Questions - AQA C4 - A-Level Maths

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AQA C4 2011 January Q1

Express $2 \sin x + 5 \cos x$ in the form $R \sin ( x + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$. Give your value of $\alpha$ to the nearest $0.1 ^ { \circ }$.
1. Write down the maximum value of $2 \sin x + 5 \cos x$.
2. Find the value of $x$ in the interval $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$ at which this maximum occurs, giving the value of $x$ to the nearest $0.1 ^ { \circ }$.

AQA C4 2011 January Q2

The polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = 9 x ^ { 3 } + 18 x ^ { 2 } - x - 2$.
1. Use the Factor Theorem to show that $3 x + 1$ is a factor of $\mathrm { f } ( x )$.
2. Express $\mathrm { f } ( x )$ as a product of three linear factors.
3. Simplify $\frac { 9 x ^ { 3 } + 21 x ^ { 2 } + 6 x } { \mathrm { f } ( x ) }$.
When the polynomial $9 x ^ { 3 } + p x ^ { 2 } - x - 2$ is divided by $3 x - 2$, the remainder is - 4 . Find the value of the constant $p$.

AQA C4 2011 January Q3

Express $\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }$ in the form $\frac { A } { 1 + x } + \frac { B } { 3 + 5 x }$, where $A$ and $B$ are integers.
Hence, or otherwise, find the binomial expansion of $\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }$ up to and including the term in $x ^ { 2 }$.
Find the range of values of $x$ for which the binomial expansion of $\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }$ is valid.
(2 marks)

AQA C4 2011 January Q4

4 A curve is defined by the parametric equations $$x = 3 \mathrm { e } ^ { t } , \quad y = \mathrm { e } ^ { 2 t } - \mathrm { e } ^ { - 2 t }$$

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

AQA C4 2011 January Q5

5 A model for the radioactive decay of a form of iodine is given by $$m = m _ { 0 } 2 ^ { - \frac { 1 } { 8 } t }$$ The mass of the iodine after $t$ days is $m$ grams. Its initial mass is $m _ { 0 }$ grams.

Use the given model to find the mass that remains after 10 grams of this form of iodine have decayed for 14 days, giving your answer to the nearest gram.
A mass of $m _ { 0 }$ grams of this form of iodine decays to $\frac { m _ { 0 } } { 16 }$ grams in $d$ days. Find the value of $d$.
After $n$ days, a mass of this form of iodine has decayed to less than $1 \%$ of its initial mass. Find the minimum integer value of $n$.

AQA C4 2011 January Q6

1. Given that $\tan 2 x + \tan x = 0$, show that $\tan x = 0$ or $\tan ^ { 2 } x = 3$.
2. Hence find all solutions of $\tan 2 x + \tan x = 0$ in the interval $0 ^ { \circ } < x < 180 ^ { \circ }$.
  (l mark)
1. Given that $\cos x \neq 0$, show that the equation $$\sin 2 x = \cos x \cos 2 x$$ can be written in the form $$2 \sin ^ { 2 } x + 2 \sin x - 1 = 0$$
2. Show that all solutions of the equation $2 \sin ^ { 2 } x + 2 \sin x - 1 = 0$ are given by $\sin x = \frac { \sqrt { 3 } - 1 } { p }$, where $p$ is an integer.

AQA C4 2011 January Q7

1. Solve the differential equation $\frac { \mathrm { d } x } { \mathrm {~d} t } = \sqrt { x } \sin \left( \frac { t } { 2 } \right)$ to find $x$ in terms of $t$.
2. Given that $x = 1$ when $t = 0$, show that the solution can be written as $$x = ( a - \cos b t ) ^ { 2 }$$ where $a$ and $b$ are constants to be found.
The height, $x$ metres, above the ground of a car in a fairground ride at time $t$ seconds is modelled by the differential equation $\frac { \mathrm { d } x } { \mathrm {~d} t } = \sqrt { x } \sin \left( \frac { t } { 2 } \right)$. The car is 1 metre above the ground when $t = 0$.
1. Find the greatest height above the ground reached by the car during the ride.
2. Find the value of $t$ when the car is first 5 metres above the ground, giving your answer to one decimal place.

