Practice Paper 1 (Cambridge International A-Level Mathematics)

The diagram shows the graph of $y = \text{f} ( x )$. The maximum point $B ( - 1 , 32 )$ and the $x$-axis intercepts $A ( - 3 , 0 )$ and $C ( 3 , 0 )$ are marked on the diagram.

On separate diagrams, sketch the following graphs.

(i) $y = \text{f} ( \frac{1}{2} x ) + 2$

(ii) $y = - \text{f} ( x - 1 )$

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

State whether each of the following mappings is one-one, many-one, one-many or many-many.

(i) \text{f}: x \mapsto 4x - 2

(ii) \text{f}: x \mapsto x^{2}

(iii) \text{f}: x \mapsto \dfrac{x}{4}

(iv) \text{f}: x \mapsto \sqrt{x}

A machine completes a calculation in a time, $t$ seconds, that is proportional to the number of processes, $p$, involved. For a calculation involving $10$ processes the machine takes $0 . 01$ seconds.

Show that the constant of proportionality is $0 . 001$.

Hence write down an equation linking $p$ and $t$.

Find the time it takes for the machine to complete a calculation involving $200$ processes.

How many processes are involved in a calculation taking $2 . 3$ seconds?

Solve the simultaneous equations

$y = 2 x - 1$

$x^{2} + y^{2} - 2 = 0$

A fountain is designed so that water is projected over a walkway. The path of the water is modelled by

y = x\left(5 - x\right) , 0 \leq x \leq 5

where $x$ is the horizontal distance in metres from the base of the fountain at ground level and $y$ is the height of the water in metres.

Sketch the graph of the model, labelling any intersections with the coordinate axes.

The average person is 1.7 m tall and needs a ground width of 1.2 m in order to walk comfortably.

Work out the maximum number of average-sized people who can comfortably walk under the fountain side by side without getting wet.

The functions f and g are defined by f \left( x \right) = kx^{2} + 2kx - 3 and g \left( x \right) = kx^{2} + 4kx - 16, where k is a constant.

The equation f \left( x \right) = 0 has two distinct real roots and the equation g \left( x \right) = 0 has no real roots.

Find the set of values of k.

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 $A$.

Consider the three transformations of the graph

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

$y = \text{f} ( a x )$

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

where $a$ and $b$ are constants and $a > 0$.

State which of the transformations satisfies each of the following conditions, determining the range of possible values of $a$ or $b$ where relevant:

(i) the images of the two marked points lie on opposite sides of the $x$-axis;

(ii) the image of point $B$ has coordinates \left( x, y \right), where $- 6 < x < - 3$;

(iii) the image of point $B$ has coordinates \left( x, y \right), where $0 < x < 3$.

(i) Complete the square for $x^{2} + 4 x$.

(ii) Complete the square for $y^{2} - 6 y$.

(i) Use your answers to part (a) to show that the equation $x^{2} + y^{2} + 4 x - 6 y + 4 = 0$ can be written in the form $( x + 2 )^{2} + ( y - 3 )^{2} = 9$.

(ii) Hence write down the centre and the radius of the circle with equation $x^{2} + y^{2} + 4 x - 6 y + 4 = 0$.

Three points $A$, $B$ and $C$ have coordinates $( - 8 , 1 )$, $( 4 , 4 )$ and $( 12 , 6 )$ respectively.

Find the gradients of the line segments $A B$ and $B C$.

Explain why your answer to part (a) shows that $A$, $B$ and $C$ are collinear.

The diagram below shows part of the curve with equation $y = x^{3} + 11 x^{2} + 35 x + 25$.

The curve touches the $x$-axis at $A$ and cuts the $x$-axis at $C$.

The points $A$ and $B$ are stationary points on the curve.

Using calculus, and showing all your working, find the coordinates of $A$ and $B$.

Show that $( - 1 , 0 )$ is a point on the curve and explain why those must be the coordinates of point $C$.

A curve has the equation $y = x \sqrt{x} + \frac{48}{\sqrt{x}}$, $x > 0$. Find the coordinates of the point on the curve where the gradient is $0$.

The diagram below shows a sketch of the curves with equations $y = x^{2} + 5$ and $y = 23 - x^{2}$.

The region $R$ is bounded by the $y$-axis and the two curves. This region is rotated through 360° about the $y$-axis. Find the volume of the solid formed.

The regions marked $R$ and $S$ are now to be considered as a single region bounded by the two curves.

A student calculates the volume swept out when this combined region is rotated through a full 360° about the $y$-axis, without taking the symmetry of the two curves into account. Call this value $I$.

How does $I$ compare to the answer you found in part (a)?

Be sure to fully explain your answer.