Course NotesPracticePast Paper Archive

National 5 Maths · Qualifications Scotland past paper

2026 Paper 1

Non-Calculator · 14 questions
1

Question 1

Expanding brackets
3 Marks
2026 P1 Q1
Expand and simplify
(y+4)(y2−3y+2).\displaystyle (y+4)(y^{2}-3y+2).
Show answer
y3+y2−10y+8\displaystyle y^{3}+y^{2}-10y+8
2

Question 2

Reversing a percentage change
3 Marks
2026 P1 Q2

An electric car shows that it has enough charge remaining in its battery to travel 180 miles.
The battery has 60% of its charge remaining.
Calculate the number of miles the car can travel when the battery is fully charged.

Show answer
300 miles
3

Question 3

Median/Quartiles/Interquartile RangeComparing Calculated Statistics
(3, 2)5 Marks
2026 P1 Q3

A restaurant recorded the waiting times to be seated from a sample of its customers.
The waiting times, in minutes, of ten customers were:
2  20  5  14  2  4  8  18  6  15
(a)  Calculate the median and interquartile range of the waiting times.
A café also recorded the waiting times to be seated from a sample of its customers.
The median waiting time at the café was 5 minutes and the interquartile range was 12 minutes.
(b)  Make two valid comments comparing the waiting times at the restaurant and at the café.

Show answer
(a) Median = 7 minutes, interquartile range = 11 minutes
(b) On average, the waiting times at the restaurant were longer than at the café, as 7>5.\displaystyle 7 > 5.
The waiting times at the restaurant were more consistent than at the café, as 11<12.\displaystyle 11 < 12.
4

Question 4

Linear equations and inequations
3 Marks
2026 P1 Q4
Solve, algebraically, the inequation
x+8<3(x−2)+20.\displaystyle x+8 \lt 3(x-2)+20.
Show answer
x>−3\displaystyle x \gt -3
5

Question 5

Fractions and mixed numbers
2 Marks
2026 P1 Q5
Sam walked 234\displaystyle 2\frac{3}{4} miles to a park and then walked a further 123\displaystyle 1\frac{2}{3} miles to a shop.
Calculate the total distance Sam walked.
Show answer
4512\displaystyle 4\frac{5}{12} miles
6

Question 6

Scatter GraphStraight Line Equation
(3, 1)4 Marks
2026 P1 Q6

A teacher records the marks scored by the pupils in her class for the prelim exam and the final exam.
The scattergraph shows the relationship between the marks scored in the prelim exam, P, and the final exam, F.

Scattergraph of final exam mark F against prelim exam mark P, with a line of best fit through A(30, 54) and B(90, 94)

A line of best fit is drawn.
Point A represents a pupil who scored 30 in the prelim exam and 54 in the final exam.
Point B represents a pupil who scored 90 in the prelim exam and 94 in the final exam.
(a)  Find the equation of the line of best fit in terms of F and P.
Give the equation in its simplest form.
A pupil scored 33 marks in the prelim exam.
(b)  Use your equation from part (a) to estimate their mark in the final exam.

Show answer
(a) F=23P+34\displaystyle F=\frac{2}{3}P+34
(b) 56
7

Question 7

MagnitudeSimplifying surds
3 Marks
2026 P1 Q7
Find ∣d∣\displaystyle |\mathbf{d}|, the magnitude of vector d=(4−57).\displaystyle \mathbf{d}=\begin{pmatrix}4\\ -5\\ 7\end{pmatrix}.
Express your answer as a surd in its simplest form.
Show answer
310\displaystyle 3\sqrt{10}
8

Question 8

Changing the subject of a formula
2 Marks
2026 P1 Q8
Change the subject of the following formula to T.
P=T−3L\displaystyle P=\sqrt{T-3L}
Show answer
T=P2+3L\displaystyle T=P^{2}+3L
9

Question 9

Parabola Equation from Graph
2 Marks
2026 P1 Q9

The diagram shows part of the graph of y=kx2.\displaystyle y=kx^{2}.

Part of the graph of y = kx squared, passing through the point (2, 24)

Find the value of k.

Show answer
k=6\displaystyle k=6
10

Question 10

Angles in diagrams involving circles
3 Marks
2026 P1 Q10

The diagram shows a circle with centre O.

Circle with centre O, tangent DE touching the circle at B, with points A and C on the circumference
  • DE is a tangent to the circle at the point B.
  • Angle OBC is 40∘.\displaystyle 40^{\circ}.
  • Angle ABE is 63∘.\displaystyle 63^{\circ}.

Calculate the size of the reflex angle AOC.

Show answer
226∘\displaystyle 226^{\circ}
11

Question 11

3d Coordinates
(1, 1)2 Marks
2026 P1 Q11

The diagram shows a cuboid, ABCDEFGH, relative to the coordinate axes.

Cuboid ABCDEFGH relative to 3D coordinate axes, with B at (0, 8, 4) and H at (12, 2, 0)
  • B has coordinates (0,8,4).\displaystyle (0,8,4).
  • H has coordinates (12,2,0).\displaystyle (12,2,0).
  • EH is parallel to the x-axis.

(a)  State the coordinates of G.
(b)  M is the midpoint of CD.
State the coordinates of M.

Show answer
(a) (12,8,0)\displaystyle (12,8,0)
(b) (12,5,4)\displaystyle (12,5,4)
12

Question 12

Sketch a parabola from equation
3 Marks
2026 P1 Q12

On the axes below, sketch the graph of y=(x+2)2+3.\displaystyle y=(x+2)^{2}+3.
On your sketch, show clearly the coordinates of the turning point and the point of intersection with the y-axis.

Blank x and y axes with origin O, for sketching the graph

Show answer
A parabola with a minimum turning point at (−2,3)\displaystyle (-2,3) and y-intercept at (0,7).\displaystyle (0,7).
13

Question 13

Function notationsin/cos/tan of related angles
2 Marks
2026 P1 Q13
Given that f(x)=5 cos⁡ 2x∘\displaystyle f(x)=5~\cos~2x^{\circ}, evaluate f(90).\displaystyle f(90).
Show answer
−5\displaystyle -5
14

Question 14

Quadratic equation by factorising
3 Marks
2026 P1 Q14
Solve the equation by factorising
10x2+11x−6=0.\displaystyle 10x^{2}+11x-6=0.
Show answer
x=25\displaystyle x=\frac{2}{5} and x=−32\displaystyle x=-\frac{3}{2}