Tuesday, June 28, 2022

# Oxford University Math Professor takes High School GCSE Maths Exam

Hello maths fans i'm back doing another exam but this time with a twist i'm joined by ben sparks who many of you will no doubt recognize from his excellent number file videos and the plan this time is going to be for both of us to do a gcse higher maths exam and then we're going to swap scripts and.

Mark each other's work it should be a lot of fun the paper we are going to be doing is the aqa gcse higher maths exam this is the 2019 version and this is paper one which fortunately for me is non-calculator but those of you not familiar with the uk system.

This would be taken by 16 year olds just as they were ready to leave high school there are a total of 80 marks available and the time allowed is 1 hour and 30 minutes let's go okay question one two right angled triangles six ten y.

Okay so these are similar triangles we've got ten increasing to fifteen because they have a sit they have the same angle x 10 is increasing to 15 1.5 times 10 6 must increase to 9. so question 1 the answer there is 9. good start question 2.

Work out the value of one and two thirds squared so one and two thirds that's three thirds that's five thirds five over three squared square both numbers that's going to be 25 over nine hopefully that's an answer oh i have to simplify it uh so i have nines that's two nines two and seven ninths.

Is that one okay good question three work out the arc length in meters of a semi-circle of radius 6. okay so a semi-circle of radius 6 and what's the arc length so if r is equal to 6 i know the total circumference.

Is pi times the diameter which is going to be 12 pi because damage of course is twice the radius so the total succumbers is 12 pi so for the semicircle arc length i think it wants half of that which would be six pi so i'm assuming it means just the curved.

Bit when it says arc length as opposed to perimeter but i'm gonna go with uh six pi and that is an answer so i'm pretty sure that one's six pi okay question four circle the fraction that is equivalent to four point six two five ooh so i know point five is a half point one.

Two five is an eight so it's five eight so that's four and five eighths okay and it wants me to convert that into eighths or quarters so four lots of 8 32 so it's 37 over eight which is there so i'm going with that seven over eight okay it's going well so far question five.

Already right what we got no right no point oh this is probably this is probably where i'm gonna go wrong now right no point no no no no no no nine seven in standard for i'm not even sure i know what standard form means i'm thinking it means something point.

Something times ten to a power so what have we got then so that's going to be 9.7 i think times 10 so to the minus 1 2 3 to the minus 4 i think because it's going to be that so if i had nine point seven to the minus four.

So it'd be naught point shift it one maybe 0.97 should it be there then it goes to there then it gets to that yes so i think it's nine point seven i have ten to the minus four i'm going with that assuming i know what standard form is okay and that was five a all right there's a.

Five b now oh work out 3 times 10 to the 5 divided by 4 times 10 to the 3. give your answer as an odd ordinary number i think that just means simplify it as a fraction i assume.

Seeing as this is non-calculator so 3 times 10 to the 5 that is um so 10 to the 3 would be 3 000 yes so 30 that's 300 000 okay so it's 300 000 i think yes and then four times ten to the three divided by four thousand.

Okay so hundred thousand divided by four thousands let's cancel the zeros so that's 300 divided by four which is 75 so it isn't a fraction i guess 75 is an ordinary number whatever that means okay i go from 75. question six anna plays a game with an ordinary fair dice oh you have to think of it more for.

This one if she rolls one she wins if she rolls two or three she loses okay so let's write this down question six okay so one equals win two or three equals lose and four five or six she rolls again okay so this is a fun game not much going on where anna lives.

What when she has to roll again oh god so okay so and roll again and if she rolls again she rolls an odd number she wins or an even number she loses so odd equals win even equals i'm really glad the question now.

Actually says draw a tree diagram because that's exactly what i was going to do anyway to figure this all out so this is perfect okay so i'm going to draw this out myself so first roll we've got um if she gets a one and that has a probability of one out of six six possibilities on the dice excellent and it's game over she wins.

Perfect then she could roll a two or a three so that's two out of the six possibilities that's one third and then a half of rolling four five or six i agree with that and then finally it wants me so after a half and she's gonna roll again nope i've written that out wrong oh get.

Your diagram right so she's rolling a four five or six with probability a half and then from there we've got a half to get odd or a half to get even okay fine i'm happy with that.

