National 5 Maths 2026 Paper 1 | Full Solutions

Hinzugefügt: 2026-05-09

[00:00:00]Qualification Scotland National Math 2026 paper one. Let's get going. Came out on 8 for me. You were doing this at 9:00 a.m. Let's see how we get on with this one. CL Math is sponsored by Leki, the educational publisher for Scotland.

[00:00:13]Now, going from National 4 to National 5 maths can be quite a difficult thing for a lot of pupils. That's why Leky have got national four to national 5 maths bridging skills book. This revises everything from national 4 to help you prepare yourself for national 5. It's got questions on every topic. It's bright and colorful and it has hints and tips and work solutions to everything as well. This is just £6.99 when you use the discount code clever maths. I highly recommend it if you are now moving on to national 5mm maths. Clear maths at leiscotland.co.uk link is in the description.

[00:00:48]Qualification Scotland National 5 math 226 paper one question one brackets expanding now it's 2x3 which is exactly what I thought it was going to be. y + 4 y^ 2 - 3 y + 2. So, we had this in the last minute live stream or very similar.

[00:01:07]Y * y^ 2 is y cubed. y * y is 3 y 2 y * 2 is 2 y and then the bottom 4 y^ 2 4 * 3 is 12 y and 4 * 2 is 8. collecting the terms backwards ones go together. So we can just do y cubed y^ 2 4 y^ 2 - 3 y^ 2 is y^ 2 positive -2 + 2 is - 10 y and we've got + 8 hanging on the end and we're done there. Qualification Scotland five maths 126 people won question two reverse percentages this one an electric car shows it has enough charge remaining in its battle battery to travel 180 mi the battery has 60% of its charge remaining calculate number of miles a car can travel when its battery is fully charged. So it has enough charge remaining in it to travel 180 mi. It's got 60% left.

[00:02:14]180 equals 60% in this case. Okay, so you're not having to take away 100 this time cuz it's just we got 6% left. So I need to find 1%. So divide by 60.

[00:02:25]Now you could just divide by six here to get 10%. It's up to you. But I'll just divide by 60. So take the zeros off. 6 freeze is 18. 3 miles is equal to 1%.

[00:02:38]And therefore 300 miles is 100%.

[00:02:44]So we're done there. of Scotland National V mass 2026 paper one question three a restaurant recorded waiting times to be seated in a sample of his customers and they are shown below calculate a median and inter quarter range of waiting times now these are not in order this time so we're going to have to put them in order so just being very careful I've got two I've got two I've got four I've got a five I've got a six I've got an Okay, I've got a 14, a 15, an 18, and I think I've got a 20 as well. Just count them at the end. 1 2 3 4 5 6 7 8 9 10. 1 2 3 4 5 6 7 8 9 10. So, I've probably done it right. Median's the middle, so I've got 10. So, 1 2 3 4 five along. The middle is in between here. So my median is equal to seven 7 minutes if you want to give the units but you usually lose a mark for not. And now I need the inter quartile range. So I need the upper quartile which the upper half is 15. The lower half 2 4 5 6 is 4. So the inter quartile range is 15 minus 4 which is 11. I'll just put minutes as well just to be on the safe side, but I don't think you'll need it.

[00:04:12]Okay, part B. So, I did a comparing here. So, we're comparing this instead of standard deviation. A cafe also record the waiting times to be seated for a sample of its customers. The median and was five and inter quarter range was 12. Make two valid comments.

[00:04:25]So, let's compare the median. Median is exactly the same as comparing means. You start off with on average.

[00:04:33]So, that's where I'm going to start. A cafe also record the waiting time against a restaurant on average. The waiting times in the restaurant I'll say in the cafe.

[00:04:49]I'll just go for the cafe. I think in the cafe were less than the restaurant.

[00:05:02]I'm just being overly dramatic about the statement here because I know that sometimes it can be a bit funny. since and we've got well 5 minutes is less than 7 minutes.

[00:05:19]So on average the waiting times in the cafe were less than the restaurant waiting times since five was less than seven. I think you'll get away with just saying on average waiting times in the cafe were less in the restaurant but just to be on the safe side and we're going to compare the interquarter range which is the same as comparing standard deviations about consistency. So the waiting times the waiting times to be seated in the cafe.

[00:06:14]where now they got 12, I got 11. So 12 is a bigger number, less consistent than waiting times in the restaurant.

[00:06:37]And the reason for that is that 12 men is bigger than 11 men. So less consistent, bigger number or more consistent, smaller number. Hopefully you find that useful. Let's move on.

[00:06:51]Qualification Scotland Nif six paper one question four. Solve algebraically with this an equation. So, I've got x + 8 on one side is less than 3x - 6 + 20.

[00:07:06]Collect my terms. So, that's 3x. 20 - 6 is 14. And I like to keep my x's positive so I don't have to flip the sign over. So, I'll take the 14 over.

