A Level Further Mathematics 9231 Paper 23, May/June 2026

Question paper and mark scheme75 marks8 questions

Solve it on ExamPal

Practise

Untimed. Write on the paper and check as you go.

Start

Exam

Timed, with a clock you set, like sitting the real paper.

Start

Opens full screen.

Download PDF Mark scheme

100%

Opening the question paper

Read as textThe questions and the mark scheme as plain text. Easier to read on a phone.

Question paper

9231/23 · May/June 2026

Download PDF

Find the roots of the equation $w^{5} + 32 = 0$ , giving your answers in the form $r e^{i θ}$ , where $r > 0$ and $- π ⩽ θ < π$ .

[5]

A curve has equation $y = \sqrt{x}$ , for $0 ⩽ x ⩽ 2$ . Find, in terms of $π$ , the area of the surface generated when the curve is rotated through $2 π$ radians about the $x$ -axis.

[5]

Let $I_{n} = \int_{0}^{a} (\ln (x + 1))^{n} d x$ , where $n$ is a non-negative integer and $a > 0$ .

By considering the derivative of $(x + 1) (\ln (x + 1))^{n}$ , or otherwise, show that for $n ⩾ 1$ $I_{n} = (a + 1) (\ln (a + 1))^{n} - n I_{n - 1} .$

[3]

Find $I_{2}$ in terms of $a$ .

[3]

Find the particular solution of the differential equation $49 \frac{d^{2} x}{d t^{2}} + 14 \frac{d x}{d t} + x = 196 e^{\frac{1}{7} t},$ given that, when $t = 0$ , $x = 7$ and $\frac{d x}{d t} = 0$ .

[10]

Starting from the definitions of $s e c h$ and $\tanh$ in terms of exponentials, prove that $1 - {s e c h}^{2} t = \tanh^{2} t .$

[3]

It is given that $x = t \sinh t and y = t \cosh t .$ Show that $\frac{d y}{d x} = \frac{t \tanh t + 1}{t + \tanh t}$ .

[3]

Show that $\frac{d^{2} y}{d x^{2}} = \frac{(t^{2} - 2) {s e c h}^{n} t}{(t + \tanh t)^{n}}$ , where $n$ is an integer to be determined.

[6]

Show that an appropriate integrating factor for $(1 - x^{2}) \frac{d y}{d x} + y = (1 - x^{2}) \sqrt{1 - x}$ is $\sqrt{\frac{1 + x}{1 - x}}$ .

[4]

Hence find the solution of the differential equation $(1 - x^{2}) \frac{d y}{d x} + y = (1 - x^{2}) \sqrt{1 - x}$ for which $y = 1$ when $x = 0$ .

[6]

Use the substitution $u = 1 - x^{2}$ to find $\int \frac{x}{\sqrt{1 - x^{2}}} d x$ .

[3]

The diagram shows the curve with equation for $y = \sin^{- 1} x$ for $0 ⩽ x ⩽ 1$ , together with a set of $n$ rectangles of width $\frac{1}{n}$ . By considering the sum of the areas of these rectangles, show that $\sum_{r = 1}^{n} \frac{1}{n} \sin^{- 1} (\frac{r}{n}) ⩾ \frac{1}{2} π - 1.$

[5]

Use a similar method to find, in terms of $n$ , an upper bound for $\sum_{r = 1}^{n} \frac{1}{n} \sin^{- 1} (\frac{r}{n})$ .

[3]

Deduce the exact value of $\lim_{n \to \infty} \sum_{r = 1}^{n} \frac{1}{n} \sin^{- 1} (\frac{r}{n})$ .

[1]

The constants $a$ and $b$ are such that the system of equations $\begin{aligned} 6 x - 12 y - 12 z & = 1, \\ a x + 4 y + z & = 2, \\ 2 x + b y - 8 z & = 3, \end{aligned}$ does not have a unique solution. Find $a$ in terms of $b$ .

[3]

Given that $a = - \frac{1}{4}$ and $b = - 32$ , show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically.

[3]

The matrix $A$ is given by $A = (\begin{array}{ccc} 6 & - 12 & - 12 \\ - 1 & 4 & 1 \\ 2 & - 8 & - 8 \end{array}) .$ Show that $(\begin{array}{c} 1 \\ 0 \\ 1 \end{array})$ is an eigenvector of $A$ and state the corresponding eigenvalue.

[2]

Find a matrix $P$ and a diagonal matrix $D$ such that $A = {P D P}^{- 1}$ .

[7]

Mark scheme

9231/23 · May/June 2026

Download PDF

1 · 5 marks

B1
Converts to exponential form.
M1
Finds one root in required form (method).
A1
Finds one root in required form (accuracy).
A1
Finds another root.
A1
Finds other three roots.

2 · 5 marks

B1
Finds $\sqrt{1 + {(\frac{d y}{d x})}^{2}}$ .
M1
Correct formula for surface area with limits. Integral must be in terms of $x$ .
A1
Integrand simplified correctly.
M1
Integrates and substitutes limits. Integrand must be of the form $(p x + a)^{n}$ with $a$ and $p$ nonzero and consistent with previous working and $n \neq 1$ .
A1
$= \frac{13}{3} π$ AEF.