Oh wow here we go is that more likely to win or lose you must work out the probability that she wins right we can do this so what do we know here let's use a different color so i know that if she rolls a one she wins so this is one six so this is a win.

And the probability of that win is one out of six she gets two or three immediately she loses so the probability of that loss is one-third now the other one so she's got a half and then a half.

What did she do if she got odd i wrote at the top odd is a win so this is a win again so this is p of a win and that's going to be a half times a half because you multiply them so that's one quarter and then it's a half times half that's gonna be another one quarter probability of this loss.

And now i'm going to check i've done it right so they should add up to one always check this on your tree diagrams so quarter quarter is a half plus a third that's so that's three out of six plus two out of six plus one six perfect total is one okay i'm happy with that so is it animal likely to win or lose.

So the total probability of winning is going to be the two added together so that's gonna be a sixth plus a quarter which i'm gonna have to work in twelfths um so six plus a quarter is two out of twelve plus three out of twelve so i think that's.

Five twelfths which means the probability of a loss should therefore be 7 out of 12 i'm going to do it properly to make sure i'm right so a quarter is 3 out of 12 plus the third is 4 out of 12 so it is 7 out of 12. so she is more likely to lose.

So more likely to lose so let's write that more likely to lose and the probability of winning is that 5 12. all right question seven three friends arrive at a party okay fun.

So three friends are arriving their arrival increases the number of people at the party by 20 percent okay so what do we know increases the number so x is the unknown people okay so x equals original size.

And we know that it's going to increase that by 20 so that means that uh x over five which would be naught point two of x or one fifth of x twenty percent of x one 1.5 of x sorry 0.2 of x or x over 5 is equal to 3 i think so i think x is 15. so that.

Means x is 15. so and then the three have arrived so i think it's i think originally there were 15 and now it's gone to 18. now let's check that makes sense so how many people are now at the party so i think now equals 18 people i think originally.

Equals 15 people because then you could say 20 percent of 15 is one fifth is three and three arrived and increased by 20 yes i would agree with that okay so how many people are now at the party.

18. question eight work out the value so it's awkward all right we can do this uh question today this is basically just me doing a calculator paper last time but now i can't use what because i couldn't use one even when i was allowed uh right work out the value of 3 to the.

12 divided by 3 to the 5 and then we're going to multi then we're going to divide that whole thing off divided by 3 squared times 3. so 3 squared times another 3 is going to be 3 cubed right so what we've really got here then is 3 to the 12 over 3 to the 5 times 3 to the 3 which is 3 to the 12.

Over 3 to the 8 which is 3 to the 4. okay probably wants it in numbers doesn't it so let's do that so 3 to the 4 3 squared is 9. so 3 to the 4 is equal to 81. so the answer is 81. all right question 9. a shaded semicircle diagram and a half.

Actually circle is inside a circle as shown okay i have to copy this out then aren't i so what have we got the shaded semicircle right so we've got a circle and then and then that's eight across there so that one is eight centimeters so that means the inside one has a.

Radius of four it's going to no doubt be helpful and the big one has a radius of 10 right okay okay how many times bigger is the unshaded area than the shaded right so let's work out the total of the outside one so the big one so big circle.

Area is pi times the radius squared so that's 100 pi the semicircle area is pi times the radius squared which here is 16 so that's 16 pi divided by 2 so that's 8.

Pi so what do we want to know how has bigger is the unshaded area so the shaded area okay so i think i think the shaded area is 8 pi.

So the unshaded is 92 pi so it's how much bigger is that so therefore 92 divided by eight times bigger probably simplify that that's 11 and a half.

I think 11 times 8 is 88 plus yes that's 11 and a half times bigger okay i hope meh let's move on question 10 the number of items n interesting made by in one hour by a.

Machine is in by n equals t is the time in minutes the machine takes to make one item sure the value of t changes for different types of item right on the grid below it says this is a weird question the grid below draw the graph of n equals 60 over t.

For values of t from one to four okay so it wants me an accurate and accurate plot as opposed to a sketch right well i'm going to have to do a sketch but i will label it to to make it clear so n equals 60 over t and he wants me to do this for t so n is 60 when t is 1. excellent so let's put these.

Labels on what's it labeled there one two three four so we want to do one we want to know two we want a three and we wanna know four right and then this is n this is t so n is sixty four t equals one uh then for two.