[00:07:21]So that gives me x 8 -4 is - 6 is less than 3x. So now take away x to both sides. - 6 is less than 2x. Divide by 2 - 3 is less than x. And you're usually happy to leave it like that. But if you did put it on the other side, you would have to flip it. So x would be greater than minus 3. Or you would get that if you had a minus x on the left and you divided by a minus which is why I try and avoid that. So x is greater than minus three or if you got minus three is less than x you're probably fine there too. Qualifications national by math 2026 paper one question five Sam walked two and 3/4 miles to a park then walked a further one and two miles to a shop calculate the total distance Sam walked but unusual for qualifications Scotland to give us a adding fractions question with some words but that's all it is. So, it's just 2 and 3/4 plus 1 and 2/3. Now, there is two ways to do this. You can add the holes, then add the fractions, or you can change them into mixed numbers to fractions first. I prefer to just add the whole parts. 2 + 1 is three. Keep that where it is. I'll do the three quarters plus the 2/3 the side. Four threes is 12.

[00:08:35]Plus in the middle, three is 9. 4 is 8.

[00:08:39]9 and 8 is 17 over 12.

[00:08:42]12 goes into 17 1's with five left over.

[00:08:46]So in total I've got three plus one and 51 12ths which is 4 and 52 and it's miles I think. So I'll write miles.

[00:08:59]That's one way to do it. The other way to do it just in case you're wondering I'll do this in blue or two fours is 8 9 10 11 over 4. One freeze is three four five.

[00:09:12]Now just do the same thing. 33 + 20 33 43 53 over 12 miles or a neater way would say be to change that into falling 512 miles. So either way is probably fine. Both cases Scotland National Math 206 paper one question six. A teacher records the mark scored by pupils in her class for her prelum and final exam.

[00:09:37]This graph shows the resp maths. A line of best fit is drawn.

[00:09:46]Point A represents 30 and 54 in the final. Point B is 90 and 94 in the final. Quite standard stuff. This find an equation of a line in terms of F and P. Exactly what I thought was going to happen. Check the live stream. m = 94 - 54 and 90 - 30. So that gives me 54 64 84 94 is 40 over 90 - 30 is 60. Simplify that to 46 which is 2/3. So just again remember that's because we go x1 y1 x2 y2 and the gradient is x2 y2 - y1 over x2 - x1. So now I usually do y - b is mx - a. You can use mx plus c if you want and you let one of the points be a and b. So I'm just going to pick the first one a and b here and there to sub in. So y - 54 is 2/3 x - 30.

[00:10:57]y - 54 then no need to take the three up because 2/3 goes into 30. 2/3 x 30 / 3 is 10 * 2 is 20. Move the 54 over. So you get 2/3 of x. 54 minus 20 because it becomes plus 54 is 34 plus we then need to switch it back into F and P 2/3 of P plus 34 and we're done there. Now we've got a part B. Part B says use your equation from A to estimate marking the final exam 33 in the pre. So we know F is 3 2/3 P which is the pre plus 34. So a sub 3 in for P. So 2/3 * 33 + 34 that equals well that's 11 3 11s is 33 * 2 is 22 + 34 34 44 54 56 and just check if there's any units attached to that% I 56% but it just says marks 56 marks and we're done there. Question seven on the qualifications Scotland National 5 math 2026 paperwork exam had this find the magnitude of a vector. So one vectors but leave your answer. So that I knew that could happen at some point. And the magnitude of a vector remember is just Pythagoras 4^ 2 + - 5 2 + 7 2 that's 16 + 25 + 49.

[00:12:34]Just taking your time adding your numbers up.

[00:12:37]25 + 16 + 49 well 49 59 69 79 89 is 90.

[00:12:44]Now 90 is not a number you can work out.

[00:12:47]However, two roots 9 * 10 is 90 square<unk> of 9 is 3. So it's 3<unk> 10 and we're done there. Scotland national 5 math 2026 paper one question 8 change the subject to t. Again, this looks pretty similar to the one that I came up with in the last minute live stream.

[00:13:07]Mine was a little bit harder. Square both sides because it's stuck under the square root sign.

[00:13:15]Now I have to change it to T. So there's only one more step. Add 3 L. P ^2 + 3 L is equal to T or more commonly T = 3^ 2 + P ^2 + 3 L. And we're done there.

[00:13:29]Scotland National Math paper one.

[00:13:31]Question nine, the diagram shows the graph of kx^2, y equals, find the value of k. Well, if you got a point, that's an x and a y. So, I can just sub that in. 24 y = k * 2 ^ 2 24 = 4 kide by 4 then 4 sixes is 24 k is six and we're done there. Scotland National Math 6 paper one question 10 diagram shows a circle center in this big diagram and we have to find the reflex angle AOC I thought big circle question would come up with like this I thought a big circle question angles would come up like this so we're looking at a radius meets a tangent at right angles so either side of that is 90 90 - 63 73 83 is 27 so I know that that's 27 But that's a radius and that's a radius.

