it's been stretched out more which would mean that this angle would therefore be bigger as well for a different Circle AB is a tangent at a c and d are on the circumference and AC equals CD here is Simon's method to work out the size of angle y so angle a d c must be 70 therefore y equal 70 is he correct now straight away using the alternate segment theorem we know that angle a c d will be 70 not angle a DC if you're not too sure about the alternate segment theorem I've made videos on circles theorems as well so have a look at those now because of that this is an isos triangle so if we do 180 take away the 70 that will give us 110 and then 110 divided 2 is 55° so Y Must Be 55° and that is our final answer so you would put in there Simon is wrong Manana decides to put £500 into an account that pays compound interest she wants to have at least 560 in the account after 3 years work out to one decimal place the minimum interest rate that she needs so the equation for compound interest is the initial investment 500 multiplied by 1 plus the interest rate divided by 100 to the power of the number of years in this case three and that must be at least 560 so if we do a greater than or equal to we can solve it just like a normal inequality so straight away we want to divide by 500 which gives us 1.12 next we will be Cube rooting to get rid of the cubed which is 1.035 then if we take away one and then finally multiply that by 100 to get our R and to one decimal place the minimum ual interest rate she needs would be 3.9% an approximate value of a root of an equation x can be found using the iterative formula the starting value is X1 = 4 work out the values of X2 and X3 straight away type four into your calculator and press equals that saves it in your ants button this is going to be very helpful at saving you a lot of time with these next you want to type in this equation to your your calculator but wherever you see xn write your ants button in instead so I'm just going to do that quickly and then if I press equals it will give me that the next value which is our x2 in this case is 4102 if I round it to three decimal places the benefit of making the four your ANS button is you can simply just press ANS again because it will override the ANS button to now be your new X2 which you can then use to work out X3 and that will give you 4.1 7 8 and those are your two X2 and X3 values by continuing the iteration show that the value of x is more than 425 so the way this works if you continue pressing equals the value in some cases will converge and settle on a certain value so if I keep pressing it and I've probably pressed the equals button about 20 30 times there but X settles to about 4.39 and it might be worth saying that kind of X4 you work out put what that is X5 put what that is and then you can say that something around x30 will be equal to your 4.39 it is only one mark So if you just simply State what the answer is then that is probably okay as well obviously clarify that that is bigger than 4.25 here are three sets of cards in a game a player has two options pick two cards from set a or pick one from B and one from C the cards are pi picked at random the player wins if the total of their two cards is exactly 10 which option gives a better chance of winning so the way I'm thinking of doing this is if you actually look at the numbers in set a which is option one if you get a one then you can't do anything with that because there's no nine likewise if you get three you can't do anything with that at the same time you can't do anything with a six or an eight because there is no four and there is no two so that means the only way you can win by picking option one is if you get a five and then another five now the chance of that happening there are eight in total and we need one of those fives so that has to be three out of eight and then assuming you've picked a five that leaves you with another two out of seven which is going to give you a chance in total of 6 over 56 or 3 over 28 that is the chance of you winning with option one if we do a similar thing for option two again you can see that in set C we only have 3 four five five and six as an option so if we choose the one the two the eight or the nine we cannot do anything with that so based off that in option two we can only win if we get a four in set b or a six now the chance of getting of four is going to be one in seven and likewise the chance of six is also going to be one in seven if we get that four we then need to get the six on set C which is a quarter if we get the six from set B we need the four in set C which again is a quarter so the probability of those two independently happening is 1 over 28 because 7th times a quarter and because that is two options they will add together to give us 1 in 14 that is your probability of winning with option two So based off of that if we were to make this over 28 you can see that that is 2 over 28 so 3 over 28 is going to be bigger than that and therefore going to be the one that you want to go for so option one gives you a better chance of winning 24 a equal 65 to the nearest integer b equals 30 to one significant figure work out the upper Bound for 2 a 2 minus b^ 2 so save yourself a lot of time with this stuff work out what you need from each of these whether you need the upper bound of one and the lower bound of the other or vice versa or both upper both lower that kind of thing now for a because this is going to be our Benchmark and we want the biggest value possible we want the biggest value of a so if it's 65 to the nearest integer that means that our upper bound would be 65.5 and just for the practice the lower bound would be 64.5 but B because we're taking away B we want B to be as small as possible because we want to keep the overall value as big as possible so 30 to one significant figure one significant figure in this case is 10 so the upper bound would be 35 and the lower bound would be 25 so as I said before we want to take the upper bound of a and the lower bound of B so two multiplied by 65.5 s take away 25 SAR is equal to 7,955 point5 that is the upper bound here show The X - 5 / x - 2 + x + 5/ x + 2 simplifies 2 this where A and B are integers so let's write it out now when we're adding fractions we want to find a common denominator the best way to do that is multiply the two bottom denominators together so now that we've done that we can cross multiply to find out what our numerators are x - 5 x + 2 + x + 5 x - 2 we can simplify this now the bottom is kind of like a difference of two squares format so that's just going to be x^2 - 4 and the two numerators we can expand out fully and you should get this when you expand them out if we now combine the numerators together we get 2x^2 the 3 X's will actually cancel out with each other and we're going to get - 20 over x^2 - 4 and as you can see that is our final answer because it fits this format where a is 2 and b equal 20 question number 26 here is a sketch of Y = x^2 the minimum point point of Y = x^2 is 0 0 write down the coordinates of the minimum point Y = x^2 + 2 if we have y = f ofx what they're asking here is a transformation which is going to be equivalent to F ofx + 2 with the plus two on the outside now that represents a movement of two spaces upwards so the new minimum point is going to be at 02 the graph is reflected in the x-axis what is the equation of the graph after this transformation reflected in the x-axis is going to mirror it underneath like that so the Y x^2 is going to become y = - x^2 and that is equivalent to a minus on the outside F ofx like that y = x^2 is now transformed to give Y = X + 3 2 describe fully this single transformation this one again if y equal F ofx to start with the new transformed version is F ofx + 3 because they've taken that X and just replaced it with the bracket that is equivalent to three spaces to the left remember if it's inside the bracket it's normally the opposite to what you would think because typically on a horizontal plane we associate positive with right but this time it is to the left and that sums up the end of the paper so well done if you've made it this far please let me know down below what you got how you found the paper any questions you might have always have happy to help thank you very much for watching and best of luck