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National 5 Maths · Qualifications Scotland past paper

2026 Paper 2

Calculator · 13 questions
1

Question 1

Appreciation and Depreciation
4 Marks
2026 P2 Q1

A van was valued at £22600.\displaystyle \pounds 22600.
Its value depreciated by 28% each year for the next three years.
Calculate the value of the van after three years.
Give your answer correct to 3 significant figures.

Show answer
£8440\displaystyle \pounds 8440
2

Question 2

Cosine rule: calculate length
3 Marks
2026 P2 Q2

In triangle XYZ:
• XZ = 5 centimetres
• YZ = 4 centimetres
• angle XZY = 106∘\displaystyle 106^{\circ}

Triangle XYZ with XZ = 5 cm, YZ = 4 cm and angle XZY = 106 degrees

Calculate the length of XY.

Show answer
7.21\displaystyle 7.21 cm
3

Question 3

Similar areas/volumes
3 Marks
2026 P2 Q3

Harriet's Hand Cream is available in two sizes of bottles which are mathematically similar.

Two mathematically similar hand cream bottles, the larger 12 cm tall and the tester 8 cm tall

The larger bottle has a height of 12 centimetres and a volume of 540 millilitres.
The tester bottle has a height of 8 centimetres.
Calculate the volume of the tester bottle.

Show answer
160 millilitres
4

Question 4

Simultaneous equations
(1, 1, 4)6 Marks
2026 P2 Q4

For a hobby, Alex makes cups and plates.
To make 5 cups and 6 plates, it takes Alex 45 minutes.
Let c be the number of minutes Alex takes to make a cup and p be the number of minutes Alex takes to make a plate.
(a)  Write down an equation in c and p to illustrate this information.
To make 7 cups and 4 plates, it takes Alex 52 minutes.
(b)  Write down an equation in c and p to illustrate this information.
(c)  Calculate, algebraically, the total number of minutes it will take Alex to make 10 cups and 8 plates.

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(a) 5c+6p=45\displaystyle 5c+6p=45
(b) 7c+4p=52\displaystyle 7c+4p=52
(c) 80 minutes
5

Question 5

Pythagoras in circle diagrams
3 Marks
2026 P2 Q5

The diagram shows a circle, centre O, with chord AC.

Circle with centre O and chord AC, with B the midpoint of AC and OB drawn

AC is 25 centimetres.
B is the midpoint of AC.
OB is 9 centimetres.
Calculate the length of the radius of the circle.

Show answer
15.4\displaystyle 15.4 cm
6

Question 6

Volume - simple shape
(2, 3)5 Marks
2026 P2 Q6

A sphere has radius 5 centimetres.

Sphere of radius 5 cm

(a)  Calculate the volume of the sphere.
This cone has the same volume as the sphere.
The base of the cone has diameter 12 centimetres.

Cone with base diameter 12 cm and height h

(b)  Calculate the height of the cone.

Show answer
(a) 523.6\displaystyle 523.6 cm3\displaystyle ^{3}
(b) 13.9\displaystyle 13.9 cm
7

Question 7

Pythagoras converse
3 Marks
2026 P2 Q7

A triangle has sides of length 88 metres, 105 metres and 137 metres.

Triangle with sides of 88 m, 105 m and 137 m

Determine whether the triangle is right-angled.
Justify your answer.

Show answer
882+1052=7744+11025=18769\displaystyle 88^{2}+105^{2}=7744+11025=18769
1372=18769\displaystyle 137^{2}=18769
The two are equal, so by the converse of Pythagoras' theorem the triangle is right-angled.
8

Question 8

Trigonometric equation
3 Marks
2026 P2 Q8
Solve the equation
5tan⁡x∘+3=4\displaystyle 5\tan x^{\circ}+3=4, for 0≤x<360.\displaystyle 0\le x\lt360.
Show answer
x=11.3\displaystyle x=11.3 and x=191.3\displaystyle x=191.3
9

Question 9

Multiply or divide Algebraic Fractions
3 Marks
2026 P2 Q9
Express
x2−815÷x+92\displaystyle \frac{x^{2}-81}{5}\div\frac{x+9}{2}, x≠−9\displaystyle x\ne-9
as a single fraction in its simplest form.
Show answer
2(x−9)5\displaystyle \frac{2(x-9)}{5}
10

Question 10

Sector areaArea of a Triangle
5 Marks
2026 P2 Q10

The vertices of a regular pentagon, ABCDE, lie on a circle, centre O.

Regular pentagon ABCDE with vertices on a circle of centre O, with one segment shaded

The radius of the circle is 11 centimetres.
Calculate the area of the shaded segment.

Show answer
18.5\displaystyle 18.5 cm2\displaystyle ^{2}
11

Question 11

Laws of indices
3 Marks
2026 P2 Q11
Simplify a17(3a4)3.\displaystyle \frac{a^{17}}{(3a^{4})^{3}}.
Show answer
a527\displaystyle \frac{a^{5}}{27}
12

Question 12

Trigonometric identities
2 Marks
2026 P2 Q12
Express the following in its simplest form:
cos⁡x∘sin⁡2x∘+cos⁡3x∘.\displaystyle \cos x^{\circ}\sin^{2}x^{\circ}+\cos^{3}x^{\circ}.
Show answer
cos⁡x∘\displaystyle \cos x^{\circ}
13

Question 13

Create equation in geometric contextQuadratic formula
(1, 2, 4)7 Marks
2026 P2 Q13

An extension with a rectangular base is being added to the rear of a house.

House with a rectangular extension added to the rear

The diagram below shows a plan of the rectangular base of the extension.

Plan of the rectangular base, with the floor 6 m long and 2 m wide and a wall x metres thick on three sides

The length of the floor is 6 metres and the width of the floor is 2 metres.
There is a wall x metres thick on three sides of the extension.
(a)  Write down an expression for the length of the extension in terms of x.
The total area of the rectangular base of the extension is 17 square metres.
(b)  Show that 2x2+10x−5=0.\displaystyle 2x^{2}+10x-5=0.
(c)  Calculate x, the thickness of the wall.
Give your answer correct to two decimal places.

Show answer
(a) 2x+6\displaystyle 2x+6
(b) (2x+6)(x+2)=17\displaystyle (2x+6)(x+2)=17 expands to 2x2+10x+12=17\displaystyle 2x^{2}+10x+12=17, giving 2x2+10x−5=0.\displaystyle 2x^{2}+10x-5=0.
(c) x=0.46\displaystyle x=0.46 m