Learn Extracted exam questions A-Level Further Mathematics 9231 Mathematics - Further November 2025 Question Paper 22
9231 Mathematics - Further November 2025 Question Paper 22
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The curve $C$ has polar equation $r = e^{\frac{1}{4}\theta}$ for $0 \leqslant \theta \leqslant \pi$. Find the length of $C$. Give your answer in an exact form.
Find the roots of the equation $z^4 = 8 - 8i\sqrt{3}$, giving your answers in the form $re^{i\theta}$, where $r > 0$ and $-\pi \leqslant \theta < \pi$.
The variables $x$ and $y$ are such that $y = 2$ when $x = -1$ and $x^2 y + (x + y)^3 = 3$.
Show that $\dfrac{dy}{dx} = \dfrac{1}{4}$ when $x = -1$.
Find the value of $\dfrac{d^2y}{dx^2}$ when $x = -1$.
By considering the binomial expansion of $\left(z + \dfrac{1}{z}\right)^6$, where $z = \cos\theta + i\sin\theta$, use de Moivre's theorem to show that $\cos^6\theta = a\cos 6\theta + b\cos 4\theta + c\cos 2\theta + d$, where $a$, $b$, $c$ and $d$ are constants to be determined.
Find the exact value of $\displaystyle\int_0^{\frac{1}{2}\pi}\cos^6 2x\,dx$.
Find the general solution of the differential equation $\dfrac{d^2y}{dx^2} + 4\dfrac{dy}{dx} + 3y = 5\cos x$.
For large positive values of $x$ and for any initial conditions, show that the solution to part (a) can be approximated by $y \approx R\sin(x + \phi)$, where the constants $R$ and $\phi$ are to be determined.
Use the substitution $x = \tfrac{1}{2}\sqrt{2}\sinh u$ to find $\displaystyle\int\dfrac{1}{\sqrt{2x^2 + 1}}\,dx$.
The diagram shows the curve with equation $y = \dfrac{1}{\sqrt{2x^2 + 1}}$ for $0 \leqslant x \leqslant 1$, together with a set of $n$ rectangles of width $\dfrac{1}{n}$. By considering the sum of the areas of these rectangles, show that $\displaystyle\sum_{r=1}^{n}\dfrac{1}{\sqrt{2r^2 + n^2}} < \tfrac{1}{2}\sqrt{2}\,\ln\!\left(\sqrt{2} + \sqrt{3}\right)$.
Use a similar method to find, in terms of $n$, a lower bound for $\displaystyle\sum_{r=1}^{n}\dfrac{1}{\sqrt{2r^2 + n^2}}$.
Deduce the exact value of $\displaystyle\lim_{n\to\infty}\sum_{r=1}^{n}\dfrac{1}{\sqrt{2r^2 + n^2}}$.
Show that $\dfrac{d}{dx}\!\left(\tfrac{1}{2}x\sqrt{4 - x^2} + 2\sin^{-1}\!\left(\tfrac{1}{2}x\right)\right) = \sqrt{4 - x^2}$.
Find the solution of the differential equation $2\dfrac{dy}{dx} + \dfrac{y}{2 + x} = 2\sqrt{2 - x}$ for which $y = \tfrac{1}{2}$ when $x = 1$. Give your answer in an exact form.
Find the values of $k$ for which the system of equations $x - y + 2kz = 1$, $kx + y + 2z = 2$, $2x - y + z = 3$ does not have a unique solution.
Given that $k = -\tfrac{1}{2}$, show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically.
Given instead that $k = -1$, show that the system of equations in part (a) is also inconsistent. Interpret this situation geometrically.
The matrix $\mathbf{A}$ is given by $\mathbf{A} = \begin{pmatrix} 1 & -1 & -2 \\ -1 & 1 & 2 \\ 2 & -1 & 1 \end{pmatrix}$. Use the characteristic equation of $\mathbf{A}$ to show that $\mathbf{A}^4 = p\mathbf{A}^2 + q\mathbf{A}$, where $p$ and $q$ are integers to be determined.