Questions - AQA C4 - A-Level Maths

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4

AQA C4 2008 January Q7

1. Express $6 \sin \theta + 8 \cos \theta$ in the form $R \sin ( \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$. Give your value for $\alpha$ to the nearest $0.1 ^ { \circ }$.
2. Hence solve the equation $6 \sin 2 x + 8 \cos 2 x = 7$, giving all solutions to the nearest $0.1 ^ { \circ }$ in the interval $0 ^ { \circ } < x < 360 ^ { \circ }$.
1. Prove the identity $\frac { \sin 2 x } { 1 - \cos 2 x } = \frac { 1 } { \tan x }$.
2. Hence solve the equation $$\frac { \sin 2 x } { 1 - \cos 2 x } = \tan x$$ giving all solutions in the interval $0 ^ { \circ } < x < 360 ^ { \circ }$.

AQA C4 2008 January Q8

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

AQA C4 2008 January Q9

9 The points $A$ and $B$ lie on the line $l _ { 1 }$ and have coordinates $( 2,5,1 )$ and $( 4,1 , - 2 )$ respectively.

1. Find the vector $\overrightarrow { A B }$.
2. Find a vector equation of the line $l _ { 1 }$, with parameter $\lambda$.
The line $l _ { 2 }$ has equation $\mathbf { r } = \left[ \begin{array} { r } 1
- 3
- 1 \end{array} \right] + \mu \left[ \begin{array} { r } 1
0
- 2 \end{array} \right]$.
1. Show that the point $P ( - 2 , - 3,5 )$ lies on $l _ { 2 }$.
2. The point $Q$ lies on $l _ { 1 }$ and is such that $P Q$ is perpendicular to $l _ { 2 }$. Find the coordinates of $Q$.

AQA C4 2009 January Q1

The polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = 4 x ^ { 3 } - 7 x - 3$.
1. Find $\mathrm { f } ( - 1 )$.
2. Use the Factor Theorem to show that $2 x + 1$ is a factor of $\mathrm { f } ( x )$.
3. Simplify the algebraic fraction $\frac { 4 x ^ { 3 } - 7 x - 3 } { 2 x ^ { 2 } + 3 x + 1 }$.
The polynomial $\mathrm { g } ( x )$ is defined by $\mathrm { g } ( x ) = 4 x ^ { 3 } - 7 x + d$. When $\mathrm { g } ( x )$ is divided by $2 x + 1$, the remainder is 2 . Find the value of $d$.

AQA C4 2009 January Q2

Express $\sin x - 3 \cos x$ in the form $R \sin ( x - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$. Give your value of $\alpha$ in radians to two decimal places.
Hence:
1. write down the minimum value of $\sin x - 3 \cos x$;
2. find the value of $x$ in the interval $0 < x < 2 \pi$ at which this minimum value occurs, giving your value of $x$ in radians to two decimal places.

AQA C4 2009 January Q3

1. Express $\frac { 2 x + 7 } { x + 2 }$ in the form $A + \frac { B } { x + 2 }$, where $A$ and $B$ are integers. (2 marks)
2. Hence find $\int \frac { 2 x + 7 } { x + 2 } \mathrm {~d} x$.
1. Express $\frac { 28 + 4 x ^ { 2 } } { ( 1 + 3 x ) ( 5 - x ) ^ { 2 } }$ in the form $\frac { P } { 1 + 3 x } + \frac { Q } { 5 - x } + \frac { R } { ( 5 - x ) ^ { 2 } }$, where $P , Q$ and $R$ are constants.
2. Hence find $\int \frac { 28 + 4 x ^ { 2 } } { ( 1 + 3 x ) ( 5 - x ) ^ { 2 } } \mathrm {~d} x$.

AQA C4 2009 January Q4

1. Find the binomial expansion of $( 1 - x ) ^ { \frac { 1 } { 2 } }$ up to and including the term in $x ^ { 2 }$.
  (2 marks)
2. Hence obtain the binomial expansion of $\sqrt { 4 - x }$ up to and including the term in $x ^ { 2 }$.
  (3 marks)
Use your answer to part (a)(ii) to find an approximate value for $\sqrt { 3 }$. Give your answer to three decimal places.

AQA C4 2009 January Q5

Express $\sin 2 x$ in terms of $\sin x$ and $\cos x$.
Solve the equation $$5 \sin 2 x + 3 \cos x = 0$$ giving all solutions in the interval $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$ to the nearest $0.1 ^ { \circ }$, where appropriate.
Given that $\sin 2 x + \cos 2 x = 1 + \sin x$ and $\sin x \neq 0$, show that $2 ( \cos x - \sin x ) = 1$.