It's going to be 60 over 230. so let's try and roughly do it to scale and it's going to be at 30. for n equals 3 60 divided by 3 is 20. and then it's going to divide by four is fifteen so if i were to sketch that and obviously as t goes to zero this goes to infinity it's gonna look something like.

This i'm just gonna go over those crosses to make sure ben can read them that one i should probably add these values in as well shouldn't i okay so this is n equals 60 this is n equals 30. this is n equals 20. and n equals 15.

Okay accurate enough right good oh there is a part b okay so that's part a my graph and then for part b what does it want me to do with this graph machine takes three minutes and 30 second three minutes and 30 seconds to make a.

Single item so i've got to estimate this from the graph oh no i didn't do it accurate didn't do an accurate graph it wants me to estimate all right well oh well this is just not even remotely accurate but i can probably estimate it from the numbers.

So i want to divide by i want to do n uh is going to be equal to 60 divided by three and a half which is seven over two so we've got a seven over two on the bottom so that's 120 over seven what's 120 over seven oh it's very close to 10 would give me 70.

Plus another 49 17. so i think 17. so n is approximately 17. that's exactly what my graph says if i add this lovely 17 value to it all right i'm going with that okay uh question 11. uh ed and faye are sharing some money.

Lucky adam faye right question 11. so they're showing 330 pounds in the ratio seven to four so we need to know elevenths of it so nice one eleventh equals thirty pounds so therefore 7 11 is going to be 210 pounds and the remainder will be 120 pounds who gets the most so i think ed has the most money.

I think if i've possibly misunderstood the question but i'm going with ed has the 210 and that means that faye has 120 and then ed gives face some so they have the same so they need to end up with 165 because that would be half of the total.

So therefore fey needs 45 pounds so i think if we did 45 like that then faye now has 165 which is half of 330 and ed is going to minus 45 from there.

Will give me 165. perfect so same smile so the answer equals 45 pounds excellent very generous of ed to give faye 45 pounds.

All right sequences question 12. this should be fun he says not knowing what's in store the next term of the sequence is made by adding the previous two terms which of the sequences follows this rule okay so minus two minus nine plus two gives you minus seven minus seven plus.

Two gives you minus five minus seven minus five using minus twelve so the first one so the answer is minus nine 2 minus 7 minus 5 minus 12. and let's just check the others fail minus 3 plus 5 would not be minus 2 that one fails 0 minus 3 would be minus 3 but then do.

Them again give you -6 minus one minus another one gives you minus two minus one gives you minus three but then minus three minus another two would not give you one all right yeah first one good first one to check okay question thirteen the triangular cross section of prism is.

An isos of isosceles right angle triangle great the volume is this use approximations to estimate the value of x or i don't know if i know the volume of a prism all right figure this out okay so all your prism.

Is going to be the cross-sectional area to the area of the triangle multiplied by the height so let's figure out the area of the triangle right so area of the triangle okay um and what do we know about the triangle it's half the base times the.

Height it's got a right angle triangle so the area is a half the base is x so it's half x squared okay so it's a half x squared multiplied by 8.14 has to give me the volume so therefore i can say 102 centimeters cubed the total volume.

Must therefore be the area of the triangle times the height which is 8.14 right so that's going to mean if i tidy this all up 102 has to equal 4.07 x squared not really sure how i meant to estimate.

This but let's try it so i want to do x squared is therefore 102 divided by 4.06 which means x is equal to the square root of that number so i think i think the idea is if i did 100 divided by 4 i get 25 and then x will be 5. so i'm guessing that's my approximation so if i say.

Therefore x is approximately the square root of 100 over 4 then that's the square root of 25 so that tells me that x is approximately 5 centimeters all right i'm happy with that question 14. i these are going down like skittles this is great i love this question 14. here.

Is a quadrilateral weird shape all right and i'm told some values okay i'm going to draw a diagram then so let's draw this funky looking thing and then include these values that they are giving me right okay.

A is going to be 90 degrees right so let's put the right angle in a to b are in the ratio five to three okay fine okay for him being the ratio five to three this is b so if a is 90 degrees um then.

What do i know i need to know what b is so we know that a b are in this ratio 5 to 3 which means that's whoops that's 30 5 to 3 which means 90 so 5 to get to 90 is 18 so 3 times 18 is 54 degrees.

