A Level Further Mathematics 9231 Paper 12, May/June 2023

Prove by mathematical induction that, for all positive integers $n$, $$2{\mathbf{A}}^n=\left(\begin{array}{cc}2\times 3^n& 0\\ 3^n-1& 2\end{array}\right)\mathrm{.}$$

Find, in terms of $n$, the inverse of ${\mathbf{A}}^n$.

Find the value of ${\alpha }^2+{\beta }^2+{\gamma }^2$.

Use standard results from the list of formulae (MF19) to show that $$\sum _{r=1}^n((\alpha +r{)}^2+(\beta +r{)}^2+(\gamma +r{)}^2)=n(n^2+an+b),$$ where $a$ and $b$ are constants to be determined.

Use the method of differences to find ${\sum }_{r=1}^n\frac{1}{(kr+1)(kr-k+1)}$ in terms of $n$ and $k$, where $k$ is a positive constant.

Deduce the value of ${\sum }_{r=1}^{\mathrm{\infty }}\frac{1}{(kr+1)(kr-k+1)}$.

Find also ${\sum }_{r=n}^{n^2}\frac{1}{(kr+1)(kr-k+1)}$ in terms of $n$ and $k$.

Show that $\mathbf{M}$ does not represent a rotation about the origin.

Find the equations of the invariant lines, through the origin, of the transformation in the $x\text{–}y$ plane represented by $\mathbf{M}$.

It is given that $\mathbf{M}$ represents the sequence of two transformations in the $x\text{–}y$ plane given by an enlargement, centre the origin, scale factor 5 followed by a shear, $x$-axis fixed, with $(0,1)$ mapped to $(5,1)$. Find $\mathbf{M}$.

The triangle $DEF$ in the $x\text{–}y$ plane is transformed by $\mathbf{M}$ onto triangle $PQR$. Given that the area of triangle $DEF$ is $12{\text{ cm}}^2$, find the area of triangle $PQR$.

Sketch $C$ and state the polar coordinates of the point of $C$ furthest from the pole.

Find the area of the region enclosed by $C$, the initial line, and the half-line $\theta =\pi$.

Show that, at the point of $C$ furthest from the initial line, $$(\theta +\frac{1}{\theta })\cot \theta -1=0$$ and verify that this equation has a root between 1.1 and 1.2.

Find the equations of the asymptotes of $C$.

Show that $C$ has no stationary points.

Sketch $C$, stating the coordinates of the intersections with the axes.

Sketch the curve with equation $y=\mid \frac{x^2-2x-15}{x-2}\mid$.

Find the set of values of $x$ for which $\mid \frac{2x^2+4x-30}{x-2}\mid <15$.

Obtain an equation of ${\mathrm{\Pi }}_1$ in the form $px+qy+rz=d$.

The plane ${\mathrm{\Pi }}_2$ has equation $\mathbf{r}\mathrm{.}(-5\mathbf{i}+3\mathbf{j}+5\mathbf{k})=4$. Find a vector equation of the line of intersection of ${\mathrm{\Pi }}_1$ and ${\mathrm{\Pi }}_2$.

The line $l$ passes through the point $A$ with position vector $a\mathbf{i}+a\mathbf{j}+(a-7)\mathbf{k}$ and is parallel to $(1-b)\mathbf{i}+b\mathbf{j}+b\mathbf{k}$, where $a$ and $b$ are positive constants. Given that the perpendicular distance from $A$ to ${\mathrm{\Pi }}_1$ is $\sqrt{2}$, find the value of $a$.

Given that the obtuse angle between $l$ and ${\mathrm{\Pi }}_1$ is $\frac{3}{4}\pi$, find the exact value of $b$.