Practice Paper 2 (Cambridge International A-Level Mathematics)

The function $\text{f}$ is defined by $\text{f} ( x ) = \sqrt{2x+1}$. Find an expression for $\text{g} ( x )$, where $\text{g} ( x )$ is obtained by applying the following sequence of transformations to $y = \text{f} ( x )$, in this order.

1. a translation by \begin{bmatrix} - 3 \\ 0 \end{bmatrix}
2. a horizontal stretch with scale factor $\frac{1}{2}$
3. a reflection in the $x$-axis

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

$\text{f} ( x ) = 4 x^{3} + 4 x^{2} - 7 x + 2$

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

Sketch the graph of $y = \text{f} ( x )$, stating the coordinates of any intersections with the axes. (You do not need to give the coordinates of any turning points.)

Solve the equation $x^{3} - x^{2} - 2 x + 4 = 4 x + 4$.

Write down the $x$-coordinates of the points of intersection of the graphs of $y = x^{3} - x^{2} - 2 x + 4$ and $y = 4 x + 4$.

Find the $y$-coordinates of these points of intersection.

On the same diagram, sketch the graphs of $y = x^{3} - x^{2} - 2 x + 4$ and $y = 4 x + 4$.

Solve the simultaneous equations

$x + y = 4$

$x^{2} - 4 x - 3 y = 0$

The number of toys, $P$, produced by a machine in one hour is modelled by

P\left(T\right) = \left(20 - T\right)\left(T - 40\right) \qquad 20 \leq T \leq 40

where the temperature of the machine is $T$ °C.

Find the number of toys produced in one hour when the temperature of the machine is 25 °C.

(i) Show that P\left(T\right) = 60T - T^{2} - 800.

(ii) By completing the square, show that P\left(T\right) = 100 - \left(T - 30\right)^{2}.

(iii) Hence find the temperature at which the machine is most productive, and the number of toys it produces in one hour at this temperature.

The function f \left( x \right) is defined by f \left( x \right) = \left( k - 1 \right) x^{2} - \left( k - 2 \right) x - 2k, $x \in ℝ$.

The function g \left( x \right) is defined by g \left( x \right) = \left( k - 1 \right) x^{2} - 3kx + k + 1, $x \in ℝ$.

k is a non-zero constant and $k \neq 1$.

The graphs of y = f \left( x \right) and y = g \left( x \right) intersect at a single point. Find the x-coordinate of the intersection, giving your answer in terms of k.

In the case when $k = 3$, find the coordinates of the point of intersection of the two graphs.

The diagram shows the graph of $y = \text{f} ( x )$. The graph meets the coordinate axes at the marked points A \left( 0, 6 \right) and B \left( 3, 0 \right), and has asymptotes with equations $y = \frac{20}{3}$ and $x = \frac{10}{3}$.

On separate diagrams, sketch the following curves, giving the coordinates of the images of $A$ and $B$ and the equations of the transformed asymptotes:

(i) $y = \text{f} ( 5 x )$

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

The graph of $y = a \text{f} ( x )$ has an asymptote with equation $y = 2$. Find the value of $a$.

The line $y + 2 x = 11$ meets the circle with equation $x^{2} + y^{2} + 6 x - 14 y = - 38$.

(i) Show that the line and the circle meet at exactly one point.

(ii) Find the coordinates of the point of intersection.

An electrician charges a fixed fee of £45 plus £22 per hour.

Defining suitable variables, write down an equation to represent the charges made by the electrician.

A rival electrician charges a fixed fee of £36 plus £25 per hour. Determine which electrician is cheapest for a job taking 4 hours.

An ice cube, of side length $x$ cm, is melting at a constant rate of $0 . 8 \text{cm}^{3} \text{s}^{-1}$.

Assuming that the ice cube remains in the shape of a cube whilst it melts, find the rate at which its surface area is decreasing at the point when its side length is 2 cm.

A curve has the equation $y = 3 x - 4 x^{-2}$, $x \neq 0$.

Find $\frac{\text{d}y}{\text{d}x}$.

Find the coordinates of the point on the curve where the gradient is $2$.

The diagram below shows part of the graph of $y = x^{3} - 2 x^{2} - x + 2$.

Find the total area of the two shaded regions.

Given that

\int_{q}^{4q} 5x\sqrt{x} \,\text{d}x = 15066 ,

find the value of the constant $q$.

The $k$th term of an arithmetic progression is given by $u_{k} = 16 + 7 k$.

Calculate the sum of the fifteenth to twenty-fifth terms of the progression, inclusive.

Calculate the sum of the first $25$ terms of the progression whose $k$th term is given by $v_{k} = u_{k} - 3$, where $u_{k}$ is defined as above.