So b therefore is 54 degrees okay so b is 54 degrees does that work ratio of five to let's just double check 5 times 18 is 90 3 times 18 is 36 it's 54 yes right so b is 54 degrees x and y are in this ratio.

1 to 3 show that b equals x ah ok so x and y so this is x and this is y okay so it's a quadrilateral so i know that the total angle has to be 360 degrees i think i think that's a fact yeah i'm going to stick with that i.

Think the total angle is 360. right so let's write that down so i think the total needs to be i'm pretty sure that's right i haven't even thought about angles in years okay i think the total angle has to be 360.

And i know a and b and x and y are in this ratio of one to three so y is clearly three times x okay so y equals three x so what that means is the two i figured out a plus b which is one four four therefore one four four.

Plus x plus y has to equal 360 but that means one four four plus four x equals 360 which is going to mean that 4x has to be 360 minus 144 is going to be 216. so therefore x should be 54 degrees and b equals 54 degrees so x equals b.

Exclamation marks all right assuming i'm pretty sure a quadrilateral based on a quadrilateral needing to have a total lag of 360. i think i think i've got that one all right well we're going on to the next question regardless okay question what are we on.

15. here is some information about the test marks the 102 i love how they put in these like teacher-based marketing questions here is information about the test marks 120 students right complete the cumulative frequency table okay i can do this so.

You just add them up so question 15 a so let's write this so we've got mark and then we're going to write it as cumulative frequency and draw a little table right so when m is less than or equal to.

10 then we want 20 up to 50. okay plus or equal to 20 um less than or equal to 30 and less than or equal to 40. and less than or equal to 50. right and then let's add in those columns so if i just add them up so we've already got that that's 20.

Plus 28 gives me 48 plus 40 gives me 88 plus another 20 is 108 and n plus 12 gives me 120 as it has to be because there's a total 120 students good so i think that's okay oh god draw another graph of course i've got to draw another graph without having the axes on my.

Doublet right draw a cumulative frequency graph so i'm going to attempt to draw this as accurately as i possibly can um sorry ben you'll have to just take my word for this um so uh mark up to 50 okay so we're gonna.

Have naught let's then go ten that's about right 20 30 40 and 50. and then that's going to be the cumulative frequency q mu relative.

Frequency down the y-axis all right then so what are we saying then this is mark m right so at 10 we know that it's 20 and the total is 120 so let's add in some scale so if that's 120 that's going to be 60. that's going to be 30 which means that's 10.

20 and i have to add all of these on 40 50. this is where i need that the little spongebob thing that pops up and says a little while litter all right 17 80 uh.

90 100 on 10. okay just about read that right so m equal to 10 has a value of 20. so at 10 the value is 20. put a little cross let's do a cross across there.

At 20 the value is 48 so that's up near 50. at 30 it's 88 so that's up there 90. saying that's about there he says 40 108 oh these are tricky numbers it's like up here and then 50 we're at 120 so that's about here.

So i'm guessing i meant to draw this in all right i'm gonna go with it so if i try and draw the graph it's gonna look something kind of coming up and then it's gonna flatten out at the top all right and then let's just go over those.

X's so that ben can see them right that's the best graph i can give you sorry then okay hopefully no there's more i thought we were done with that one okay students who scored 15 marks of fewer take another test user have to.

Estimate how many students of course i'm estimating from a sketch not drawn accurately i need square paper all right let's try and do this so you scored 50 marks or fewer take another test so let's find the midpoint of this we're going to go up and then we're going to go across.

And this is going to be terribly terribly inaccurate now let's just look at those numbers so we're going from 20 to 48 so it's going to be somewhere between the two it's kind of increasing steepest at this section not quite yet steepest it's steeper on.

The next section it's going to be somewhere around 40. i think it's somewhere around 40 if i'm gonna have a complete guess it's definitely more than half is the point because it's steeper because we've gone from 20 in that first section to 28 in a second so the gradient's like increased.

I'm going to say approximately students need to do another test sad face because that's quite a lot of students plus all right thankfully we're now done with question 15. did not enjoy that one right 16. simplify oh this looks fun i.

Love some algebra 4x minus 8x squared divided by 12x minus 6. okay so factorize out the top we've got i can take out 4x and then i'm left with a 1 minus that's going to be 2x and then on the bottom.

