Practice Paper 2 (Edexcel IAL Maths)

The graph of $y=f(x)$ where $f(x)=2x-3$ is shown below.

Determine the coordinates of the points marked $A$ and $B$.

(i) On the diagram above sketch the graph of $y = \vert \mathrm{f} ( x - 1 ) \vert$.

(ii) Determine the coordinates of the image of the points $A$and $B$ under the transformation in part (i).

State whether the following mappings are one-to-one, many-to-one, one-to-many or many-to-many.

(i) $\mathrm{f} : x \rightarrow \text{from} \text{bar} x^{2}$

(ii) $\mathrm{f} : x \rightarrow \text{from} \text{bar} 3 x + 1$

(iii) $\mathrm{f} : x \rightarrow \text{from} \text{bar} ( x + 1 )^{3}$

(iv) $\mathrm{f} : x \rightarrow \text{from} \text{bar} \pm \sqrt{x}$

Simplify fully $\frac{x^{2}+x-2}{x^{3}+4x^{2}-4x-1}$

Simplify fully $\frac{3x+9}{x+2}\times\frac{x^{2}+6x+8}{x+3}$

Simplify fully $\frac{x^{2}+8x-9}{x^{2}+7x+12}\div\frac{x^{2}+11x+18}{2x^{2}+7x-4}$

Differentiate $(x^{3}-2x)\mathrm{ln}x$with respect to x.

Differentiate $e^{x}\mathrm{cos}2x$ with respect to x.

A particle is travelling with velocity, $v$ ms^-1, at time $t$ seconds. The velocity of the particle is modelled $v=0.3e^{kt}$, where $k$ is a constant.

Write down the initial velocity of the particle.

Find an expression (in terms of $k$ and $t$) for the acceleration of the particle.

After 12 seconds the velocity of the particle is 0.9 ms^-1. Find the value of $k$, giving your answer to 3 significant figures.

State a problem with the model for large values of $t$

Show that the equation

$y=2x^{\frac{3}{4}}$

can be written as

$\mathrm{log}y=0.75\mathrm{log}x+\mathrm{log}2$

Rewrite the equation $\mathrm{log}y=4.7\mathrm{log}x+\mathrm{log}12$ in the form $y=Ax^{b}$.

Find

$\int \sqrt{1 + \cot ^{2} x} \mathrm{d} x$

The game of Logball is played on a flat table.

A player rolls a ball from a fixed point, at any angle, with the aim of it coming to rest within a winning zone.

A particular player decides to roll the ball at an angle of 45°, as illustrated in the graph below, with the ball being rolled from the origin and the shaded area being the winning zone.

The lower boundary of the winning zone has equation $y=3-\mathrm{ln}(x+1)^{2}$, $x,y\geq0$

The upper boundary of the winning zone has equation $y = 4 - \ln ( x + 1 )^{2}$ , $x,y\geq0$

Using an appropriate iterative formula with initial value $x_{0}=1.8,$, find the minimum distance this player’s ball needs to travel to stop within the winning zone. Give your answer to two significant figures.

Using another iterative formula with initial value $x_{0}=2.5,$, find the maximum distance this player’s ball can travel yet remain within the winning zone. Give your answer to two significant figures.

Sketch the graph of $y=\mathrm{tan}2θ$ for $0\leqθ\leq2π$.

Label the points at which the graph intersects the coordinate axes.

Show that the equation cosec² $x=2$ cosec $x-1$ can be written as

$( \text{cosec} x - 1 ) ^{2} = 0$

Hence, or otherwise, solve the equation

cosec² $x=2$ cosec $x-1,-2π\leq x\leq2π$

Using the double angle formula sin $2A\equiv2$ sin $A$ cos $A$, find the solutions to the equation

sec $x$ cosec $x\quad -75=5$ cosec $2x$

in the range $-π<x\leqπ$. Give your answers correct to 3 significant figures.

Show that

sin $3θ+$sin $θ\equiv4$ sin $θ-4$sin³ $θ$

The diagram below shows the graph of $y=f(x)$. The stationary points are marked on the diagram.

On separate diagrams, sketch the graphs with equation

(i) $y=2f(x)-4$

(ii) $y=f(x+1)+3$

On each diagram, state the coordinates of the images of the points $A$and $B$ under the given transformation.

State whether the following mappings are one-to-one, many-to-one, one-to-many or many-to-many.

(i) $\mathrm{f} : x \rightarrow \text{from} \text{bar} \tan x$

(ii) $\mathrm{f} : x \rightarrow \text{from} \text{bar} \vert \frac{1}{x} \vert$

(iii) $\mathrm{f} : x \rightarrow \text{from} \text{bar} \sqrt{x^{2}}$

(iv) $\mathrm{f} : x \rightarrow \text{from} \text{bar} \pm \sqrt{25 - x^{2}}$

Simplify

(i) $\frac{x}{2x}$

(ii) $\frac{x+1}{x(x+1)}$

(iii) $\frac{6x+12}{x^{2}+2x}$

A is the point on the graph of $y=\mathrm{arctan}x$such that the tangent to the graph at $A$ passes through the point $( 0 , 1 \frac{1}{2} )$. Show that the x-coordinate of A satisfies the equation

$x - \tan ( \frac{( 1 + x )^{2}}{2 ( 1 + x^{2} ) } ) = 0$