National 5 Maths 2026 Paper 2 | Full Solutions

追加: 2026-05-15

[00:00:00]for Scotland National Maths 2026 paper 2 full solutions. Let's go and see how we got on. Clever Math is sponsored by Leky, the educational publisher for Scotland. Today we're going to look at practice workbooks for National 5 and Higher Maths. These are just £559 when you use the discount code clever math at leiscotland.co.uk.

[00:00:19]And these are excellent because they have lots of questions topic by topic by topic. So as you're working through the year, you can revise and practice any topic you want and make sure you are comfortable with that topic. But also when it gets towards exam season, a practice exam paper right at the end to make sure you're ready for your exams.

[00:00:37]Just £599 when you use a discount code clever maths. The links are in the description and down below. Okay.

[00:00:42]Scotland National 5 math 2026 paper question one VA has valued at 22,600.

[00:00:49]It depreciated by 28% in next 3 years.

[00:00:52]find the value band after 3 years to three significant figures, but it's similar to what I thought. So, we've got 22,600 and we need to do 100 minus 28%.

[00:01:05]So, you've always got a calculator, remember? So, if you can't work it out, just get your calculator and see what it says.

[00:01:14]So, 100us 28 is 72%.

[00:01:19]Change that to a decimal. So times 0.72 and the power is 3 years.

[00:01:27]22600 * 0.72 to the^ 3 is 8435.448 8435.44 8435.448 Now three significant figures the third number I'm cutting it. So it's a pound sign. 8 4 3 goes up to four and the five becomes a zero. We stop at a decimal point and we're done there.

[00:01:55]Qualification Scotland massify math six paper two question two. In the triangle XY Z you've got XZ is 5 Y Z is 4 angle is 106. Between them calculate X Y. XY is a length. So I've got one length two length. Find the third. So it's a cosine rule because the cosine rules for three sides. So have to find that one. So start of the exam paper. We write down the cosine rule for a side which is a 2 = b^ 2 + c^ 2 - 2 b c cos a. Now some people will change them out but I don't bother. You just need to remember that the sides making up the angle are your b and your c and your angles in between.

[00:02:42]So it's five and four or four and five.

[00:02:44]It doesn't matter which. So I just sub in 5^ 2 + 4^ 2 * 5 * 4 * cos 106.

[00:02:54]Get a calculator. Just be careful with your calculator here. If you're going to do it in one, then you're going to do 5^ 2 + 4^ 2 - bracket 2 * 5 * 4 * cos 106 52.02549. 02549 and so on. That's a squared though. So I want a so it's a square root of 52.02549 dot dot dot to show I've no rounded square<unk> 52.02 549 say 7.212 212.

[00:03:39]And at this point, you would probably round, but you're not getting a mark for rounding this one. So, I'll just say that A is equal to about 7.2. I'll put my units in cm.

[00:03:51]And we're done there. Qualification Scotland National by math 26 paper two question three. Harriet's hand cream is available in two sizes of bottles and a mathematical cinema. The height of the large is 12 and the volume is 540. I'll just take a note of that. 540 ml.

[00:04:08]The test bottle has a height of eight.

[00:04:10]Find the volume of the test bottle.

[00:04:12]Pretty much exactly the same as the question I did in the National Vide.

[00:04:17]So scale factor.

[00:04:21]I want it in the new shape over the old.

[00:04:23]New over old.

[00:04:26]The new is eight. The old is 12. There's no need to actually simplify that because you're going to use it anyway.

[00:04:32]But you can if you want. And if you did, you would get 2/3. So now your volume scale factor is just the linear scale factor cubed. Again, you can work that out if you want to get 28 over 27 or I'd probably just leave it as 2/3 cubed or even 8 12 cubed. Now your new volume is now your volume is your scale factor cubed times your old volume which was 540.

[00:05:02]So 2 over 3 to the^ 3 * 540 is 160. You should be getting a smaller number milliliters.

[00:05:15]And we're done there. Qualification Scotland National Math 26 paper two question four hobby. Alex makes cups and plates. Five cups and six plates is 45 minutes. Let C be the number of minutes to make a cup.

[00:05:29]P be the number of minutes to make a plate. Write an equation for C and P. 5 C 6 P = 45. So we're going to sum equations of course. Part B. To make seven cups and four plates, it takes 52.

[00:05:46]7 cups, four plates, 52. Is that what I thought would come up? And now we've got find the total number of minutes it takes Alex to make 10 cups and eight plates. Don't be worried about that.

[00:05:57]find one cup in one plate and then just scale up. So let me just write these together.

[00:06:06]So we've got these equations.