I can't take out 4x whoops well i've committed this so i can take out there a what can i take out a 6. so if i take out a 6 then i've got 2x minus 1. ah it worked we're good because now i can say i've got cancel those as long as i have the minus on the bottom so minus 4x over 6. all right minus 4x over 6. yes.

I'm going with that right toby is forming and solving equations good on you toby the the product of oh okay think about this the product of half a number let's write this down the product of half of a number and three more than a number.

Is the same as the square of the number toby uses y i was going to use x damn you toby toby uses y to represent the number okay so the product of half of a number so y over two is half the number and three more than the number multiplies it so that gives me the product of half of.

Y and three more yeah it's the same as a square so that's equals y squared fab all right i'm going with that that's an equation toby could form toby forms another equation of course he does so that's part a and now he's obviously using x.

Which says x is nine over eight x okay x equals nine divided by eight x right what are we doing toby he wants to work out the values of x here is working oh gotta find an error literally marking again they love this examiners are clearly teachers and they clearly don't like marking i.

Want to make the students experience marky he wants to work out the values of x here is his working okay so x equals 9 over 8x so if i multiply both sides how would i solve this let's i'm going to solve it and then compare mine with toby's so what i would do is multiply up.

And say 8x squared is equal to 9 i then say x squared equals 9 over 8 and then i'd say x is plus or minus 3 divided by the square root of 8 which is 2 root 2. that's what i would say ah so he's not done that so he said 8x squared equals 9. and then what he's done so toby.

Has square rooted the nine and the x squared but not the eight sorry toby he's his error there was he said 8x squared is 9 and then he's square rooted.

To get plus or minus 3 but the left-hand side needs to be root 8 times x okay good number 18. here is an identity right well let's write that down question 18. here's an identity x squared minus y squared ah good old difference of two squared i love difference of two squares really.

Helpful remember that x plus y times x minus y use the identity to work out the value of this alrighty so what we're going to say here is let x equal 193 y is going to be 7.

Therefore i can say 193 squared minus 7 squared equals 193 plus 7 multiplied by 193 minus seven so that's equal to 200 times is the first bracket 193 minus seven is going to be 186. yep uh and he wants the answer so you.

Got to do that fine or so 200 times 186 so let's double it and then add two zeros so 186 times two would be 200 36372 he says with no conviction yes that's right three seven two and then add two zeros.

So thirty seven thousand two hundred is the answer i hope oh now factorizing right so that was a okay and then b we want to factorize 100 a squared okay good a squared minus 81 b squared nice difference of two squares again because this is just 10 a.

All squared minus 9b all squared so i'm guessing we can now use the result from above now because x is going to be 10a and y will be 9b from what we had above so i know this is equal to 10a plus 9b multiplied by 10a.

Minus 9b and i think that's it 10a plus 9b multiplied by 10a minus 9. yeah all right cool going with that question 19. circle the fraction this is equivalent to 0.1 forever hooray classic.

Right so 0.11111 is going to be 1 9 i know that but i'm guessing they want some working maybe they do maybe they don't um it's equal to 1 9 and i can show that by saying uh let x equal 0.111 forever then you can say 10x therefore is 1.11111 forever.

So 9x is equal to 1 minus the other is equal to 1. so x equals 1 9. smiley face because that's what i thought it was awesome right number 20. i'm hoping we're near the end how many questions could they have a b and c are points on the circle and cd is a diagram a diagram that i have to draw.

Less accurately than what they have though i like that all of their diagrams say not draw it accurately such a get out right let's draw the circle and let's add in these things so we've got a going out there this one then comes down and it's tangent to c that one goes across to b.

And then we're going up to a right wicket and then we've also got this diagonal okay cool so this is c this one up here is going to be d this one is a and down here we've got b and let's just draw big dots so i can make my diagram a lot more accurate.

And look like they all actually crossed especially this one right okay assume the triangle abc is isosceles that's nice of them to tell me and ac equals bc so.

That one equals that one and that's isosceles okay so a b c equals i sus sleeves so that means this angle let's call it x is the same as this angle also x and then we know this angle is 180 minus x cool i do know that the angles in a triangle.

For certain are 180 less still less convinced about the quadrilateral one i'm pretty sure it's 360 though prove that a b is parallel to dc ah and we also nice cd is a tangent so we know that that.