AQA C4 2011 January Q8

8 The coordinates of the points $A$ and $B$ are $( 3 , - 2,4 )$ and $( 6,0,3 )$ respectively.
The line $l _ { 1 }$ has equation $\mathbf { r } = \left[ \begin{array} { r } 3
- 2
4 \end{array} \right] + \lambda \left[ \begin{array} { r } 2
- 1
3 \end{array} \right]$.

1. Find the vector $\overrightarrow { A B }$.
2. Calculate the acute angle between $\overrightarrow { A B }$ and the line $l _ { 1 }$, giving your answer to the nearest $0.1 ^ { \circ }$.
The point $D$ lies on $l _ { 1 }$ where $\lambda = 2$. The line $l _ { 2 }$ passes through $D$ and is parallel to $A B$.
1. Find a vector equation of line $l _ { 2 }$ with parameter $\mu$.
2. The diagram shows a symmetrical trapezium $A B C D$, with angle $D A B$ equal to angle $A B C$.
  \includegraphics[max width=\textwidth, alt={}, center]{5fe2527a-33da-4076-b3fa-4cab545336ec-9_620_675_1197_726} The point $C$ lies on line $l _ { 2 }$. The length of $A D$ is equal to the length of $B C$. Find the coordinates of $C$.

AQA C4 2012 January Q1

Express $\frac { 2 x + 3 } { 4 x ^ { 2 } - 1 }$ in the form $\frac { A } { 2 x - 1 } + \frac { B } { 2 x + 1 }$, where $A$ and $B$ are integers. (3 marks)
Express $\frac { 12 x ^ { 3 } - 7 x - 6 } { 4 x ^ { 2 } - 1 }$ in the form $C x + \frac { D ( 2 x + 3 ) } { 4 x ^ { 2 } - 1 }$, where $C$ and $D$ are integers.
(3 marks)
Evaluate $\int _ { 1 } ^ { 2 } \frac { 12 x ^ { 3 } - 7 x - 6 } { 4 x ^ { 2 } - 1 } \mathrm {~d} x$, giving your answer in the form $p + \ln q$, where $p$ and $q$ are rational numbers.
(5 marks)

AQA C4 2012 January Q2

2 Angle $\alpha$ is acute and $\cos \alpha = \frac { 3 } { 5 }$. Angle $\beta$ is obtuse and $\sin \beta = \frac { 1 } { 2 }$.

1. Find the value of $\tan \alpha$ as a fraction.
  (1 mark)
2. Find the value of $\tan \beta$ in surd form.
Hence show that $\tan ( \alpha + \beta ) = \frac { m \sqrt { 3 } - n } { n \sqrt { 3 } + m }$, where $m$ and $n$ are integers.
(3 marks)

AQA C4 2012 January Q3

Find the binomial expansion of $( 1 + 6 x ) ^ { \frac { 2 } { 3 } }$ up to and including the term in $x ^ { 2 }$.
(2 marks)
Find the binomial expansion of $( 8 + 6 x ) ^ { \frac { 2 } { 3 } }$ up to and including the term in $x ^ { 2 }$.
(3 marks)
Use your answer from part (b) to find an estimate for $\sqrt [ 3 ] { 100 }$ in the form $\frac { a } { b }$, where $a$ and $b$ are integers.
(2 marks)

AQA C4 2012 January Q4

4 A scientist is testing models for the growth and decay of colonies of bacteria. For a particular colony, which is growing, the model is $P = A \mathrm { e } ^ { \frac { 1 } { 8 } t }$, where $P$ is the number of bacteria after a time $t$ minutes and $A$ is a constant.

This growing colony consists initially of 500 bacteria. Calculate the number of bacteria, according to the model, after one hour. Give your answer to the nearest thousand.
For a second colony, which is decaying, the model is $Q = 500000 \mathrm { e } ^ { - \frac { 1 } { 8 } t }$, where $Q$ is the number of bacteria after a time $t$ minutes. Initially, the growing colony has 500 bacteria and, at the same time, the decaying colony has 500000 bacteria.
1. Find the time at which the populations of the two colonies will be equal, giving your answer to the nearest 0.1 of a minute.
2. The population of the growing colony will exceed that of the decaying colony by 45000 bacteria at time $T$ minutes. Show that $$\left( \mathrm { e } ^ { \frac { 1 } { 8 } T } \right) ^ { 2 } - 90 \mathrm { e } ^ { \frac { 1 } { 8 } T } - 1000 = 0$$ and hence find the value of $T$, giving your answer to one decimal place.
  (4 marks)

AQA C4 2012 January Q5

5 A curve is defined by the parametric equations $$x = 8 t ^ { 2 } - t , \quad y = \frac { 3 } { t }$$

Show that the cartesian equation of the curve can be written as $x y ^ { 2 } + 3 y = k$, stating the value of the integer $k$.
(2 marks)
1. Find an equation of the tangent to the curve at the point $P$, where $t = \frac { 1 } { 4 }$.
2. Verify that the tangent at $P$ intersects the curve when $x = \frac { 3 } { 2 }$.