[00:14:26]So that's 27 cuz isoclesles. So 27 * 2 is 54.

[00:14:32]180us 54 is 180 130 126 I think. 126 126 130 180. The other side I've got 40 already.

[00:14:51]So that's 40 because of isoclesles triangle again two 40s is 80 so that's 100. Let's go find my final answer which says state your answer below 126 + 100 is 226° there Scotland national mass 2026 paper one question 11 we've got the diagram shows a cuboid relative to coordinate axis this looks pretty similar as well to what I did B is coordinate 084 H is 12 2 E H is power to the X ais state the coordinates of G So I need to find G. So I start with what I know. I know that that is zero. So that's zero. Then it goes along N8.

[00:15:40]So that must go N8.

[00:15:42]But how do I get to the first number?

[00:15:45]Well, the first number I can look at this one. That goes along 12. So that one must go along 12. So it's 12.

[00:15:52]I then go in the same distance as B, which is eight.

[00:15:57]And I then don't go up off the floor. So the last number is the same as H, which is Z 1280.

[00:16:03]B M is the midpoint CD. State M. So M is here. So let's start off with H. H is down here. So we need to go along to H, which is 12 one. So H, we need to go halfway into here. Now we already know G, if I write it above here, is 1280. So that middle starts at two and goes to eight. So I need to find the middle of two and eight. 8 + 2 is 10 / two is five. So I need to go up five. In other words, it has to go three and three. So that's five. And then the height is just the same as the height of any other one.

[00:16:42]Four. So it's 124.

[00:16:45]And we're done there. Okay. Scotland National Math 2026 paper one question 12. On the axis below, sketch a graph of x + 2 + 3. Stay clear of the coordinates turning point and intersection with y- axis. So turning point is easy.

[00:17:00]It is just change keep - 2 3. Always remember change keep if it's already completely square. And to get the intercept with the yaxis x = 0.

[00:17:16]So I've got 0 + 2 + 3.

[00:17:21]Two twos is four + 3 is 7. So min - 2 3 maybe about there.

[00:17:30]And it's a happy face because it's a positive x + 2 and it's a seven is going through. It doesn't matter how accurate your sketch is. It could be moved down a little bit or whatever. As long as it's a curve, it goes through seven and minus 2, three.

[00:17:42]And we're done there. I'll probably write that as 07 just in case we want it as a full coordinate, but putting seven there should be fine. Question of 13.

[00:17:51]Given that f ofx is 5 cos 2x evaluate f of 90. So it's trig a little bit with functions. f of 90 means a sub 90 in 5 cos 2 * 90.

[00:18:06]That's 5 cos 180. So how am I going to know the cost of 180? You need to know a cost graph. So it's really knowing do you know your graphs? It goes to minus1 and 1 0 90 180 270 then 360. There's my 180 there which is at -1.

[00:18:25]So that means I've got 5 * -1 which is -5 and we're done there. 14. Solve this equation by factorizing 10 x^2 + 11 x - 6 is 0. So we're going to factoriize a quadratic. Check if there's a common factor first. There isn't. So I need to do a method. Now here's my best method.

[00:18:44]10 * - 6 that's -60.

[00:18:48]So I'm looking for two numbers at times to make -60 but add to make the middle number which is 11. So I want numbers at times to make 60 but add to make 11. So you can do one and 60. Well that doesn't work. Two and 30. Well that doesn't work. Three and 20. Well that doesn't work. Four and 15. Ah now there we go.

[00:19:09]Four 15s is 60.

[00:19:12]I need to make -6 div and + 11 -5 + 4.

[00:19:19]There we go. Or + 15 - 4 + 15 - 4 is 11.

[00:19:25]And now I do my grid again. So I've got 10 x^2 in one place plus 15x in another place - 4x and then we still got - 6.

[00:19:40]And we just factoriize the line. 10 and 15 has a common factor of 5 x 5 * 2 is 10 x^ 2ar 5 * 3 is 15 and then 2 * something is 4. 2 * 2 is 4 but is minus.

[00:19:56]So now I've got my brackets.

[00:19:59]So my brackets are 2x + 3 5x - 2. And if it was factorizing we would stop there.

[00:20:06]But that equals zero. So that means that both brackets equal 0. So 2x + 3 = 0. 5x - 2 = 0.

[00:20:17]To resolve that, I can just move the 2x over to get - 3. So x is -3 / 2 is one solution. Or I can move the two over to get positive2.

[00:20:29]So x is 2 over 5 is the other solution.

[00:20:32]And we're done there. It's been CL Mast with the Qualification Scotland 2026 National 5 math paper one. Hopefully you found that useful.