Practice Paper 2 (Edexcel IAL Further Maths)

$2z+z^{\ast}=\frac{3+4i}{7+i}$

Find $z$, giving your answer in the form $a+bi$, where $a$ and $b$ are real constants. You must show all your working.

$f(z)=z^{4}+6z^{3}+76z^{2}+az+b$

where $a$ and $b$ are real constants.

Given that $-3+8i$ is a complex root of the equation $f(z)=0$

write down another complex root of this equation.

Hence, or otherwise, find the other roots of the equation $f(z)=0$

Show on a single Argand diagram all four roots of the equation $f(z)=0$

The parabola $C$ has equation $y^{2}=4ax$, where $a$ is a positive constant.

The line $l$ with equation $3x-4y+48=0$ is a tangent to $C$ at the point $P$.

Show that $a=9$

Hence determine the coordinates of $P$.

Given that the point $S$ is the focus of $C$ and that the line $l$ crosses the directrix of $C$ at the point $A$,

determine the exact area of triangle $PSA$.

The elements of each matrix should be expressed in exact numerical form.

(a) Write down the $2\times2$ matrix that represents a rotation of $210^{\circ}$ anticlockwise about the origin.

(b) Write down the $2\times2$ matrix that represents a stretch parallel to the $y$-axis with scale factor 5.

The transformation $T$ is a rotation of $210^{\circ}$ anticlockwise about the origin followed by a stretch parallel to the $y$-axis with scale factor 5.

(c) Determine the $2\times2$ matrix that represents $T$.

$M = ( \begin{gathered} k k + 3 \\ - 5 & 1 - k \end{gathered} ) \text{where } k \text{ \text{is} a constant}$

(a) Find det $M$, giving your answer in simplest form in terms of $k$.

A closed shape $R$ is transformed to a closed shape $R'$ by the transformation represented by the matrix $M$.

Given that the area of $R$ is 2 square units and that the area of $R'$ is $16k$ square units,

(b) determine the possible values of $k$.

$P = ( \begin{gathered} 1 \frac{1}{2} & - \frac{ \sqrt{3} }{2} \\ \frac{ \sqrt{3} }{2} & 1 \frac{1}{2} \end{gathered} )$

The matrix $P$ represents the transformation $U$

Give a full description of $U$ as a single geometrical transformation.

The transformation $V$ , represented by the 2×2 matrix $Q$ , is a reflection in the line $y=-x$

Write down the matrix $Q$

The transformation $U$ followed by the transformation $V$ is represented by the matrix $R$

Determine the matrix $R$

The transformation $W$ is represented by the matrix $3R$

The transformation $W$ maps a triangle $T$ to a triangle $T'$

The transformation $W'$ maps the triangle $T'$ back to the original triangle $T$

Determine the matrix that represents $W'$

$f(x)=x^{3}+4x-6$

(a) Show that the equation $f(x)=0$ has a root $α$ in the interval $[ 1 , 1.5 ]$

(b) Taking $1.5$ as a first approximation, apply the Newton–Raphson process twice to $f(x)$ to obtain an approximate value of $α$.Give your answer to 3 decimal places. Show your working clearly.

$g(x)=4x^{2}+x-\mathrm{tan}x$

where $x$ is measured in radians.

The equation $g(x)=0$ has a single root $β$ in the interval $[ 1.4 , 1.5 ]$.

Use linear interpolation on the values at the end points of this interval to obtain an approximation to $β$. Give your answer to 3 decimal places.

Prove by induction that for $n\inℕ$

$\sum_{r = 1}^{n} r^{2} = \frac{n}{6} ( n + 1 ) ( 2 n + 1 )$

Hence show that

$\sum_{r = 1}^{n} ( r^{2} + 2 ) = \frac{n}{6} ( a n^{2} + b n + c )$

where $a$, $b$ and $c$ are integers to be found.

Using your answers to part (b), find the value of

$\sum_{r = 10}^{25} ( r^{2} + 2 )$

The quadratic equation

$4x^{2}+3x+k=0$

where equation$k$ is an integer, has roots $α$ and $β$

Write down, in terms of $k$ where appropriate, the value of $α+β$ and the value of $αβ$

Determine, in simplest form in terms of $k$ , the value of$\frac{α}{β^{2}}+\frac{β}{α^{2}}$

Determine a quadratic equation which has root

$\frac{α}{β^{2}}\text{and}\frac{β}{α^{2}}$

giving your answer in the form $px^{2}+qx+r=0$ where equation$p$, $q$ and $r$ are integer values in terms of $k$

Use the standard results for $\sum_{r = 1}^{n} r$ and for $\sum_{r = 1}^{n} r^{3}$ to show that, for all positive integers $n$,

$\sum_{r = 1}^{n} r \left(r^{2} - 3\right) = \frac{n}{4} \left(n + a\right) \left(n + b\right) \left(n + c\right)$

where $a$, $b$ and $c$ are integers to be found.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

$z_{1}=3+2iz_{2}=2+3iz_{3}=a+bia,b\inℝ$

Determine the exact value of $\vert z_{1} + z_{2} \vert$

Given that $w=\frac{z_{2}z_{3}}{z_{1}}$

Determine $w$in terms of $a$ and $b$, giving your answer in the form $x+iy$ , where $x,y\inℝ$

Given also that $w=\frac{4}{13}+\frac{58}{13}i$

Determine the value of $a$ and the value of $b$

Determine $\mathrm{arg}w$, giving your answer in radians to 4 significant figures.