[00:06:10]Now I always scale up the second number and I never think about it. I mean there might be I could times by three or something. I just times by the bottom four and times by the top six. Unless it was an on calculator then I'd maybe think harder cuz I've got a calculator anyway. So that's 20 C. 4 sixes is 24p.

[00:06:28]4590 is 180. 67s is 42.

[00:06:35]67s is 42. 6 4 is 24 and 6 52s. Use a calculator. But 62s is 12. So it's 312.

[00:06:46]So now I can just do one minus the other. So if I call that one and two, say, I can do one minus two. Doesn't matter what you call them. 42 - 20 is 22. So I'm taking away because the signs are the same on the 24. 312 - 180. Well, I'm not going to give myself a headache.

[00:07:06]It's 132.

[00:07:10]So that means that C is 132 / 22 6. So then we try and find P. Now just pick an equation. So I'll just pick one of the originals. 5 C + 6 P is 45.

[00:07:28]5 C 5 * 6 + 6 P = 45.

[00:07:37]30 + 6 P = 45. So 6 P 45 - 30 is 15.

[00:07:45]P is 15 / 6 or 15 / 6. So I've got a calculator.

[00:07:52]2.5 minute.

[00:07:58]So normally you would then say in context it's a number of minutes to make cups and plates. So you would say something like it takes six men to make a cup and it takes 2.5 men to make a plate.

[00:08:24]But what do we actually want for this question? If you read it, 10 cups, eight plates. So 10 cups plus eight plates.

[00:08:35]That's 10 * 6 + 8 * 2.5. Then 60 + 8 * 2.5 is 80. So it takes 80 minutes for 10 cups and eight points. Now 80 minutes is a bit of a funky number. So you might want to say that that's 1 hour 20 minutes but I think it doesn't really ask you to do anything with it. So 80 minutes I think is a fine answer. Qualification Scotland nify math 226 paper 2 question five. We have got a circle with some lines in it and we have to find some missing sides.

[00:09:22]So it's probably Pythagoras. Let's just check. AC is 25. So that's from here to there. B is a midpoint of AC.

[00:09:31]OB is nine. Find the length of the radius. So if to find the length of a radius, I need to write a radius in there. So that's a radius there. We know we don't know that, but half of 25 is 12.5.

[00:09:45]So that's 12.5 cm.

[00:09:48]That's 9 from O to B. That's a right angle. So it's just Pythagoras. Quite basic actually. You're not adding or taking away anything at the end, which is quite unusual.

[00:10:00]So always get a mark for drawing a right angle triangle. So put a right angle 12.5 opposite the right angle is your missing side which is your radius. Call it rx whatever you want. If it's opposite the right angle you're adding in Pythagoras. If it's not opposite you're taken away. So I can just do 9^2 + 12.5^2 = r 2. And always do your right angle triangle because if you mess it up you got to mark your right angle triangle. But if you mess it up you lose too much.

[00:10:28]9^2 + 12.5^ 237.25 is R 2. So R 237.25 square rooted 15.4 four roughly cm and we're done there. Okay, Scotland National Math 2026 paper 2 question six.

[00:11:00]A sphere has a radius five. Find the volume of a sphere. So far so normal.

[00:11:05]Volume of a sphere is 4/3 p<unk> r cubed.

[00:11:08]And the r in this case is 5. So we're just doing 4/3 *<unk> * 5 cubed. Make sure you use cubed for a sphere. 4 / 3 *<unk> * 5 ^ 3 523.59877 523.59877 a reasonable place to round that to I suppose I'll write 523.6 6 cm cubed.

[00:11:36]Done.

[00:11:38]Part B for question six. The cone has the same volume. The base of the cone is 12. Calculate the height. Right? So volume of a cone is 1/3 pi r²h.

[00:11:50]It's got the same volume as my sphere, which is 523.6.

[00:11:54]So it's a working backwards question.

[00:11:58]If you couldn't do part A, you should have guessed part A and done part B. But chess I could do party. Now if I'm working backwards, it's just I can times by three to start with 3 * 523.6.

[00:12:12]And then I want to isolate h.

[00:12:16]So the height is just 3 * 523.6 over p<unk> r 2. So what is the radius?

[00:12:25]Well, we've given us a diameter of 12.

[00:12:27]So r is six. So, we're just going to sub in that.

[00:12:32]3 * 523.6 over<unk> * 6^ 2. And you could have worked bits of it out as you went. 3 * 523.6 /<unk> * 6^ 2 13.888 888 or about 13.9 cm and we're done there. Triangle has size of length 88 105307 m. Determine if it's right angled. The converse of Pythagoras literally just did one of these in the last minute live stream. So let's to go.