Angle we know the angle to the center is therefore 90 degrees but that doesn't necessarily say anything else oh oh this is tricky this one's actually tricky um all right so we want to show that a b.

Is parallel to d c so that means showing the angles are the same right and i know that one is x and that's x that's 118 minus x now what do i and don't i know the tangent thing's got to be key they wouldn't tell me.

That cd was a tangent unless it was important so what do i know about circles and tangents so i'm pretty sure i want to say there's something in my head saying that if you have a circle and a tangent.

To the circle so it just touches it i'm pretty sure that that forms a right angle with the center of the circle but we don't seem to have any centers on our diagram okay so what else can we figure out then um.

There's got to be something about that tangent oh there's got to be something about that tangent wreck your brains tom it's in too long circles and angles so cd is a tangent it's a tangent so it's touching the edge of the circle.

Okay and then it's going across to b and that's isosceles and i know the ac and bc are the same right but i don't know anything else about these angles okay hmm do anything else about those angles no i'm gonna have to call it i don't know i.

Don't know my circle theorem sorry everybody dead i'm so sorry i i don't think i know my circle theorems uh i think that they should be parallel right the diagram makes them look parallel i'm convinced that they are parallel but as i don't think i can argue it so i'm out all right we're gonna have to.

Move on you have to move on four marks as well ouch all right fingers crossed the rest of it's all right all right let's stick with part b oh we're changing it so in fact triangle abc is now going to be equilateral.

Right so i don't like this question so now they're all the same so let's do them all in orange two three one two three they're all the same that's xx so this one will also be x so i know if it's equilateral the angles are 60 degrees right so i now.

Know that x is 60 degrees and this is for part b in orange okay take the two boxes for the statements that must be correct a b is parallel to d c do i know that um i feel like that should still be true there's nothing stopping that from being.

True because it's still if it's equal lateral it's still isosceles and it was true for isosceles so yes so a b parallel to dc yep that's correct and ac by sex angle bcd maybe.

Ac bisex angle b a d a d if it's equilateral would it bisect the tangent angle or would it bisect the one in the top corner given that these are parallel so if these two lines i've drawn a terrible.

Diagram these two lines are now parallel then i want to say it's going to bisect which one i'm guessing at this point i'm really bad at angles i want to say ac will bisect it's more likely surely to be.

Which one would help if i had an accurate diagram if i was actually let's look at their diagram so if i was trying to draw this which one would be correct if it was equilateral bad right i'm going with bad it's a bit of a guess as we've seen i.

Can't do i can't do circle theorems i think ac bisex angle bisex um bad right i think they're the two correct ones really don't like that question.

Let's very swiftly and very happily move on to question 21 oh awesome this is going to make me feel better solve some simultaneous equations right here we go so two x plus three y equals five p and we're told that y equals two x plus p.

So i can rearrange that to say minus 2x plus y equals p so if i add those two equations together that tells me that 4y equals 6p so y is equal to 3 over 2 times p and therefore therefore we can say.

3 over 2 p equals 2x plus p minus p from both sides says that p over two equals two x so x is therefore p over four right and let's just check that makes sense in the other one so two x would be p over two plus three y would be 9p over 2 12p over 2.

Oh let's just check that 2x would be half of p three lots of it would be four and a half so you add four and a half ps to half p you get five p perfect so that's correct so y is three over two p x appeared before and it was so much better for four marks than that silly circle i'm not happy about that.

Put out your mind tom all right move on good exam technique we move on we ignore it we forget about it 22 right abc and ac it's another triangle question abc and acd are triangles and k is a constant show the vector cd.

Is 6a plus 4.5 b right so in that case i'm going to draw this diagram so what we got we've got a we've got b going to a with some vector we've got some other vector going over here we've got this.

One and then we've got something coming down right so we're saying that that is k times the vector a fine this is c this is going to be three lots of b sure this one is six a plus seven point five.

B going out to the point d and that's going in from a to d so if i want to go from let's do a different color if i want to go from c to d so it's asked me for i've got to travel minus equals minus 3.

And then plus a to d okay so then plus 6a plus 7.5 b so putting that all together that's going to be 6a plus 4.5 b which was the answer hooray okay good uh bcd is a straight line okay good.

Work out the value of k sure so if bc d is a straight line that means it has to have the same direction okay so therefore can i work out bc.

