Questions - AQA C4 - A-Level Maths

Find the distance between $A$ and $B$.
Find the acute angle between the lines $A B$ and $l$. Give your answer to the nearest degree.
The points $B$ and $C$ lie on $l$ such that the distance $A C$ is equal to the distance $A B$. Find the coordinates of $C$.

The number of fish in a lake is decreasing. After $t$ years, there are $x$ fish in the lake. The rate of decrease of the number of fish is proportional to the number of fish currently in the lake.
1. Formulate a differential equation, in the variables $x$ and $t$ and a constant of proportionality $k$, where $k > 0$, to model the rate at which the number of fish in the lake is decreasing.
2. At a certain time, there were 20000 fish in the lake and the rate of decrease was 500 fish per year. Find the value of $k$.
The equation $$P = 2000 - A \mathrm { e } ^ { - 0.05 t }$$ is proposed as a model for the number of fish, $P$, in another lake, where $t$ is the time in years and $A$ is a positive constant. On 1 January 2008, a biologist estimated that there were 700 fish in this lake.
1. Taking 1 January 2008 as $t = 0$, find the value of $A$.
2. Hence find the year during which, according to this model, the number of fish in this lake will first exceed 1900.

Use the Remainder Theorem to find the remainder when $3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 5$ is divided by $3 x - 1$.
Express $\frac { 3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 5 } { 3 x - 1 }$ in the form $a x ^ { 2 } + b x + \frac { c } { 3 x - 1 }$, where $a , b$ and $c$ are integers.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$.
Find an equation of the normal to the curve at the point where $t = 1$.
Show that the cartesian equation of the curve can be written in the form $$x ^ { 2 } - 2 x y + k = 0$$ where $k$ is an integer.

Find the binomial expansion of $( 1 - x ) ^ { - 1 }$ up to and including the term in $x ^ { 2 }$.
1. Express $\frac { 3 x - 1 } { ( 1 - x ) ( 2 - 3 x ) }$ in the form $\frac { A } { 1 - x } + \frac { B } { 2 - 3 x }$, where $A$ and $B$ are integers.
2. Find the binomial expansion of $\frac { 3 x - 1 } { ( 1 - x ) ( 2 - 3 x ) }$ up to and including the term in $x ^ { 2 }$.
Find the range of values of $x$ for which the binomial expansion of $\frac { 3 x - 1 } { ( 1 - x ) ( 2 - 3 x ) }$ is valid.

1. Write down the value of $A$.
2. Show that the value of $k$ is 0.984025 , correct to six decimal places.
The value of this car first dropped below $\pounds 5000$ during the $n$th month from new. Find the value of $n$.

1. Show that the equation $3 \cos 2 x + 7 \cos x + 5 = 0$ can be written in the form $a \cos ^ { 2 } x + b \cos x + c = 0$, where $a , b$ and $c$ are integers.
2. Hence find the possible values of $\cos x$.
1. Express $7 \sin \theta + 3 \cos \theta$ in the form $R \sin ( \theta + \alpha )$, where $R > 0$ and $\alpha$ is an acute angle. Give your value of $\alpha$ to the nearest $0.1 ^ { \circ }$.
2. Hence solve the equation $7 \sin \theta + 3 \cos \theta = 4$ for all solutions in the interval $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$, giving $\theta$ to the nearest $0.1 ^ { \circ }$.
1. Given that $\beta$ is an acute angle and that $\tan \beta = 2 \sqrt { 2 }$, show that $\cos \beta = \frac { 1 } { 3 }$.
2. Hence show that $\sin 2 \beta = p \sqrt { 2 }$, where $p$ is a rational number.

Solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 150 \cos 2 t } { x }$$ given that $x = 20$ when $t = \frac { \pi } { 4 }$, giving your solution in the form $x ^ { 2 } = \mathrm { f } ( t )$. (6 marks)
The oscillations of a 'baby bouncy cradle' are modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 150 \cos 2 t } { x }$$ where $x \mathrm {~cm}$ is the height of the cradle above its base $t$ seconds after the cradle begins to oscillate. Given that the cradle is 20 cm above its base at time $t = \frac { \pi } { 4 }$ seconds, find:
1. the height of the cradle above its base 13 seconds after it starts oscillating, giving your answer to the nearest millimetre;
2. the time at which the cradle will first be 11 cm above its base, giving your answer to the nearest tenth of a second.
  (2 marks)

Questions — AQA C4 (160 questions)