Higher Maths 2026 Paper 1 — Questions & Video Solutions

1

Question 1

Completing the square

3 Marks

2026 P1 Q1
Express

\displaystyle 2x^{2}+20x+3

in the form

\displaystyle p(x+q)^{2}+r.

Show answer

\displaystyle 2(x+5)^{2}-47

2

Question 2

Integrate (definite or indefinite): polynomial

4 Marks

2026 P1 Q2
Find

\displaystyle \int\left(15x^{\frac{2}{3}}+\frac{7}{x^{2}}\right)dx,

\displaystyle x \gt 0.

Show answer

\displaystyle 9x^{\frac{5}{3}}-\frac{7}{x}+c

3

Question 3

Equation of a tangent to a circle at a point

4 Marks

2026 P1 Q3

A circle has equation $\displaystyle x^{2}+y^{2}+2x-4y-20=0.$
Point A(2,6) lies on the circle.

Circle with centre marked and the tangent drawn at point A on the circumference

Find the equation of the tangent to the circle at A.

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\displaystyle 3x+4y-30=0

4

Question 4

Apply compound angle formula to simplify or evaluate

(1, 4, 2, 1)8 Marks

2026 P1 Q4

The diagram shows two right-angled triangles with angles p and q as marked.

Two right-angled triangles, one with sides 3 and 4 and angle p, the other with sides 13 and angle q

(a)  Determine the exact value of $\displaystyle \sin q.$
(b)  Find the exact value of:
    (i)   $\displaystyle \sin(p+q)$
    (ii)   $\displaystyle \cos(p+q).$
(c)  Hence find the exact value of $\displaystyle \tan(p+q).$

Show answer

(a)

\displaystyle \frac{12}{13}

(b)(i)

\displaystyle \frac{56}{65}

(b)(ii)

\displaystyle -\frac{33}{65}

(c)

\displaystyle -\frac{56}{33}

5

Question 5

Composite functionsDomain and range

(2, 1, 1)4 Marks

2026 P1 Q5

Functions f and g are defined on $\displaystyle \mathbb{R}$ , the set of real numbers, by
$\displaystyle f(x)=2x^{2}+5$ and $\displaystyle g(x)=x+3.$
(a)  Find an expression for:
    (i)   $\displaystyle f(g(x))$ and
    (ii)   $\displaystyle g(f(x)).$
(b)  State the range of $\displaystyle g(f(x)).$

Show answer

(a)(i)

\displaystyle 2x^{2}+12x+23

(a)(ii)

\displaystyle 2x^{2}+8

(b)

\displaystyle y \ge 8

6

Question 6

Vector pathways in geometric diagrams

2 Marks

2026 P1 Q6

ABCD, EFGH is a cuboid.
$\displaystyle \overrightarrow{AB}=\mathbf{u}$ , $\displaystyle \overrightarrow{BC}=\mathbf{v}$ and $\displaystyle \overrightarrow{GC}=\mathbf{w}.$
The point M is the mid-point of HG.
The point N divides EA in the ratio 1:2.

Cuboid ABCD EFGH with vectors u, v and w marked, M the midpoint of HG and N on EA

Express the vector $\displaystyle \overrightarrow{MN}$ in terms of $\displaystyle \mathbf{u}$ , $\displaystyle \mathbf{v}$ and $\displaystyle \mathbf{w}.$

Show answer

\displaystyle -\frac{1}{2}\mathbf{u}-\mathbf{v}+\frac{1}{3}\mathbf{w}

7

Question 7

Differentiate or evaluate derivative: polynomial

4 Marks

2026 P1 Q7
Determine the gradient of the tangent to the curve with equation

\displaystyle y=2x+4\sqrt{x},

\displaystyle x \gt 0

at the point where

\displaystyle x=9.

Show answer

\displaystyle \frac{8}{3}

8

Question 8

Solving equations containing a logarithm

4 Marks

2026 P1 Q8
Solve

\displaystyle \log_{2}x+\log_{2}(x+2)=3

, where

\displaystyle x \gt 0.

Show answer

\displaystyle x=2

9

Question 9

Collinearity (in 3d or 2d)Ratio in which one point divides two others

(3, 2)5 Marks

2026 P1 Q9

(a) Show that the points $\displaystyle D(-1,6,1)$ , $\displaystyle E(1,2,7)$ and $\displaystyle F(2,0,10)$ are collinear.
Point G is such that F divides DG in the ratio 3:4.
(b) Find the coordinates of G.

Show answer

(a)

\displaystyle \overrightarrow{DE}=\begin{pmatrix}2\\ -4\\ 6\end{pmatrix}

and

\displaystyle \overrightarrow{EF}=\begin{pmatrix}1\\ -2\\ 3\end{pmatrix}

, so

\displaystyle \overrightarrow{DE}=2\overrightarrow{EF}.

The vectors are parallel and share the common point E, so D, E and F are collinear.
(b)

\displaystyle G(6,-8,22)

10

Question 10

Integrate (definite or indefinite): trigonometric expression

4 Marks

2026 P1 Q10
Find

\displaystyle \int_{\frac{\pi}{6}}^{\frac{\pi}{3}}6\sin\left(3x-\frac{\pi}{2}\right)dx.

Show answer

2

11

Question 11

Cubic/quartic expressions/equations: factorise or solveDiscriminant and Quadratics

(2, 2, 4)8 Marks

2026 P1 Q11

(a)
    (i)  Show that $\displaystyle (x+2)$ is a factor of $\displaystyle x^{3}+7x^{2}+18x+16.$
    (ii)  Explain why $\displaystyle (x+2)$ is the only linear factor of $\displaystyle x^{3}+7x^{2}+18x+16.$
    Give a reason for your answer.
(b)  Find the coordinates of the point(s) where the curves with equations
$\displaystyle y=2x^{3}+20x^{2}+27x+9$ and $\displaystyle y=6x^{2}-9x-23$
intersect.

Show answer

(a)(i) Synthetic division by

\displaystyle -2

gives a remainder of

\displaystyle 0

, so

\displaystyle (x+2)

is a factor.
(a)(ii) The remaining factor is

\displaystyle x^{2}+5x+8

, which has discriminant

\displaystyle 5^{2}-4(1)(8)=-7.

Since the discriminant is negative there are no further real roots, so

\displaystyle (x+2)

is the only linear factor.
(b)

\displaystyle (-2,19)

12

Question 12

Inverse functionsGraphs of logarithmic or exponential functions

(2, 2, 1)5 Marks

2026 P1 Q12

An exponential function, f, is defined for $\displaystyle x\in\mathbb{R}.$
The diagram shows the graph of $\displaystyle y=f(x).$

Graph of the exponential function y = f(x) passing through (0, 3), (1, 1) and (2, 5)

The inverse function, $\displaystyle f^{-1}(x)$ , exists.
(a)  On the diagram in your answer booklet, sketch the graph of the inverse function.
The inverse function is of the form $\displaystyle f^{-1}(x)=\log_{a}(x+b).$
(b)
    (i)  Determine the values of a and b.
    (ii)  State the domain of the inverse function, $\displaystyle f^{-1}(x).$

Show answer

(a) The graph of

\displaystyle y=f(x)

reflected in the line

\displaystyle y=x

, with a vertical asymptote at

\displaystyle x=-4

and passing through

\displaystyle (-3,0)

,

\displaystyle (-1,1)

and

\displaystyle (5,2).

(b)(i)

\displaystyle a=3

,

\displaystyle b=4

(b)(ii)

\displaystyle x \gt -4