Practice Paper 2 (Edexcel IAL Maths)

On the same diagram, sketch the graphs of $y=x(x+2)(x-1)$ and $y=\frac{1}{x}$.

Use your graph to determine the number of solutions to the equation

$x(x+2)(x-1)=\frac{1}{x}$.

The equation $kx^{2}+2kx+4=0$, where $k$ is a constant, has two distinct real roots.

Find the possible value(s) of k.

Simplify the following expressions, giving your answers in the form $ax^{n}$ where $a$ and $n$ are rational numbers and any fractions are in lowest terms.

$7x^{-3}\div2x^{-\frac{1}{3}}$

$\frac{2}{3}x^{\frac{7}{6}}\times\frac{5}{2}x^{-\frac{5}{3}}$

$( 3 x^{- \frac{2}{3} } )^{2} \div 6 x^{- \frac{1}{6} }$

Given that $(2x-3y)^{2}(y-2x)=ax^{3}+bx^{2}y+cxy^{2}+dy^{3}$, where $a$,$b$,$c$ and $d$ are constants, find the values of $a$,$b$,$c$ and $d$.

Solve the equation $2x^{2}+x-6=0$.

Find the coordinates of the turning point on the graph of $y=2x^{2}+x-6$.

Sketch the graph of $y=2x^{2}+x-6$, labelling the turning point and any points where the graph crosses the coordinate axes.

Solve the simultaneous equations

$5x-2y-16=0$ $3x+7y+15=0$

A curve has equation $y = \mathrm{f} ( x )$.

Write down the equations of the curves, in terms of $f\quad ( x )$, given by the following transformations:

(i) Translation by the vector $( \begin{gathered} 3 \\ 0 \end{gathered} )$,

(ii) Horizontal stretch, scale factor 2,

(iii) Vertical stretch, scale factor $\frac{1}{3}$,

(iv) Reflection in the $y$-axis.

Find the length of the straight line segments connecting the following points:

(i) (2 , 4) and (5 , 8),

(ii) (3 , - 6) and (-2, -14),

(iii) (5, -13) and (2, -7).

For each of the following, find $\frac{dy}{dx}$ in terms of x :

$y=-\frac{5}{4}x^{3}+\frac{3}{5}x^{2}-x\sqrt{2}+π$

$y=\frac{3}{2}x^{\frac{4}{5}}-\frac{10}{3}x^{-\frac{4}{5}}$

Show that $(2-x)^{3}=8-12x+6x^{2}-x^{3}$

Hence, or otherwise, work out $\int ( 2 - x )^{3} d x$

A triangle ABC has side lengths AB = 5 cm, BC = 4 cm and AC = 7 cm. Calculate the size of the angle BAC, giving your answer to the nearest degree.

A triangle PQR has side lengths PQ = 12 cm, QR = 8 cm and angle PQR = 60°. Calculate the length of RP, giving your answer to three significant figures.

The canopy of a parachute and the outermost suspension lines (cords) form a sector of a circle as shown in the diagram below, with the parachutist modelled as a particle at point $O$ .

The area of the sector $OAB$ is $\frac{49π}{150}m^{2}$.

The parachute is made with 40 equal length suspension lines.

Disregarding any additional lengths of cord that might be required for knots, fastenings, etc., find the total length of the suspension lines required to make the parachute.

On the same set of axes, sketch the graphs of $y=\mathrm{tan}\frac{1}{2}θ$ and y = tan ( theta - pi / 6 ) in the interval $-2π\leqθ\leq2π$. Label the coordinates of points of intersection with the coordinate axes.

Within the interval $-2π\leqθ\leq2π$, determine the coordinates of the two points where tan 1 half theta = tan ( theta - pi / 6 ). Give your answer in surd form.

Express $2x^{3}+2x^{2}-12x$ in the form $ax(x+b)(x+c)$, where $a,b\mathrm{and}c$ are integers to be found.

Hence sketch the graph of $y=2x^{3}+2x^{2}-12x$ labelling any points where the graph intersects the coordinate axes.

Find the discriminant for the quadratic function $x^{2}+8x+15$.

Write down the number of real solutions to the equation $x^{2}+8x+15=0$.

Solve the equation

$\frac{1}{9}x^{\frac{4}{5}}=9$

Expand and simplify $(x+y)(x-y)(y-x)(-x-y).$

A cuboid has a length of $(2x-3y+3)$units , a width of $(2x+3y-3)$units, and a height of $(x-y)$units. Find an expression for the volume of the cuboid in terms of $x$ and $y$.

Solve the equation $5\sqrt{x}+3=2x$.

Solve $x^{\frac{2}{3}}+2x^{\frac{1}{3}}=8$.

Solve the equation $2^{2x}+64=20(2^{x})$.