Practice Paper 2 (Cambridge International A-Level Mathematics)

The function $\text{f} ( x )$ is given by

$\text{f} ( x ) = ( 1 - p x )^{-4}$

where $p$ is an integer.

Find the coefficient of $x^{3}$ in the binomial expansion of $\text{f} ( x )$, in terms of $p$.

The graph of $y = \text{f} ( x )$, where $\text{f} ( x ) = 2 x - 3$, is shown below.

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

(i) On the diagram above, sketch the graph of y = |\text{f}(x - 1)|.

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

Let

$\text{f} ( x ) = \frac{x^{2}-5}{x^{3}-x^{2}-17x-15}$

Express $\text{f} ( x )$ in partial fractions.

The polynomial $\text{f} ( x )$ is defined by $\text{f} ( x ) = x^{3} + 9 x^{2} + r x + s$.

Given that $\text{f} ( 2 ) = 0$ and $\text{f} ( - 1 ) = - 54$, find the values of $r$ and $s$.

Factorise $\text{f} ( x )$ completely.

The solutions of the cubic equation $z^{3} - 11 z^{2} + 36 z - 26 = 0$ are $z_{1}$, $z_{2}$ and $z_{3}$, where $z_{3}$ is real.

Given that $z_{1} = 5 + i$, work out the other two solutions of the equation.

(i) On an Argand diagram, show the complex numbers $z_{1}$, $z_{2}$, $- 3 z_{3}$ and $z_{1} - z_{2}$ as position vectors.

(ii) Describe the geometric transformation that maps $z_{1}$ to $z_{2}$.

For a complex number $z$, a square root of $z$ is a complex number $x + i y$ where $x$ and $y$ are real numbers and where

$( x + i y )^{2} = z$

Given that $z = - 3 + 4 i$, show that $x^{2} - y^{2} = - 3$ and $2 x y = 4$.

(i) Solve the equations in part (a) to find the possible values of $x$ and $y$.

(ii) Hence determine the two square roots of $z = - 3 + 4 i$.

Express the numbers $3$ and $- i$ in exponential form.

The complex number $z$ is given by $z = r e^{iθ}$.

Hence express each of the following in exponential form, in terms of $r$ and $θ$:

(i) $3 z$

(ii) $- i z$

The point $z$, where |z| \neq 0, is represented on an Argand diagram.

Describe the geometrical transformation that maps $z$ to each of the following points:

(i) $3 z$

(ii) $- i z$

The diagram shows the curve $C$ with parametric equations

$x = 3 \text{sin} 3 θ$, $y = 6 \text{cos} 2 θ$,

for $- \frac{π}{2} \leq θ \leq \frac{π}{2}$.

(i) Write down the value of $\frac{\text{d}y}{\text{d}θ}$ at the point $( 0 , 6 )$.

(ii) Write down the value of $\frac{\text{d}x}{\text{d}θ}$ at the points $( - 3 , 3 )$ and $( 3 , 3 )$.

Find an expression for $\frac{\text{d}y}{\text{d}x}$ in terms of $θ$.

(i) Find the values of $x$, $y$ and $\frac{\text{d}y}{\text{d}x}$ at the point where $θ = \frac{π}{12}$.

(ii) Hence show that the equation of the tangent to $C$ at the point where $θ = \frac{π}{12}$ is

2\sqrt{2}x + 3y - \left( 9\sqrt{3} + 6 \right) = 0

Find the derivative of the function \text{f}(x) = \text{sin}\left(\text{cos}\left(\text{ln}\left(\dfrac{1}{x}\right)\right)\right), $x > 0$.

The equation of a curve is $\text{ln} y - 2 x y^{3} = 8$.

Show that $\frac{\text{d}y}{\text{d}x} = \frac{2y^{4}}{1-6xy^{3}}$.

Find the equation of the normal to the curve at the point where $y = 1$. Give your answer in the form $a x + b y + c = 0$, where $a$, $b$ and $c$ are integers.

Given that $y > - 1$, solve the differential equation

$\frac{\text{d}y}{\text{d}x} = 2 x y + 2 x - y - 1$

giving $y$ in terms of $x$.

Show that

\int_{0}^{1} \frac{3x\text{e}^{-3x^{2}}}{3 - 2\text{e}^{-3x^{2}}} \, \text{d}x = \frac{1}{4}\,\text{ln} \; \left(3 - 2\text{e}^{-3}\right)

Simplify

$2 \text{ln} 3^{4} + \text{ln} 3^{3} - \text{ln} 9$

giving your answer in the form $a \text{ln} b$, where $a$ and $b$ are integers to be found.

Write

2\text{log}_{a} x + 3\text{log}_{a}\left(x + 1\right) - \text{log}_{a} 4\left(x + 2\right)

as a single logarithm.

(i) Sketch the graph of $y = 0 . 4^{x}$.

(ii) State whether this graph indicates exponential growth or exponential decay.

Find the value of $x$ when $y = 0 . 064$.