Higher Maths · Qualifications Scotland past paper

2026 Paper 2

Calculator · 15 questions

Question 1

Discriminant and Quadratics
3 Marks
2026 P2 Q1
The equation x2+kx+(k+3)=0\displaystyle x^{2}+kx+(k+3)=0 has equal roots.
Find algebraically the values of k.
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k=6\displaystyle k=6 and k=−2\displaystyle k=-2

Question 2

Scalar productCalculating an angle using the scalar product
(1, 4)5 Marks
2026 P2 Q2

Vectors u\displaystyle \mathbf{u} and v\displaystyle \mathbf{v} are defined as
u=(326)\displaystyle \mathbf{u}=\begin{pmatrix}3\\ 2\\ 6\end{pmatrix} and v=(−5−24).\displaystyle \mathbf{v}=\begin{pmatrix}-5\\ -2\\ 4\end{pmatrix}.
(a)  Find u⋅v.\displaystyle \mathbf{u}\cdot\mathbf{v}.
(b)  Calculate the acute angle between u\displaystyle \mathbf{u} and v.\displaystyle \mathbf{v}.

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(a) 5
(b) 83.9∘\displaystyle 83.9^{\circ}

Question 3

Inverse functions
3 Marks
2026 P2 Q3
A function, g, is defined on R\displaystyle \mathbb{R}, the set of real numbers, by
g(x)=15x+6.\displaystyle g(x)=\frac{1}{5}x+6.
Find the inverse function, g−1(x).\displaystyle g^{-1}(x).
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g−1(x)=5x−30\displaystyle g^{-1}(x)=5x-30

Question 4

Circle equation from radius/centre or vice versa
2 Marks
2026 P2 Q4
A circle has centre (−2,5).\displaystyle (-2,5).
The point (3,7)\displaystyle (3,7) lies on the circle.
Find the equation of the circle.
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(x+2)2+(y−5)2=29\displaystyle (x+2)^{2}+(y-5)^{2}=29

Question 5

Find a specific term of a recurrence relationLimits of recurrence relations
(1, 1, 2)4 Marks
2026 P2 Q5

A sequence is defined by the recurrence relation un+1=1.125un+5\displaystyle u_{n+1}=1.125u_{n}+5, u0=2.\displaystyle u_{0}=2.
(a)  Calculate the value of u1.\displaystyle u_{1}.
(b)  Explain why this sequence does not approach a limit as n→∞.\displaystyle n\rightarrow\infty.
(c)  Determine the smallest value of n for which un>30.\displaystyle u_{n} \gt 30.

Show answer
(a) 7.25\displaystyle 7.25
(b) A limit exists only when −1<a<1.\displaystyle -1 \lt a \lt 1. Here a=1.125\displaystyle a=1.125, which is greater than 1, so the sequence diverges and has no limit.
(c) n=5\displaystyle n=5

Question 6

Identifying/sketching graphs of related functions (non-trigonometric)
3 Marks
2026 P2 Q6

The diagram shows the graph of a cubic function y=f(x)\displaystyle y=f(x) with stationary points at (−1,0)\displaystyle (-1,0) and (1,−4).\displaystyle (1,-4).

Graph of a cubic function with stationary points at (-1, 0) and (1, -4)

On the diagram in your answer booklet, sketch the graph of y=−f(x)+1.\displaystyle y=-f(x)+1.

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The graph of y=f(x)\displaystyle y=f(x) reflected in the x-axis and then translated 1\displaystyle 1 unit upwards.
Stationary points at (−1,1)\displaystyle (-1,1), now a minimum, and (1,5)\displaystyle (1,5), now a maximum.
The graph passes through (2,1).\displaystyle (2,1).

Question 7

Areas using integration
4 Marks
2026 P2 Q7

The diagram shows the curve with equation y=x2−x−2.\displaystyle y=x^{2}-x-2.

Curve y = x squared minus x minus 2, with the region between the curve and the x-axis shaded

Calculate the shaded area.

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4.5\displaystyle 4.5 square units

Question 8

Perpendicular and parallel linesAngles and straight lines: m = tanθ
(4, 2)6 Marks
2026 P2 Q8

A and B are the points (−2,11)\displaystyle (-2,11) and (10,−13).\displaystyle (10,-13).

Points A, B and C with the line L, the perpendicular bisector of AB, and the line AC

(a)  Find the equation of L, the perpendicular bisector of AB.
The equation of AC is y+x=9.\displaystyle y+x=9.
(b)  Calculate the size of the obtuse angle between L and AC.

