Practice Paper 1 (Cambridge International A-Level Mathematics)

A gardener wants to model the number of hours of daylight his allotment receives at different times of the year using a function of the form

$h ( t ) = a + b \text{sin} ( \frac{2π}{365} t )$

$t \geq 0$

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

(i) Given that the model needs to predict a maximum of 17 hours daylight and a minimum of 7 hours daylight, find the values of $a$ and $b$.

(ii) Explain the significance of the value $\frac{2π}{365}$ in the model.

(iii) Suggest, with a reason, the date of the year that the model starts on.

Expand

\left(1 - \frac{1}{3}x\right)^{-1}(2 - x)^{-2}

in ascending powers of $x$, up to and including the term in $x^{2}$, simplifying the coefficients.

State the set of values of $x$ for which the expansion in part (a) is valid.

The function $\text{f}$ is defined by \text{f}: x \mapsto |3x - 2| for $x \in ℝ$.

Explain why the inverse of $\text{f} ( x )$ does not exist.

Suggest an adaptation to the domain of $\text{f} ( x )$ so that its inverse exists, while also producing the maximum possible range for $\text{f} ( x )$.

Using your adaptation from part (b), find an expression for $\text{f}^{-1} ( x )$ and state its domain and range.

It is given that

\text{f} \left(x\right) = x^{3} - 13 x + 12 .

(i) Show that \text{f} \left(1\right) = 0.

(ii) Hence write down a factor of \text{f} \left(x\right).

(i) Show that \text{f} \left(3\right) = 0 .

(ii) Hence write down another factor of \text{f} \left(x\right).

Fully factorise \text{f} \left(x\right).

Express

$\frac{3x^{2}-13}{x^{3}-13x+12}$

in partial fractions.

Given that $( x - 4 )$ is a factor of $x^{3} - k x^{2} - 4 x + 16$, find the value of $k$.

$\text{f} ( z ) = 2 z^{3} - 13 z^{2} + 60 z - 100$

Given that $\frac{5}{2}$ is a root of the equation $\text{f} ( z ) = 0$, use algebra to solve $\text{f} ( z ) = 0$ completely.

Show all three solutions on an Argand diagram.

Given that $z_{1} = a - 6 i$, $z_{2} = 1 + b i$ and $z_{1} z_{2} = - 17 - 9 i$, where $a$ and $b$ are real numbers, find the possible values of $a$ and $b$.

Using your answers to part (a), state values for $c$ and $d$ that will satisfy the equation

$- ( 3 + i ) ( c + d i ) = - 17 - 9 i$

The points representing the complex numbers 1 and $z$, where $z \neq 0$, are marked on an Argand diagram.

Explain how to find the point representing $z^{2}$ by geometrical construction, and provide a sketch to illustrate your answer.

Explain how to find the point representing \left( 2 - i \right) z by geometrical construction, and provide a sketch to illustrate your answer.

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 $t$.

Differentiate $\frac{\text{cos}x}{\text{sin}x}$ with respect to $x$.

Differentiate $\frac{2x^{2}-3x+4}{\text{sin}3x}$ with respect to $x$.

The equation of a curve is $3 y^{2} - 2 x^{3} = 10$.

Find the gradient of the curve at the point $( 1 , 2 )$.