[00:13:15]We've got for the converse of Pythagoras. You need to be aware that you need to be careful and keep it separate. So identify your two smallest sides.

[00:13:25]And then your big side and just go cut 888 2 + 105 squared. Other side is 137 squared. Do not write equals. Work them out. You don't need a square root at the end. You're just checking if the squares are equal. 18769 and 137 squared I think is 18769 off the top of my head. I'm joking. I don't know it. 18769 as well.

[00:13:54]And now you can do your statement.

[00:13:56]They're either the same or different.

[00:13:57]This time they're the same. So since 888 2 + 105^ 2 = 137^ squared 88. Yeah.

[00:14:08]by the converse of Pythagoras is a right angle right angled triangle. Now sometimes with these questions there's obviously another bit where it's like a it's a thing and you say therefore it's north or east or south or whatever it is but for this it's just it's a right angle triangle that's it. So we're done there.

[00:14:38]Ques Scotland math 26 paper 2 question 8. Solve the equation 5 tan x + 3 = 4.

[00:14:45]So I'll take away three for both sides because it's a trig equation. I want tan x on its own. So 5 tan x is equal to just 1. So tan x is 15.

[00:14:56]So inverse tan over 5th. Inverse tan of 15. and a calculator shift tan a fifth is 11.3° 11.39 so 11.3 is fine and then I do a cast diagram to find all the angles so our tan was positive so it is a so that's one of them and it's t so that means that x is 11.3 and in t it's 180 + 11.3 3 using a calculator you get 191.3° and we're done there. Qualifications Scotland National Math 2026 paper 2 question 9 express x^2 - 81 over 5 / x + 9 /2 as a fraction. So we're putting in dividing fractions which is completely fine. It just turns into times x^2 - 81 over 5 divide becomes times by the reciprocal 2x + 9. Now you need to somehow do something with that or times the top times the bottom because it's just tsing, right? So that is 2 x^2 - 81 over 5 x + 9. And now I need to somehow simplify that. So it means factorizing.

[00:16:17]This is an easy one cuz it's a difference of two squares on the top. 9 and 9 make 81. So I get - 9 and + 9. 5 x + 9 by to x + 9 then leaving me with just 2 x - 9 on top over 5. And there's no need to expand that top because it's already factorized. So that is simplest but you could write it as 2x - 18 over 5 if you want. There's nothing else to do.

[00:16:46]So we're done there. Qualification Scotland National Math 2026 paper 2 question 10. The vertices of a regular pentagon lie on a circle. The radius is 11. Calculate your segment. This seems more complicated than it is. If it's a pentagon, the angle in the center is 360 / 5. So, we can work that out to get an angle. So, it looks like an angle question, but it really isn't. That's 72° and that's 11 cm. If I drew that outside because I only want that bit, I would have a triangle with an arc on the top.

[00:17:26]And that means that if I want the area of this bit, which is a bit like a we cone, it isn't a cone. So don't get worried about that.

[00:17:36]Then I can do the area of the whole sector minus the area of the triangle.

[00:17:43]So that's exactly the approach we need to take with this. So we've got area of sector minus area of triangle. So a sector is angle of 360 which is 72 in this case*<unk> r² which is 11. So I'm doing that minus now triangle start exam paper half a b sin c.

[00:18:13]So that's a half * 11 * 11 * sin 72.

[00:18:22]So that's our two sums. So I think a lot you will do that separately. You can do it all in one if you want. So the first one 72 over 360 *<unk> * 11^ 2 is 76.026 026 with other numbers minus my other one a half * 11 * 11 * sin 72 57.538 we have the numbers again dot dot dot to show I haven't rounded 76.026 026 minus 57.538 is 18.488 and at this point I would round to 18.5 and my units it's an area cm squared and we're done there. So just watch rounding in the middle because if you round in the middle and then you round at the end you probably end up double rounding which ends up giving you the wrong answer. So just make sure if you are just go about three decimal places and dot dot dot it then take away qualification Scotland National 5 math 26 paper 2 question 11 simplify this it's the laws of indices so this is there's a little trap to this if it's 3 a4 cubed the three needs cubed as well so I need to do 3 * 3 * 3 and you've got to calculate remember so if you don't know that it's 27 not three and then power power means times 4 * 3 is 12. So you end up with that and then divide means take away. So I've got a 17 - 12.

[00:20:07]Got 17 - 12 is five. Now that's on the top cuz you took away and it's sitting on the top. But the 27 is still on the bottom. So I need it to be over 27 to finish me off. And I'm done there. If I put the 27 up the top again, that's going to lose me a mark. So you get a mark for getting the 27, a mark for getting the 12, and a mark for getting the final answer usually. Okay, a little trig identity. Went over these last night. And there's two types. Remember, sin / cos is tan. Sin^ square + cos square is 1. So if I see any squares, I think it's probably going to be that.

