Practice Paper 1 (Cambridge International AS Mathematics)

A gardener is modelling the number of hours of daylight his allotment receives at different times of the year using the function

h \left(t\right) = 12 + 5 \text{sin} \; \left(\frac{2 \pi t}{365}\right)

$t \geq 0$

where $h$ is the number of hours of daylight on a given day, and $t$ is the time measured in whole days. Note that $t = 0$ corresponds to the first day of the model.

(i) Find the number of hours of daylight on the 100^th day.

(ii) Write down the maximum and minimum number of daylight hours the model predicts.

(iii) Assuming the allotment is located in the UK, give a reason why the first day of the model most likely does not correspond to 1^st January.

Solve the equation \left|3 x - 2\right| = 7.

Given that \left(x^{2} - 8 x - 20\right) \div \left(x - 2\right) = A x + B + \frac{C}{x - 2}, where $A , B$ and $C$ are integer constants, in terms of $A , B$ and/or $C$ as appropriate:

(i) write down the divisor,

(ii) write down the quotient,

(iii) write down the remainder.

Find the values of $A , B$ and $C$.

Given

$x = e^{t}$ and $y = 2 t^{3} + 3 t$

find $\frac{\text{d}x}{\text{d}t}$ and $\frac{\text{d}y}{\text{d}t}$.

Hence, or otherwise, find $\frac{\text{d}y}{\text{d}x}$ in terms of $t$.

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 $3 x^{2} - 2 y = x y$.

Find an expression for $\frac{\text{d}y}{\text{d}x}$.

Hence show that the stationary points of the curve lie on the line $y = 6 x$.

Find

\int \text{sin} \; x \, \; \text{d}x

Find the exact value of

\int_{1}^{4} \frac{1}{x} \, \; \text{d}x

Find an expression for $y$ given that

y = \int 7\text{e}^{7x} \, \; \text{d}x

Show that $4 - \text{ln} 16$ can be written in the form 4\text{ln}\left(\dfrac{e}{2}\right).

Sketch the graph of $y = 4 e^{x}$ for $x \geq 0$.

Label any points of intersection with the coordinate axes. Write down the equations of any asymptotes.

Find the gradient of $y = 4 e^{x}$ at the point where $x = 3$, giving your answer correct to 3 significant figures.

The population growth of population, $P$, at time $t$ years, is modelled by the equation $P = 4 e^{t}$.

Write down the initial population.

An exponential growth model for the number of bacteria in an experiment is of the form

$N = N_{0} a^{kt}$

$N$ is the number of bacteria and $t$ is the time in hours since the experiment began. $N_{0}$, $a$ and $k$ are constants.

A scientist records the number of bacteria at various points over a six-hour period. The results are in the table below.

Use your calculator to evaluate $\text{log}_{5} N$ for each value of $N$ in the table, giving your values correct to three significant figures.

Plot the observations on the graph below, plotting $\text{log}_{5} N$ against $t$. Draw a line of best fit.

Find an equation for your line of best fit in the form $\text{log}_{5} N = m t + \text{log}_{5} c$.

Estimate the values of $N_{0}$, $a$ and $k$.

The diagram shows part of the curve $y = \text{f} ( x )$, where $\text{f} ( x ) = 2 x \text{cos} ( 3 x ) - 1$ and $x$ is in radians.

(i) Find $\text{f} ( 1 . 6 )$ and $\text{f} ( 1 . 7 )$, giving your answers correct to 3 significant figures.

(ii) State what your results tell you about the equation $\text{f} ( x ) = 0$.

One of the solutions to the equation $\text{f} ( x ) = 0$ is $x = 2 . 55$, correct to 3 significant figures.

(i) Write down the lower and upper bounds of $2 . 55$.

(ii) Hence use the change of sign rule to confirm that $2 . 55$ is a solution of $\text{f} ( x ) = 0$, correct to 3 significant figures.

The diagram below shows two right-angled triangles. Angles $A$ and $B$ have been labelled.

Given that $α = A + B$, find the exact values of $\text{sin} α$, $\text{cos} α$ and $\text{tan} α$.