AQA C4 2012 January Q6

Use the Factor Theorem to show that $4 x - 3$ is a factor of $$16 x ^ { 3 } + 11 x - 15$$
Given that $x = \cos \theta$, show that the equation $$27 \cos \theta \cos 2 \theta + 19 \sin \theta \sin 2 \theta - 15 = 0$$ can be written in the form $$16 x ^ { 3 } + 11 x - 15 = 0$$
Hence show that the only solutions of the equation $$27 \cos \theta \cos 2 \theta + 19 \sin \theta \sin 2 \theta - 15 = 0$$ are given by $\cos \theta = \frac { 3 } { 4 }$.

AQA C4 2012 January Q7

7 Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = y ^ { 2 } x \sin 3 x$$ given that $y = 1$ when $x = \frac { \pi } { 6 }$. Give your answer in the form $y = \frac { 9 } { \mathrm { f } ( x ) }$.

AQA C4 2012 January Q8

8 The points $A$ and $B$ have coordinates $( 4 , - 2,3 )$ and $( 2,0 , - 1 )$ respectively. The line $l$ passes through $A$ and has equation $\mathbf { r } = \left[ \begin{array} { r } 4
- 2
3 \end{array} \right] + \lambda \left[ \begin{array} { r } 1
5
- 2 \end{array} \right]$.

1. Find the vector $\overrightarrow { A B }$.
2. Find the acute angle between $A B$ and the line $l$, giving your answer to the nearest degree.
The point $C$ lies on the line $l$ such that the angle $A B C$ is a right angle. Given that $A B C D$ is a rectangle, find the coordinates of the point $D$.

AQA C4 2013 January Q1

1 The polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 8 x - 7$.

Use the Remainder Theorem to find the remainder when $\mathrm { f } ( x )$ is divided by $( 2 x + 1 )$.
(2 marks)
The polynomial $\mathrm { g } ( x )$ is defined by $\mathrm { g } ( x ) = \mathrm { f } ( x ) + d$, where $d$ is a constant.
1. Given that $( 2 x + 1 )$ is a factor of $\mathrm { g } ( x )$, show that $\mathrm { g } ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 8 x - 4$.
  (1 mark)
2. Given that $\mathrm { g } ( x )$ can be written as $\mathrm { g } ( x ) = ( 2 x + 1 ) \left( x ^ { 2 } + a \right)$, where $a$ is an integer, express $\mathrm { g } ( x )$ as a product of three linear factors.
3. Hence, or otherwise, show that $\frac { \mathrm { g } ( x ) } { 2 x ^ { 3 } - 3 x ^ { 2 } - 2 x } = p + \frac { q } { x }$, where $p$ and $q$ are integers.
  (3 marks)

AQA C4 2013 January Q2

2 It is given that $\mathrm { f } ( x ) = \frac { 7 x - 1 } { ( 1 + 3 x ) ( 3 - x ) }$.

Express $\mathrm { f } ( x )$ in the form $\frac { A } { 3 - x } + \frac { B } { 1 + 3 x }$, where $A$ and $B$ are integers.
(3 marks)
1. Find the first three terms of the binomial expansion of $\mathrm { f } ( x )$ in the form $a + b x + c x ^ { 2 }$, where $a$, $b$ and $c$ are rational numbers.
  (7 marks)
2. State why the binomial expansion cannot be expected to give a good approximation to $\mathrm { f } ( x )$ at $x = 0.4$.
  (1 mark)

AQA C4 2013 January Q3

1. Express $3 \cos x + 2 \sin x$ in the form $R \cos ( x - \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, giving your value of $\alpha$ to the nearest $0.1 ^ { \circ }$.
  (3 marks)
2. Hence find the minimum value of $3 \cos x + 2 \sin x$ and the value of $x$ in the interval $0 ^ { \circ } < x < 360 ^ { \circ }$ where the minimum occurs. Give your value of $x$ to the nearest $0.1 ^ { \circ }$.
1. Show that $\cot x - \sin 2 x = \cot x \cos 2 x$ for $0 ^ { \circ } < x < 180 ^ { \circ }$.
2. Hence, or otherwise, solve the equation $$\cot x - \sin 2 x = 0$$ in the interval $0 ^ { \circ } < x < 180 ^ { \circ }$.

