Practice Paper 1 (Edexcel IAL Further Maths)

Given that

$\frac{3z-1}{2}=\frac{λ+5i}{λ-4i}$

where $λ$ is a real constant,

determine $z$, giving your answer in the form $x+yi$, where $x$ and equation$y$ are real and in terms of $λ$.

Given also that $\mathrm{arg}z=\frac{π}{4}$

find the possible values of equation$λ$.

$f(z)=z^{4}+az^{3}+bz^{2}+cz+d$

where $a$, $b$, $c$ and $d$ are integers.

The complex numbers $3+i$ and $-1-2i$ are roots of the equation $f(z)=0$

Write down the other roots of this equation.

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

Determine the values of $a$, $b$, $c$ and $d$.

The parabola $C$ has equation $y^{2}=\frac{4}{3}x$

The point $P ( 1 \frac{1}{3} t^{2} ,   \frac{2}{3} t )$ , where $t\neq0$,lies on $C$.

Use calculus to show that the normal to $C$ at $P$ has equation

$3tx+3y=t^{3}+2t$

The normal to $C$ at the point where $t=9$ meets $C$ again at the point $Q$.

Determine the exact coordinates of $Q$.

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

Determine the matrix $A^{2}$

Describe fully the single geometrical transformation represented by the matrix $A^{2}$

Hence determine the smallest positive integer value of $n$ for which $A^{n}=I$

The matrix B represents a stretch scale factor 4 parallel to the $x$-axis.

Write down the matrix B

The transformation represented by matrix A followed by the transformation represented by matrix B is represented by the matrix C

Determine the matrix C

The parallelogram $P$ is transformed onto the parallelogram $P'$ by the matrix C

Given that the area of parallelogram $P'$ is 20 square units, determine the area of parallelogram $P$

$A = ( \begin{gathered} 1 0 \\ 0 3 \end{gathered} )$

(a) Describe fully the single transformation represented by the matrix $A$.

The matrix $B$ represents a rotation of $45^{\circ}$ clockwise about the origin.

(b) Write down the matrix $B$ , giving each element of the matrix in exact form.

The transformation represented by matrix $A$ followed by the transformation represented by matrix $B$ is represented by the matrix $C$.

(c) Determine $C$.

The trapezium $T$ has vertices at the points $(-2,0)$, $(-2,k)$, $(5,8)$ and $(5,0)$, where $k$ is a positive constant. Trapezium $T$ is transformed onto the trapezium $T'$ by the matrix

$( \begin{gathered} 5 1 \\ - 2 & 3 \end{gathered} )$

Given that the area of trapezium $T'$ is 510 square units, calculate the exact value of $k$.

Show that the equation $4x-2\mathrm{sin}x-1=0$, where $x$is in radians, has a root in $α$ the interval$[ 0.2 , 0.6 ]$

Starting with the interval $[ 0.2 , 0.6 ]$,use interval bisection twice to find an interval of width $0.1$ in which $α$ lies.

Prove, by induction, that for $n\inℤ$, n greater-than or slanted equal to 2

$4^{n}+6n-10$

is divisible by 18

The quadratic equation

$2x^{2}-5x+7=0$

has roots $α$ and $β$.

Without solving the equation,

write down the value of $(α+β)$ and the value of $αβ$

determine, giving each answer as a simplified fraction, the value of

(i) $α^{2}+β^{2}$

(ii) $α^{3}+β^{3}$

find a quadratic equation that has roots

$\frac{1}{α^{2}+β}\text{and}\frac{1}{β^{2}+α}$

giving your answer in the form $px^{2}+qx+r=0$ where $p$,$q$ and $r$ are integers to be determined.