Learn Extracted exam questions A-Level Further Mathematics 9231 Mathematics - Further November 2025 Question Paper 24
9231 Mathematics - Further November 2025 Question Paper 24
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It is given that $I_n = \displaystyle\int_0^1 x^n \sinh x\,dx$, where $n$ is a non-negative integer.
Show that, for $n \geqslant 2$, $I_n = \cosh 1 - n\sinh 1 + n(n-1)I_{n-2}$.
Find the exact value of $I_2$.
Find the exact values of $x$ for which $\cosh 2x = 6\sinh^2 x$, giving your answers in logarithmic form.
Sketch the curves $C_1: y = \cosh 2x$ and $C_2: y = 6\sinh^2 x$ on the same diagram.
Use de Moivre's theorem to show that $\sin 5\theta = 16\sin^5\theta - 20\sin^3\theta + 5\sin\theta$.
Hence obtain the roots of the equation $32x^5 - 40x^3 + 10x - \sqrt3 = 0$ in the form $\sin q\pi$, where $q$ is a rational number.
The diagram shows the curve with equation $y = \tfrac12(3^x)$ for $0 \leqslant x \leqslant 1$, together with a set of $N$ rectangles each of width $\tfrac1N$.
By considering the sum of the areas of these rectangles, show that $\displaystyle\int_0^1 \tfrac12(3^x)\,dx < U_N$, where $U_N = \dfrac{3^{1/N}}{N(3^{1/N} - 1)}$.
Use a similar method to find, in terms of $N$, a lower bound $L_N$ for $\displaystyle\int_0^1 \tfrac12(3^x)\,dx$.
By simplifying $U_N - L_N$, show that $\displaystyle\lim_{N\to\infty}(U_N - L_N) = 0$.
Find the general solution of the differential equation $\dfrac{d^2y}{dx^2} + \dfrac{dy}{dx} + y = \sin 2x + \cos 2x$.
For large positive values of $x$ and for any initial conditions, show that the solution to part (a) can be approximated by $y \approx R\sin(2x - \phi)$, where $R$ and $\phi$ are positive constants to be determined.
Find the particular solution of the differential equation $\dfrac{dy}{dx} + \dfrac{1+x}{1 - 2x - x^2}y = 1$, given that $y = \pi$ when $x = 0$. Give your answer in an exact form.
The curve $C$ has parametric equations $x = 8\ln(\tan\tfrac12 t) - 3\cot t - 3t$, $y = 6\ln(\tan\tfrac12 t) + 4\cot t + 4t$, for $\tfrac16\pi \leqslant t \leqslant \tfrac13\pi$.
Show that $\dfrac{dx}{dt} = 8\,\mathrm{cosec}\,t + 3\cot^2 t$.
Find $\dfrac{dy}{dt}$.
Find the exact value of the length of $C$.
Show that $\dfrac{d^2y}{dx^2} = \dfrac{a\cot t\,\mathrm{cosec}\,t(\mathrm{cosec}^2 t + 1)}{(b\,\mathrm{cosec}\,t + c\cot^2 t)^n}$, where $a$, $b$, $c$ and $n$ are integers to be determined.
Find the set of values of $a$ for which the system of equations $5x + ay = 3$, $20x - 5y = 2$, $2x - 3y + z = 1$ has a unique solution and interpret this situation geometrically.
Given that $a = -\tfrac54$, show that the system of equations in part (a) is inconsistent and interpret this situation geometrically.
The matrix $\mathbf{A}$ is given by $\mathbf{A} = \begin{pmatrix} 5 & 0 & 0 \\ 20 & -5 & 0 \\ 2 & -3 & 1 \end{pmatrix}$. Find a matrix $\mathbf{P}$ and a diagonal matrix $\mathbf{D}$ such that $(\mathbf{A} + 3\mathbf{I})^2 = \mathbf{P}\mathbf{D}\mathbf{P}^{-1}$.