Practice Paper 2 (Cambridge International AS Mathematics)

The graph of $y = \text{f} ( x )$ is transformed. In each case, write down, in terms of $\text{f} ( x )$, the equation of the resulting graph.

(i) a translation by \begin{pmatrix} -2 \\ 3 \end{pmatrix}

(ii) a horizontal stretch with scale factor 2, followed by a vertical stretch with scale factor 3

The functions $\text{f}$ and $\text{g}$ are defined by

$\text{f} ( x ) = x^{2}$

$\text{g} ( x ) = 4 x - 3$

for $x \in ℝ$.

Write down the range of $\text{f}$.

Find an expression for

(i) $\text{fg} ( x )$,

(ii) $\text{gf} ( x )$.

Solve the equation $\text{f} ( x ) = \text{g} ( x )$.

On the axes below, sketch the graphs of $y = x^{2} + 2 x - 3$ and $y = x - 1$.

Using your graphs, or otherwise, solve the simultaneous equations $y = ( x + 3 ) ( x - 1 )$ and $x - y = 1$.

The cross section of a tunnel is in the shape of the region defined by the inequalities

On the axes below show the region satisfying the inequalities

Given that $x$ and $y$ are in metres, write down the height and the maximum width of the tunnel.

Find the area of the cross-section of the tunnel.

It has often been said that, to avoid relegation from football's Premier League, a team should aim to score at least 40 points in a season. Each team plays 38 games; a win is rewarded with 3 points and a draw with 1 point. No points are awarded for a loss.

Using $W$ and $D$ as the number of wins and draws respectively, write down two inequalities: one relating to the number of points a team needs to avoid relegation, and one relating to the number of games played.

Explain why $W \geq 0$ and $D \geq 0$ must also be conditions.

A team has won 4 games and drawn 4 after playing 17 games. Write down two updated inequalities for the number of wins and draws required during the remainder of the season in order to avoid relegation.

The equation of a curve is $y = 3 x^{2} + 2 p x + 4 q$, where $p$ and $q$ are constants.

Given that the equation $3 x^{2} + 2 p x + 4 q = 0$ has no real roots, show that $p^{2} < 12 q$.

Given that the curve passes through $( - 2 , 6 )$ and $( 2 , 6 )$, find the values of $p$ and $q$.

The point P \left( 2, 6 \right) lies on the curve with equation $y = \text{f} ( x )$.

State the coordinates of the image of $P$ on each of the following curves:

(i) $y = \text{f} ( x ) + 1$

(ii) $y = - \text{f} ( x )$

(iii) y = \text{f}\left( \tfrac{1}{4}x \right)

Find the centre and the radius of the circle with equation $x^{2} + y^{2} + x - 3 y + 2 = 0$.

A quadrilateral $A B C D$ has vertices with coordinates $A ( - 1 , 6 )$, $B ( - 3 , 2 )$, $C ( 0 , - 4 )$ and $D ( 2 , 0 )$.

Find the equation of each of the four lines that form the quadrilateral, and state the mathematical name of the quadrilateral.

A curve has equation $y = 3 x - \frac{1}{2} x^{2}$.

Find the gradient of the tangent to the curve at the point where $x = 5$.

(i) Find the gradient of the normal to the curve at the point where $x = 5$.

(ii) Hence find the equation of the normal to the curve at this point.

The function $\text{f}$ is defined by $\text{f} ( x ) = x^{3} - 4 x^{2} + 6 x - 9$. Show that there are no solutions to the equation $\text{f} ' ( x ) = 0$.

Starting with the equation of a semicircle of radius $r$, $y = \sqrt{r^{2}-x^{2}}$ (where $r > 0$), use calculus to show that the general formula for the volume $V$ of a sphere of radius $r$ is

$V = \frac{4}{3} π r^{3}$

Use calculus to find

\int \left(3x^{\frac{1}{2}} + 2x^{-\frac{1}{2}}\right) \,\text{d}x .

An arithmetic progression has first term $3$ and common difference $4$.

Find the sum of the first $20$ terms of the progression.

In the expansion of \left(3a + \dfrac{1}{2}x\right)^6, the coefficient of $x^{3}$ is equal to the coefficient of $x^{5}$. Find the values of $a$, giving your answers in the form $\frac{\sqrt{m}}{n}$, where $m$ and $n$ are integers to be found.

The second and third terms of a geometric progression are $( x - 1 )$ and $( x^{2} - 1 )$, where $x$ is a real number not equal to $1$ or $- 1$.

Given that the progression is convergent, find the range of possible values of $x$.

Given that the sum to infinity of the progression is $- 6$, find the two possible values of $x$.

A model maker is constructing part of a model building using a series of hollow tubes, stacked next to each other as illustrated below.

Each tube is one-tenth shorter than the one to its left. All the tubes are the same width, as they are all cut from one longer tube.

Show that, no matter how many tubes the model maker uses, the longer tube they are cut from need not be any longer than ten times the height of the tallest tube.

In a different part of the model building, tubes are required to be stacked on top of each other, as shown below.

The tallest tube is the same length as the tallest tube in part (a). Each of the other tubes is $1$ cm shorter than the tube immediately below it. The longer tube that all the tubes are cut from is ten times the height of the tallest tube in the stack, and all of it is used.

Given that the length of the tallest tube is $a$ cm and there are $n$ tubes in the stack, show that $n^{2} - ( 2 a + 1 ) n + 20 a = 0$.