So let's work out b c so b to c is equal to k a uh so i go from b to a and then go from a to c so i'm going to be positive plus 3b right so b to c is that and that needs to have the same direction.

As cd so it needs to be in the same ratio of the amounts of a and b so therefore so what i can write this we need same ratio of um amounts of a and b uh so therefore.

We can say um what have we got a is or what ratio are they in so 4.5 b so a is four thirds of b so therefore k equals four because then multiply 4 by 1.5 you get your 6a multiply 3 by 1.5 and you get 4.5 b.

So k equals 4. okay i'm sure you're working yes just did right question 23 simplify eight to the fourth divided by 32 to the two-fifths sure thing uh so and he wants this in powers of two.

I was gonna do that anyway so that's 2 cubed to the power 4 divided by 32 is 2 to the 5 i think 2 cubed is 8 yes 2 to the 5 for 32 to the power of 2 fifths so this whole thing is therefore 2 to.

The 12 because you multiply the powers in the bracket over 2 squared so that equals 2 to the 10. all right d to the 10. good uh question 24. we're told that f of x equals sine of x minus 90 degrees or degrees before radians.

Circle the value of f of zero right okay so f of zero degrees so we want when x minus 90 degrees equals zero so x equals 90 degrees okay ah no my bad that's completely wrong don't do.

That thomas uh let's go back and let's actually answer the question so if f of x is that and i want to know f of zero yes so therefore if that's f of x f of 0 is sine of minus 90. okay so sine of 90 is plus one so that's minus one.

All right answer is minus one gotta be getting there question 25 i really should have looked how many questions all right more diagrams um okay p48 is a point on the circle center oh the tangent of p intersects at these.

Positions short so the gradient of the tangent is minus a half right how would i do this so what have i got i've got a circle we've got an x axis and a y axis we've got a tangent coming in like so and i know that this line is passing.

Through the point four eight great so therefore the gradient of the line um the gradient is simply the uh change in y over the change in x um do i know that yeah maybe i don't know that yet oh okay.

Um hmm think about this uh so what have i got to show so the gradient of the tangent is minus a half right um right what do i know i know the circle let's do the circle equation i think i.

Assume this is gcse that's how i'm going to do it so the circle therefore has equation right so i know that 4 8 lies on the circle and that's equation is x squared plus y squared.

Equals let's call it capital r the radius squared um i'm hoping this will help it might not so what do we know we know that x is equal to 4 so we know that 16 plus 64 equals r squared so we know the radius is therefore the square root of 80. um.

Don't know if that helps um okay uh what else can i say what else can i say so if i know a point that a line passes through it's got a negative slope based on the picture wait can we not say negative slope.

From picture and then just say because of the value it passes i'm pretty sure we can just do this so because it passes through the 0.48 therefore the change so therefore the gradient is just equal to.

The change in y over the change in x so if i were to kind of draw something down like so ah so then i know what do i know okay so let's pretend it was the positive one.

And then i know that it's minus the inverse of the positive one right so i can say okay so i'm going to say this is so not the way to do this question i'm going to say that the perpendicular gradient.

We know that right so if i know the perpendicular gradient i'm going to say that's the change in y which here is clearly eight perpendicular gradient is two eight over four which equals two and then you say the product of gradients is minus one.

Therefore the gradient we want equals minus half absolutely not how you do that question but i stand by it okay um 35b work out the length a b right ah and i worked out the radius now it's helpful to know the radius okay.

So let's try and add this to the diagram so i know these points a and b so what do we know a is at the bottom and b is at the top i know that the radius of the circle oh okay so i know the radius of the circle is.

The square root of 80 sure and what am i trying to get here i'm trying to find the length a to b more geometry questions i'm so bad at geometry questions apparently okay what do i know i want the length a b.

I know it has a gradient and i know the point it goes through right let's work out its equation let's do that okay we're going to get that and then it's a triangle problem i thought i was going to be able to do that one as well okay so what we know then is.

And this is going to be part b so 25 b we know that m the gradient of our line is minus half so if i try and do y equals mx plus c approach i know i'm going through the point p which is given by 4 8.

So i sub them in i find c so therefore y uh which is 8 equals minus a half times four plus c this will give me c so eight equals minus that so minus two plus c so c is ten right so the line i've got therefore line equation.