[00:20:42]But you always think about, can you expand brackets? Can you factoriize?

[00:20:44]There something I can do. Well, I can see that cos x is a common factor.

[00:20:49]So cos x * sin^ 2 x for the first term.

[00:20:53]And cos x * cos^ 2 x is cos cubed. So I've got a plus in between.

[00:20:58]But because sin^ square + cos square is 1, I can just write that that equals cos x * 1. I need to really show that I know that that's one. So that equals cos x.

[00:21:11]And with there you'll get a mark for factorizing a mark for your final answer. Qualification Scotland National Math 2026 paper two question 13. The final one. An extension of a rectangular base has been answered for wheel of a house. diagram shows a plan of a rectangular base of the extension. We don't really need that diagram. It's just to show you what it is cuz below the diagram below shows a plan of the rectangular base of the extension. And it's going to have write an expression for something then show that it's a quadratic. So then solve it. So if you can't do part A or B, jump to part C and do the quadratic formula. But let's try.

[00:21:47]So part A first.

[00:21:51]It says lots of words here. The length of the four is six. The width is two.

[00:21:56]It's all in the diagram anyway. There's a wall of x meters fixed around the three sides. Write an expression for the length of the extension in terms of x.

[00:22:03]Well, our length goes all the way along the top. So, I just need to add all the bits that that takes up. So, for part a, it's x plus and then it goes six from here to here.

[00:22:17]Six plus and then another x. So x I have a bit then six.

[00:22:23]I'm just going to get my white pen to make that neater. X + X is 2 X + 6. Now let's look at what part B is asking us.

[00:22:34]The total area is 17. Show that that is true. It's mentioned area. So I want to work out what the area of this thing is.

[00:22:42]So area is length times breadth.

[00:22:47]The length of four is six. Width is two.

[00:22:49]There's a wall. So the wall isn't part of the is is part of the area of course.

[00:22:55]Yeah. So that means that I've got the whole length times the whole breadth to do. The whole length is 2x + 6.

[00:23:03]The whole breadth is this height and that height is going to be 2 + x. So I just need to say area = 2 + x * 2x + 6. And I really can't do it if I can't do part A. And that equals our 17. So now we expand our brackets.

[00:23:26]You can use a grid if you want.

[00:23:29]2 + x on one side, 2x + 6 on the other.

[00:23:34]2's is 4 x 2 sixes is 12. 2 x^2 and 6 x. So I write them in order. I've got 2 x^2.

[00:23:46]6 and 4 is 10 x + the 12 = 17 and finally take away 17 you get 2x^2 + 10 x - 5 is 0. So that's how you go about that question. But if you can't do it part C calcate x the width of the wall two decimal places so it must be quadratic formula because I can't factoriize it if it's two decimal places. So start of the exam paper.

[00:24:12]There's the quadratic formula.

[00:24:16]So we need to sub in. So a is 2 and I went through this in the last minute live stream. b is 10. C is -5.

[00:24:26]So x = - 10 plus or minus the<unk> of 10^ 2 - 4 * 2 * - 5 all over 2 * a 2 * 2. So usually you would do the bit in the square root next. So - 10 plus or minus.

[00:24:45]So I do 10^ 2 minus bracket 4 * 2 * - 5 is 140 all over 4. Separate that out into 2 - 10 +<unk> 140 / 4 - 10 -<unk> 140 over all of it over 4.

[00:25:08]Do them separately. So you've got - 10 +<unk> 140 and then divide by 4 0.458 458 or - 10 - 140 over 4 - 5.458.

[00:25:44]However, it is the width of the wall. We have to work out the thickness of the wall. Sorry. So, you need to reject reject reject that one cuz you can't have a negative bit of a wall. It makes no sense. So, you've only got that one.

[00:25:58]So, the answer is x equals how many decimal places does it want? Two decimal places. So, I went to 34 six. It's probably meters, isn't it?

[00:26:10]Meters.

[00:26:14]And we're done there. So, just remember this one. Write down your unrounded answers first because it's a mark for rounding. And if you don't do the rounding correctly, if you wrote down straight away in your calculator 0.45, you'll lose two marks because you've never got the actual answers and you're never going to get done properly. So just watch out. Whereas if you get the actual answers, that's a mark and then round bad. You just lose one mark. More math. Today we've done national five math 2026 paper two qualifications Scotland. It's first year not too bad a paper some tricky ones somewhere.

[00:26:45]Hopefully you found that useful.

[00:26:48]Stay safe, take care, and I'll see you