AQA C4 2009 January Q6

6 A curve is defined by the equation $x ^ { 2 } y + y ^ { 3 } = 2 x + 1$.

Find the gradient of the curve at the point $( 2,1 )$.
Show that the $x$-coordinate of any stationary point on this curve satisfies the equation $$\frac { 1 } { x ^ { 3 } } = x + 1$$ (4 marks)

AQA C4 2009 January Q7

A differential equation is given by $\frac { \mathrm { d } x } { \mathrm {~d} t } = - k t \mathrm { e } ^ { \frac { 1 } { 2 } x }$, where $k$ is a positive constant.
1. Solve the differential equation.
2. Hence, given that $x = 6$ when $t = 0$, show that $x = - 2 \ln \left( \frac { k t ^ { 2 } } { 4 } + \mathrm { e } ^ { - 3 } \right)$.
  (3 marks)
The population of a colony of insects is decreasing according to the model $\frac { \mathrm { d } x } { \mathrm {~d} t } = - k t \mathrm { e } ^ { \frac { 1 } { 2 } x }$, where $x$ thousands is the number of insects in the colony after time $t$ minutes. Initially, there were 6000 insects in the colony. Given that $k = 0.004$, find:
1. the population of the colony after 10 minutes, giving your answer to the nearest hundred;
2. the time after which there will be no insects left in the colony, giving your answer to the nearest 0.1 of a minute.

AQA C4 2009 January Q8

8 The points $A$ and $B$ have coordinates $( 2,1 , - 1 )$ and $( 3,1 , - 2 )$ respectively. The angle $O B A$ is $\theta$, where $O$ is the origin.

1. Find the vector $\overrightarrow { A B }$.
2. Show that $\cos \theta = \frac { 5 } { 2 \sqrt { 7 } }$.
The point $C$ is such that $\overrightarrow { O C } = 2 \overrightarrow { O B }$. The line $l$ is parallel to $\overrightarrow { A B }$ and passes through the point $C$. Find a vector equation of $l$.
The point $D$ lies on $l$ such that angle $O D C = 90 ^ { \circ }$. Find the coordinates of $D$.

AQA C4 2010 January Q1

1 The polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = 15 x ^ { 3 } + 19 x ^ { 2 } - 4$.

1. Find $\mathrm { f } ( - 1 )$.
2. Show that $( 5 x - 2 )$ is a factor of $\mathrm { f } ( x )$.
Simplify $$\frac { 15 x ^ { 2 } - 6 x } { f ( x ) }$$ giving your answer in a fully factorised form.

AQA C4 2010 January Q2

Express $\cos x + 3 \sin x$ in the form $R \cos ( x - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$. Give your value of $\alpha$, in radians, to three decimal places.
1. Hence write down the minimum value of $\cos x + 3 \sin x$.
2. Find the value of $x$ in the interval $0 \leqslant x \leqslant 2 \pi$ at which this minimum occurs, giving your answer, in radians, to three decimal places.
Solve the equation $\cos x + 3 \sin x = 2$ in the interval $0 \leqslant x \leqslant 2 \pi$, giving all solutions, in radians, to three decimal places.

AQA C4 2010 January Q3

1. Find the binomial expansion of $( 1 + x ) ^ { - \frac { 1 } { 3 } }$ up to and including the term in $x ^ { 2 }$.
  (2 marks)
2. Hence find the binomial expansion of $\left( 1 + \frac { 3 } { 4 } x \right) ^ { - \frac { 1 } { 3 } }$ up to and including the term in $x ^ { 2 }$.
Hence show that $\sqrt [ 3 ] { \frac { 256 } { 4 + 3 x } } \approx a + b x + c x ^ { 2 }$ for small values of $x$, stating the values of the constants $a , b$ and $c$.

AQA C4 2010 January Q4

4 The expression $\frac { 10 x ^ { 2 } + 8 } { ( x + 1 ) ( 5 x - 1 ) }$ can be written in the form $2 + \frac { A } { x + 1 } + \frac { B } { 5 x - 1 }$, where $A$ and $B$ are constants.

Find the values of $A$ and $B$.
Hence find $\int \frac { 10 x ^ { 2 } + 8 } { ( x + 1 ) ( 5 x - 1 ) } \mathrm { d } x$.

AQA C4 2010 January Q5

5 A curve is defined by the equation $$x ^ { 2 } + x y = \mathrm { e } ^ { y }$$ Find the gradient at the point $( - 1,0 )$ on this curve.

AQA C4 2010 January Q6

1. Express $\sin 2 \theta$ and $\cos 2 \theta$ in terms of $\sin \theta$ and $\cos \theta$.
2. Given that $0 < \theta < \frac { \pi } { 2 }$ and $\cos \theta = \frac { 3 } { 5 }$, show that $\sin 2 \theta = \frac { 24 } { 25 }$ and find the value of $\cos 2 \theta$.
A curve has parametric equations $$x = 3 \sin 2 \theta , \quad y = 4 \cos 2 \theta$$
1. Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $\theta$.
2. At the point $P$ on the curve, $\cos \theta = \frac { 3 } { 5 }$ and $0 < \theta < \frac { \pi } { 2 }$. Find an equation of the tangent to the curve at the point $P$.