3a · 3 marks

B1
CAO (AEF) or CAO (Condone missing limits).
M1
Integrates both sides using the limits given or expression needs to be fully integrated and in terms of $a$ and $I_{n - 1}$ .
A1
Substitutes limits and rearranges. AG / Substitutes limits, convincing work to AG.

3b · 3 marks

B1
$I_{1} = (a + 1) \ln (a + 1) - a$
M1
Applies reduction formula to find $I_{2}$ .
A1
$I_{2} = (a + 1) \ln^{2} (a + 1) - 2 (a + 1) \ln (a + 1) + 2 a$

4 · 10 marks

M1
Auxiliary equation.
A1
Complementary function.
B1
Particular integral and its derivatives.
M1
Substitutes and equates coefficients.
A1
$k = 49$
A1
General solution.
M1
General solution must be of correct form for any further marks.
M1
Substitutes initial conditions to find $A$ and $B$ (method).
A1
Substitutes initial conditions to find $A$ and $B$ (accuracy).
A1
$x = e^{- \frac{1}{7} t} (- 13 t - 42) + 49 e^{\frac{1}{7} t}$

5a · 3 marks

B1
$sech t = \frac{2}{e^{t} + e^{- t}} \tanh t = \frac{e^{t} - e^{- t}}{e^{t} + e^{- t}}$
M1
Expands, gets to $\frac{e^{2 t} + 2 + e^{- 2 t} - 4}{(e^{t} + e^{- t})^{2}}$ for M1, AG.
A1
Variable used must be consistent throughout. A0 if a mixture of variables is used.

5b · 3 marks

B1
$\frac{d y}{d t} = t \sinh t + \cosh t, \frac{d x}{d t} = t \cosh t + \sinh t$
M1
Uses chain rule.
A1
Divides numerator and denominator by $\cosh t$ , AG.

5c · 6 marks

M1
Applies quotient rule (method).
A1
Applies quotient rule (accuracy).
M1
Simplifies numerator to the form $(t^{2} - 2) {sech}^{n} t$ by applying $\tanh^{2} t = 1 - {sech}^{2} t$ .
A1
$= \frac{(t^{2} - 2) {sech}^{2} t}{(t + \tanh t)^{2}}$
M1
Uses chain rule, multiplies by their $\frac{d t}{d x}$ for M1.
A1
Convincing argument to final result for A1.

6a · 4 marks

B1
Divides through by $1 - x^{2}$ .
M1
Finds integrating factor. Allow $e^{\int \frac{a}{1 - x^{2}} d x}$ for M1.
A1
Integrates to find $e^{\tanh^{- 1} x}$ or $e^{\frac{1}{2} \ln (\frac{1 + x}{1 - x})}$ .
A1
AG. Must have obtained all previous marks.

6b · 6 marks

M1
Correct form on LHS and RHS (method).
A1
Correct form on LHS and RHS (accuracy).
M1
Integrates RHS (method).
A1
Integrates RHS (accuracy).
M1
Substitutes initial conditions.
A1
$y = \frac{2}{3} (1 + x) (1 - x)^{\frac{1}{2}} + \frac{1}{3} (1 + x)^{- \frac{1}{2}} (1 - x)^{\frac{1}{2}}$ . Final answer must be in terms of $x$ and not be in terms of integrating factor.

7a · 3 marks

M1
Applies given substitution. Allow $\int \frac{a}{\sqrt{u}} d u$ for M1.
A1
Correct integration in terms of $u$ .
A1
Accept without '+C'. Must be in terms of $x$ .

7b · 5 marks

M1
M1 correct comparison (no limits required)
A1
A1 correct limits on both sides
M1
Integrates by parts.
A1
CAO with correct signs
A1
Clear substitution of limits to given result

7c · 3 marks

M1
Forms the sum of the areas of appropriate rectangles.
A1
Adjusts summation with reference to missing term.
A1
CAO

7d · 1 mark

B1
$\frac{1}{2} π - 1$

8a · 3 marks

M1
Finds determinant in terms of $a$ and $b$ (method) or row eliminates $z$ from two of the equations and forms an equation from resulting coefficients being in ratio.
A1
Finds determinant in terms of $a$ and $b$ (accuracy) or correct ratio equation.
A1
$a = - \frac{b + 20}{2 (b + 8)} [b \neq - 8.]$ CAO

8b · 3 marks

B1
Must show an inconsistent system.
B1
Two distinct parallel planes.
B1
Other plane not parallel.

8c · 2 marks

M1
Multiplies $A$ with given eigenvector.
A1
$- 6$

8d · 7 marks

M1
Characteristic equation.
A1
$λ = 2, 6, - 6$
M1
Uses vector product (or equations) to find corresponding eigenvectors.
A1
Eigenvector for $λ = 2$ .
A1
Eigenvector for $λ = 6$ .
M1
Forms matrix $P$ and matrix $D$ (method).
A1
Forms matrix $P$ and matrix $D$ (accuracy). Or correctly matched permutations of columns. M0A0 if a zero eigenvector is included.

Questions

1
Question 15 marks
Mark scheme ›
2
Question 25 marks
Mark scheme ›
3
Question 36 marks
Mark scheme ›
4
Question 410 marks
Mark scheme ›
5
Question 512 marks
Mark scheme ›
6
Question 610 marks
Mark scheme ›
7
Question 712 marks
Mark scheme ›
8
Question 815 marks
Mark scheme ›