Is equal to y equals minus half x plus 10. so if i draw that i don't need anything to do with the circle for this if i draw that i know it's going through these two points and i know that when x is 0 y is 10.

So i know that length is 10 at y equals naught i can say that half x is 10 so i know that x equals 20. that seems big but sure so this one is 20 and i know that's a right angle so that's 20 that's 10.

So pythagoras don't think so you might do it again but i'm going for it so pythagoras tells me that 20 squared 400 plus 100 equals let's call it c squared no it's called d squared because i've already used c so d.

Is the square root of 500 and the square root of 500 is oh you want me to factor it out so d is equal to 10 root 5. all right so therefore a b equals 10 root five.

We hope all right question 26 the turning point of a graph has this x coordinate right so x is going to be x plus a squared plus b and i'm told that the x will okay the exponent is minus two when it turns oh okay i kind of want to differentiate i think.

I'm supposed to differentiate but i'm gonna totally differentiate okay so if this is my graph i therefore totally cheating here but i'm doing it it's allowed i know this diy dx is therefore going to be 2 x plus a right and then i can say from that.

Therefore turning point occurs at when two times x plus a is equal to zero so when x equals minus a okay so x equals minus a and i'm told that it has x-coordinate minus two so therefore a is equal to two right excellent so if a is two.

So i can now say y equals x plus two squared plus b and this has to pass through the point three pass through this point at 3 1.

So we're going to plug those in so 1 equals 25 3 plus 2 is 5 squared 25 plus b so b equals 24 okay and then we want to know the y coordinate of the turning point so the equation is y equals x plus 2.

Squared plus 24 and the turning point therefore occurs so and then you say turning point at x equals minus 2 which means in the equation y is equal to b which is 24. so the answer.

Is 24. question 27 i hope this isn't never ending and it's just like some kind of sick joke the questions never end okay angle x is acute so that means it's less than 90. i have to remember that okay so x is less than 90. so let's note that one down before i forget.

So x is less than or equal to 90 degrees okay and then we're told that cos of x is going to equal sine of 60 degrees times tan of 30 degrees nice okay and i need to know x all right i can work this out uh so graphs um sine of 60. so let's draw the sine graph so the.

Sine graph goes up and looks like this so 60 it's increasing so it's quite near the top but not quite at one yet so i want to say it is sine of 60 degrees what's that equal to it's not quite one but it's bigger than the 30 degree one.

I think it's root three over two i'm going to say it's root three over two okay yeah he says yeah root three over two i'm gonna say sine of sixty is root three over two tan of thirty okay so to figure out tan of thirty.

I need to know turn of thirty degrees is going to be sine of 30 over cos of 30 so that's sine of 30 again based on my graph is going to be oh sine of 30.

Is that one is that one not one over root two is that one over root two is that root two over two oh i can't remember i can't remember i can't remember i can't remember sine of 30 i think is that root 2.

No it's not is it's a half that's a half sine of 30 i think is a half he says with no conviction okay i think it's a half over cos of 30 so let's draw the graph of cos so that one comes down and then obviously it does whatever so it starts at one and when you're at 30.

Degrees you're gonna be like here so that's the big one so that's root three over two so they cancel so that's one over root three does that sound right tan of 31 over root 3 i buy that i think that's right.

So that's 1 over root 3. so therefore cos of x is equal to sine of 60 root 3 over 2 times 1 over root 3. this looks promising so that's going to equal a half so what value of x makes cos a half i think it's 60 degrees all right.

Yeah because it's the lowest yes has to be he says i'm going with it 60 degrees okay next question ah yes end of questions absolute sight for sore eyes right there 20 set.

One odd number of questions to end on 27 questions down i have finally finished my gcse maths exam well i didn't answer i think it was it question 20. the one with this i can't believe i forgot my circle theorems ben i'm so sorry and you know maybe ben forgot his circle.

Theorems i have no idea we'll find out when we mark each other's scripts so we're going to record all of the marking process and then in a follow-up video we're going to have a discussion about how we both did presenting each other with our final marks so that should be fun and also talking through maybe some of the different approaches that we used.

When trying to get our solutions so do keep your eyes peeled for the marking video as for this one thank you as always for watching please do subscribe if you've enjoyed the video and i'll see you all soon take care you

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