AQA C4 2010 January Q7

7 Solve the differential equation $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { y } \cos \left( \frac { x } { 3 } \right)$, given that $y = 1$ when $x = \frac { \pi } { 2 }$.
Write your answer in the form $y ^ { 2 } = \mathrm { f } ( x )$.

AQA C4 2010 January Q8

8 The points $A , B$ and $C$ have coordinates $( 2 , - 1 , - 5 ) , ( 0,5 , - 9 )$ and $( 9,2,3 )$ respectively. The line $l$ has equation $\mathbf { r } = \left[ \begin{array} { r } 2
- 1
- 5 \end{array} \right] + \lambda \left[ \begin{array} { r } 1
- 3
2 \end{array} \right]$.

Verify that the point $B$ lies on the line $l$.
Find the vector $\overrightarrow { B C }$.
The point $D$ is such that $\overrightarrow { A D } = 2 \overrightarrow { B C }$.
1. Show that $D$ has coordinates $( 20 , - 7,19 )$.
2. The point $P$ lies on $l$ where $\lambda = p$. The line $P D$ is perpendicular to $l$. Find the value of $p$.

AQA C4 2010 January Q9

9 A botanist is investigating the rate of growth of a certain species of toadstool. She observes that a particular toadstool of this type has a height of 57 millimetres at a time 12 hours after it begins to grow. She proposes the model $h = A \left( 1 - \mathrm { e } ^ { - \frac { 1 } { 4 } t } \right)$, where $A$ is a constant, for the height $h$ millimetres of the toadstool, $t$ hours after it begins to grow.

Use this model to:
1. find the height of the toadstool when $t = 0$;
2. show that $A = 60$, correct to two significant figures.
Use the model $h = 60 \left( 1 - \mathrm { e } ^ { - \frac { 1 } { 4 } t } \right)$ to:
1. show that the time $T$ hours for the toadstool to grow to a height of 48 millimetres is given by $$T = a \ln b$$ where $a$ and $b$ are integers;
2. show that $\frac { \mathrm { d } h } { \mathrm {~d} t } = 15 - \frac { h } { 4 }$;
3. find the height of the toadstool when it is growing at a rate of 13 millimetres per hour.
  (1 mark)

AQA C4 2005 June Q1

Express $2 \sin x + \cos x$ in the form $R \sin ( x + \alpha )$ where $R$ is a positive constant and $\alpha$ is an acute angle. Give your value of $\alpha$ to the nearest $0.1 ^ { \circ }$.
Solve the equation $2 \sin x + \cos x = 1$ for $0 ^ { \circ } \leqslant x < 360 ^ { \circ }$.

AQA C4 2005 June Q2

Express $\frac { 3 x - 5 } { ( x + 3 ) ( 2 x - 1 ) }$ in the form $\frac { A } { x + 3 } + \frac { B } { 2 x - 1 }$.
Hence find $\int \frac { 3 x - 5 } { ( x + 3 ) ( 2 x - 1 ) } \mathrm { d } x$.

AQA C4 2005 June Q3

Find the remainder when $2 x ^ { 3 } - x ^ { 2 } + 2 x - 2$ is divided by $2 x - 1$.
Given that $\frac { 2 x ^ { 3 } - x ^ { 2 } + 2 x - 2 } { 2 x - 1 } = x ^ { 2 } + a + \frac { b } { 2 x - 1 }$, find the values of $a$ and $b$.

AQA C4 2005 June Q4

Find the binomial expansion of $( 1 + x ) ^ { - \frac { 1 } { 2 } }$ up to the term in $x ^ { 2 }$.
Hence, or otherwise, obtain the binomial expansion of $\frac { 1 } { \sqrt { 1 + 2 x } }$ up to the term in $x ^ { 2 }$, in simplified form.
Use your answer to part (b) with $x = - 0.1$ to show that $\sqrt { 5 } \approx 2.23$.

AQA C4 2005 June Q5

5 A curve is defined by the parametric equations $$x = 2 t + \frac { 1 } { t } , \quad y = \frac { 1 } { t } , \quad t \neq 0$$

Find the coordinates of the point on the curve where $t = \frac { 1 } { 2 }$.
Show that the cartesian equation of the curve can be written as $$x y - y ^ { 2 } = 2$$
Show that the gradient of the curve at the point $( 3,2 )$ is 2 .

Questions — AQA C4 (160 questions)

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