AQA C4 2013 January Q4

A curve is defined by the equation $x ^ { 2 } - y ^ { 2 } = 8$.
1. Show that at any point $( p , q )$ on the curve, where $q \neq 0$, the gradient of the curve is given by $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { p } { q }$.
  (2 marks)
2. Show that the tangents at the points $( p , q )$ and $( p , - q )$ intersect on the $x$-axis.
  (4 marks)
Show that $x = t + \frac { 2 } { t } , y = t - \frac { 2 } { t }$ are parametric equations of the curve $x ^ { 2 } - y ^ { 2 } = 8$.
(2 marks)

AQA C4 2013 January Q5

Find $\int x \sqrt { x ^ { 2 } + 3 } \mathrm {~d} x$.
(2 marks)
Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x \sqrt { x ^ { 2 } + 3 } } { \mathrm { e } ^ { 2 y } }$$ given that $y = 0$ when $x = 1$. Give your answer in the form $y = \mathrm { f } ( x )$.

AQA C4 2013 January Q6

The points $A , B$ and $C$ have coordinates $( 3,1 , - 6 ) , ( 5 , - 2,0 )$ and $( 8 , - 4 , - 6 )$ respectively.
1. Show that the vector $\overrightarrow { A C }$ is given by $\overrightarrow { A C } = n \left[ \begin{array} { r } 1
  - 1
  0 \end{array} \right]$, where $n$ is an integer.
2. Show that the acute angle $A C B$ is given by $\cos ^ { - 1 } \left( \frac { 5 \sqrt { 2 } } { 14 } \right)$.
Find a vector equation of the line $A C$.
The point $D$ has coordinates $( 6 , - 1 , p )$. It is given that the lines $A C$ and $B D$ intersect.
1. Find the value of $p$.
2. Show that $A B C D$ is a rhombus, and state the length of each of its sides.

AQA C4 2013 January Q7

7 A biologist is investigating the growth of a population of a species of rodent. The biologist proposes the model $$N = \frac { 500 } { 1 + 9 \mathrm { e } ^ { - \frac { t } { 8 } } }$$ for the number of rodents, $N$, in the population $t$ weeks after the start of the investigation. Use this model to answer the following questions.

1. Find the size of the population at the start of the investigation.
2. Find the size of the population 24 weeks after the start of the investigation. your answer to the nearest whole number.
3. Find the number of weeks that it will take the population to reach 400 . Give your answer in the form $t = r \ln s$, where $r$ and $s$ are integers.
1. Show that the rate of growth, $\frac { \mathrm { d } N } { \mathrm {~d} t }$, is given by $$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { N } { 4000 } ( 500 - N )$$
2. The maximum rate of growth occurs after $T$ weeks. Find the value of $T$.

AQA C4 2010 June Q1

The polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = 8 x ^ { 3 } + 6 x ^ { 2 } - 14 x - 1$.
Find the remainder when $\mathrm { f } ( x )$ is divided by $( 4 x - 1 )$.
The polynomial $\mathrm { g } ( x )$ is defined by $\mathrm { g } ( x ) = 8 x ^ { 3 } + 6 x ^ { 2 } - 14 x + d$.
1. Given that $( 4 x - 1 )$ is a factor of $\mathrm { g } ( x )$, find the value of the constant $d$.
2. Given that $\mathrm { g } ( x ) = ( 4 x - 1 ) \left( a x ^ { 2 } + b x + c \right)$, find the values of the integers $a , b$ and $c$.
  (3 marks)

AQA C4 2010 June Q2

2 A curve is defined by the parametric equations $$x = 1 - 3 t , \quad y = 1 + 2 t ^ { 3 }$$

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 $t = 1$.
Find a cartesian equation of the curve.

Questions — AQA C4 (160 questions)

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AQA C4 2011 January Q1

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