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(a) y=12x−3\displaystyle y=\frac{1}{2}x-3, or x−2y−6=0\displaystyle x-2y-6=0
(b) 108.4∘\displaystyle 108.4^{\circ}

Question 9

Increasing/decreasing: show that, or find values for which
4 Marks
2026 P2 Q9
Determine the range of values of x for which the function f(x)=4x3−12x2+7\displaystyle f(x)=4x^{3}-12x^{2}+7 is strictly decreasing.
Justify your answer.
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f′(x)=12x2−24x\displaystyle f'(x)=12x^{2}-24x
Strictly decreasing when f′(x)<0\displaystyle f'(x) \lt 0, that is 12x(x−2)<0.\displaystyle 12x(x-2) \lt 0.
0<x<2\displaystyle 0 \lt x \lt 2

Question 10

Solving a trigonometric equation using formula for cos(2x)
5 Marks
2026 P2 Q10
Solve the equation 3cos⁡2x∘+10cos⁡x∘=1\displaystyle 3\cos 2x^{\circ}+10\cos x^{\circ}=1 for 0≤x<360.\displaystyle 0\le x \lt 360.
Show answer
x=70.5\displaystyle x=70.5 and x=289.5\displaystyle x=289.5

Question 11

Deriving relationship y = ab^x or y = ax^b from straight line
5 Marks
2026 P2 Q11

Two variables x and y are connected by the equation y=abx.\displaystyle y=ab^{x}.
The graph of log⁡4y\displaystyle \log_{4}y against x is a straight line as shown.

Straight line graph of log base 4 of y against x

Find the values of a and b.

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a=64\displaystyle a=64, b=16\displaystyle b=16

Question 12

Find stationary points and determine natureOptimisation on a closed interval
(3, 3)6 Marks
2026 P2 Q12

A function f is defined on the set of real numbers by
f(x)=x3+2x2−4x+1.\displaystyle f(x)=x^{3}+2x^{2}-4x+1.
(a)  Find the x-coordinates of the stationary points on the curve with equation y=f(x).\displaystyle y=f(x).
(b)  Hence determine the maximum and minimum values of f(x)\displaystyle f(x) in the interval −1≤x≤1.\displaystyle -1\le x\le 1.

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(a) x=23\displaystyle x=\frac{2}{3} and x=−2\displaystyle x=-2
(b) Maximum value 6\displaystyle 6, minimum value −1327\displaystyle -\frac{13}{27}

Question 13

Wave function (y = asin x ± bcosx)Trig equation involving compound angle
(4, 3)7 Marks
2026 P2 Q13

(a)  Express 2sin⁡x∘−7cos⁡x∘\displaystyle 2\sin x^{\circ}-7\cos x^{\circ} in the form ksin⁡(x−a)∘\displaystyle k\sin(x-a)^{\circ}, where k>0\displaystyle k \gt 0 and 0<a<360.\displaystyle 0 \lt a \lt 360.
The diagram shows part of the curve with equation y=2sin⁡x∘−7cos⁡x∘\displaystyle y=2\sin x^{\circ}-7\cos x^{\circ} and the line with equation y=3.\displaystyle y=3.
The curve and the line intersect at the point P.

Curve y = 2 sin x - 7 cos x meeting the line y = 3 at the point P

(b)  Find the x-coordinate of P.

Show answer
(a) 53sin⁡(x−74.1)∘\displaystyle \sqrt{53}\sin(x-74.1)^{\circ}
(b) −130.3\displaystyle -130.3

Question 14

Solving equations where the unknown is in the exponent
(1, 4)5 Marks
2026 P2 Q14

During a metal-working process a heated steel rod is cooled by dropping it into a container of oil.
The temperature of the rod is given by
T=800e−0.124t+25,\displaystyle T=800e^{-0.124t}+25,
where T is the temperature of the rod, in degrees Celsius, t seconds after it is dropped into the oil.
(a)  Determine the temperature of the rod as it is dropped into the oil.
(b)  Calculate the time taken for the temperature of the rod to cool to 150∘\displaystyle 150^{\circ}C.

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(a) 825∘\displaystyle 825^{\circ}C
(b) 15.0\displaystyle 15.0 seconds

Question 15

Apply double angle formula to simplify or evaluate
3 Marks
2026 P2 Q15
Express (3sin⁡θ+cos⁡θ)(sin⁡θ+3cos⁡θ)\displaystyle (3\sin\theta+\cos\theta)(\sin\theta+3\cos\theta) in the form a+bsin⁡cθ\displaystyle a+b\sin c\theta, where a, b and c are integers.
Show answer
3+5sin⁡2θ\displaystyle 3+5\sin 2\theta, so a=3\displaystyle a=3, b=5\displaystyle b=5 and c=2.\displaystyle c=2.