Practice Paper 1 (Cambridge International AS Mathematics)

The diagram shows the graph of $y = \text{f} ( t )$, where $\text{f} ( t ) = \text{cos} t$ for $0 ^{\circ} \leq t \leq 360 ^{\circ}$.

(i) Write down the maximum value of $y = - 2 \text{f} ( 3 t )$.

(ii) Write down the value of $t$ at which this maximum first occurs.

Find, in terms of $\text{f} ( t )$, a combination of transformations that maps the graph of $y = \text{f} ( t )$ onto the graph of $y = 2 - 4 \text{sin} t$ for $0 ^{\circ} \leq t \leq 360 ^{\circ}$.

The function $\text{f}$ is defined by

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

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

Sketch the graph of $y = \text{f} ( x )$, stating the coordinates of any intersections with the axes and the equations of any asymptotes.

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

A machine completes a calculation in a time, $t$ seconds, that is proportional to the cube root of the number of processes, $p$, involved. For a calculation involving $8$ processes the machine takes $6 . 4 \times 10^{-4}$ seconds.

How many processes are involved in a calculation taking $1 . 28 \times 10^{-3}$ seconds?

Find the time it takes for the machine to complete a calculation involving $250$ processes.

By eliminating $y$ from the equations

$x^{2} - 8 y = - 40$

$3 x + 2 y = 4$

show that $x^{2} + 12 x + 24 = 0$.

Hence solve the simultaneous equations

$x^{2} - 8 y = - 40$

$3 x + 2 y = 4$

giving $x$ and $y$ in the form $a \pm b \sqrt{3}$, where $a$ and $b$ are integers.

The Blue Blades Aerobatic Display Team use white and blue smoke trails in their flying display. At the start of the display, each aeroplane holds a tank containing 360 litres of white smoke fluid and a separate tank of blue smoke fluid.

The amount of white smoke fluid remaining, $w$ litres, after $t$ seconds of use is modelled by

$w = \frac{k}{t+p} - q$

where $k$, $p$ and $q$ are positive constants.

(i) Given that $q = 90$ and that it takes 50 seconds for the amount of white smoke fluid to halve, find the values of $k$ and $p$.

(ii) Hence find the number of minutes of use that a tank of 360 litres of white smoke fluid will last.

A similar model for the blue smoke fluid is $b = \frac{8400}{t+40} - 30$. Find the initial amount of blue smoke fluid in the tank, and determine how many minutes of use the blue smoke fluid will last.

Find the minimum value of the function f \left( x \right) = x^{2} + 8x + c, giving your answer in terms of c.

Find the values of c for which the function f \left( x \right) = x^{2} + 8x + c has no real roots.

The diagram shows the graph of $y = \text{f} ( x )$. The marked point B \left( 4, 8 \right) lies on the graph, and the graph meets the origin at the marked point $A$.

On separate diagrams, sketch the following curves, giving the coordinates of the images of $A$ and $B$ on each:

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

(ii) $y = \text{f} ( 4 x )$

On the graph of $y = a \text{f} ( x )$ the image of one of the two marked points has a $y$-coordinate of $4$. Find the value of $a$.

Show that $x^{2} + y^{2} + 5 x - 2 y - 5 = 0$ can be written in the form $( x - a )^{2} + ( y - b )^{2} = r^{2}$, where $a$, $b$ and $r$ are constants to be found.

Hence write down the centre and the radius of the circle with equation $x^{2} + y^{2} + 5 x - 2 y - 5 = 0$.

The coordinates of the ends of the diameter of a circle are $( - 3 , 5 )$ and $( 3 , - 3 )$.

Find the length of the diameter.

Find the centre of the circle.

A company manufactures food tins in the shape of cylinders which must have a constant volume of $150 π \text{cm}^{3}$. To lessen material costs the company would like to minimise the surface area of the tins.

By first expressing the height $h$ of the tin in terms of its radius $r$, show that the surface area of the cylinder is given by $S = 2 π r^{2} + \frac{300π}{r}$.

Use calculus to find the minimum value for the surface area of the tins. Give your answer correct to 2 decimal places.

For each of the following, find $\frac{\text{d}y}{\text{d}x}$ in terms of $x$:

$y = 4 x^{2} - 3 x + 19$

$y = x^{3} - 5 x^{2} + 14 x - 1$

$y = 4 x^{\frac{3}{2}} - 3 x^{-1}$