A-Level Further Maths – Core Pure

Given that $1 + 2\text{i}$ is a root of $z^3 – 5z^2 + 11z – 15 = 0$, find the other two roots.

Let $A = \begin{pmatrix}2 & 1\\3 & 4\end{pmatrix}$.(a) Find $A^{-1}$.(b) The matrix $A$ represents a transformation $T$. Describe the effect of $T$ on the area of a shape.(c) Find the eigenvalues and corresponding eigenvectors of $M = \begin{pmatrix}4 & 1\\2 & 3\end{pmatrix}$.

The roots of $z^3 – 2z^2 + 3z – 1 = 0$ are $\alpha$, $\beta$ and $\gamma$.(a) Write down $\alpha + \beta + \gamma$, $\alpha\beta + \beta\gamma + \gamma\alpha$ and $\alpha\beta\gamma$.(b) Find $\alpha^2 + \beta^2 + \gamma^2$.(c) Find $\alpha^3 + \beta^3 + \gamma^3$.

Prove by induction that $7^n + 5$ is divisible by 6 for all positive integers $n$.

Let $z = -2 + 2\sqrt{3}\,\text{i}$.(a) Write $z$ in the form $r\text{e}^{\text{i}\theta}$, $-\pi < \theta \leq \pi$.(b) Hence find $z^4$ in the form $a + b\text{i}$.(c) Find the four roots of $w^4 = -16$, giving your answers in the form $a + b\text{i}$.

(a) Solve $3\cosh x – 2\sinh x = 3$, giving answers in exact logarithmic form.(b) Find $\displaystyle\int \frac{1}{\sqrt{x^2 + 4x + 13}}\,\text{d}x$.

(a) Use standard results to show that $\displaystyle\sum_{r=1}^{n} (2r – 1)^2 = \frac{n(2n-1)(2n+1)}{3}$.(b) Use the method of differences to find $\displaystyle\sum_{r=1}^{n} \frac{1}{r(r+1)}$ and hence state the sum to infinity.

Let $B = \begin{pmatrix}1 & 2 & 0\\0 & 1 & 3\\2 & 0 & 1\end{pmatrix}$.(a) Show that $\det B = 13$.(b) Find $B^{-1}$.(c) Hence solve the system $x + 2y = 3$, $y + 3z = 4$, $2x + z = 5$.

(a) Find the Maclaurin series of $\ln(1 + 2x)$ up to and including the term in $x^3$, and state the range of $x$ for which it is valid.(b) Hence find an approximation for $\ln 1.2$ to 4 decimal places and compare it with the calculator value.

The curve $C$ has polar equation $r = 2 + \cos\theta$, $0 \leq \theta < 2\pi$.(a) Find the area enclosed by $C$.(b) Find the exact values of $r$ at the points where the tangent to $C$ is parallel to the initial line, other than where $\theta = \pi$. [Hint: consider $y = r\sin\theta$.]

(a) The region bounded by $y = \text{e}^{-x}$, the $x$-axis and the lines $x = 0$ and $x = \ln 2$ is rotated through $2\pi$ radians about the $x$-axis. Find the exact volume of the solid formed.(b) Find the mean value of $\cosh x$ on the interval $[0, \ln 3]$, giving your answer in exact form and to 3 significant figures.(c) Evaluate $\displaystyle\int_0^1 \frac{1}{x^2 + 2x + 5}\,\text{d}x$ exactly.