Questions - AQA C4 - A-Level Maths

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AQA C4 2014 June Q4

11 marks Moderate -0.3

4 A painting was valued on 1 April 2001 at $\pounds 5000$.
The value of this painting is modelled by $$V = A p ^ { t }$$ where $\pounds V$ is the value $t$ years after 1 April 2001, and $A$ and $p$ are constants.

Write down the value of $A$.
According to the model, the value of this painting on 1 April 2011 was $\pounds 25000$. Using this model:
1. show that $p ^ { 10 } = 5$;
2. use logarithms to find the year in which the painting will be valued at $\pounds 75000$.
A painting by another artist was valued at $\pounds 2500$ on 1 April 1991. The value of this painting is modelled by $$W = 2500 q ^ { t }$$ where $\pounds W$ is the value $t$ years after 1 April 1991, and $q$ is a constant.
1. Show that, according to the two models, the value of the two paintings will be the same $T$ years after 1 April 1991, $$\text { where } T = \frac { \ln \left( \frac { 5 } { 2 } \right) } { \ln \left( \frac { p } { q } \right) }$$
2. Given that $p = 1.029 q$, find the year in which the two paintings will have the same value.
  [0pt] [1 mark]

AQA C4 2014 June Q5

15 marks Standard +0.3

1. Express $3 \sin x + 4 \cos x$ in the form $R \sin ( x + \alpha )$ where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, giving your value of $\alpha$ to the nearest $0.1 ^ { \circ }$.
2. Hence solve the equation $3 \sin 2 \theta + 4 \cos 2 \theta = 5$ in the interval $0 ^ { \circ } < \theta < 360 ^ { \circ }$, giving your solutions to the nearest $0.1 ^ { \circ }$.
1. Show that the equation $\tan 2 \theta \tan \theta = 2$ can be written as $2 \tan ^ { 2 } \theta = 1$.
2. Hence solve the equation $\tan 2 \theta \tan \theta = 2$ in the interval $0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }$, giving your solutions to the nearest $0.1 ^ { \circ }$.
1. Use the Factor Theorem to show that $2 x - 1$ is a factor of $8 x ^ { 3 } - 4 x + 1$.
2. Show that $4 \cos 2 \theta \cos \theta + 1$ can be written as $8 x ^ { 3 } - 4 x + 1$ where $x = \cos \theta$.
3. Given that $\theta = 72 ^ { \circ }$ is a solution of $4 \cos 2 \theta \cos \theta + 1 = 0$, use the results from parts (c)(i) and (c)(ii) to show that the exact value of $\cos 72 ^ { \circ }$ is $\frac { ( \sqrt { 5 } - 1 ) } { p }$ where $p$ is an integer.
  [0pt] [3 marks]

AQA C4 2014 June Q6

10 marks Moderate -0.3

6 The line $l _ { 1 }$ has equation $\mathbf { r } = \left[ \begin{array} { r } 4 \\ - 5 \\ 3 \end{array} \right] + \lambda \left[ \begin{array} { r } - 1 \\ 3 \\ 1 \end{array} \right]$.
The line $l _ { 2 }$ has equation $\mathbf { r } = \left[ \begin{array} { r } 7 \\ - 8 \\ 6 \end{array} \right] + \mu \left[ \begin{array} { r } 2 \\ - 3 \\ 1 \end{array} \right]$.
The point $P$ lies on $l _ { 1 }$ where $\lambda = - 1$. The point $Q$ lies on $l _ { 2 }$ where $\mu = 2$.

Show that the vector $\overrightarrow { P Q }$ is parallel to $\left[ \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right]$.
The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $R ( 3 , b , c )$.
1. Show that $b = - 2$ and find the value of $c$.
2. The point $S$ lies on a line through $P$ that is parallel to $l _ { 2 }$. The line $R S$ is perpendicular to the line $P Q$. \includegraphics[max width=\textwidth, alt={}, center]{9f03a5f3-7fea-4fb7-b3bd-b4c0cdf662a2-16_887_1159_1320_443} Find the coordinates of $S$. $7 \quad$ A curve has equation $\cos 2 y + y \mathrm { e } ^ { 3 x } = 2 \pi$.
  The point $A \left( \ln 2 , \frac { \pi } { 4 } \right)$ lies on this curve.

AQA C4 2014 June Q8

11 marks Standard +0.3

Express $\frac { 16 x } { ( 1 - 3 x ) ( 1 + x ) ^ { 2 } }$ in the form $\frac { A } { 1 - 3 x } + \frac { B } { 1 + x } + \frac { C } { ( 1 + x ) ^ { 2 } }$.
Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 16 x \mathrm { e } ^ { 2 y } } { ( 1 - 3 x ) ( 1 + x ) ^ { 2 } }$$ where $y = 0$ when $x = 0$.
Give your answer in the form $\mathrm { f } ( y ) = \mathrm { g } ( x )$.
[0pt] [7 marks]

AQA C4 2015 June Q1

9 marks Moderate -0.8

1 It is given that $\mathrm { f } ( x ) = \frac { 19 x - 2 } { ( 5 - x ) ( 1 + 6 x ) }$ can be expressed as $\frac { A } { 5 - x } + \frac { B } { 1 + 6 x }$, where $A$ and $B$ are integers.

Find the values of $A$ and $B$.
Hence show that $\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x = k \ln 5$, where $k$ is a rational number.
[0pt] [6 marks]

AQA C4 2015 June Q2

8 marks Standard +0.3

Express $2 \cos x - 5 \sin x$ in the form $R \cos ( x + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$, giving your value of $\alpha$, in radians, to three significant figures.
1. Hence find the value of $x$ in the interval $0 < x < 2 \pi$ for which $2 \cos x - 5 \sin x$ has its maximum value. Give your value of $x$ to three significant figures.
2. Use your answer to part (a) to solve the equation $2 \cos x - 5 \sin x + 1 = 0$ in the interval $0 < x < 2 \pi$, giving your solutions to three significant figures.
  [0pt] [3 marks]

AQA C4 2015 June Q3

9 marks Moderate -0.3

The polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = 8 x ^ { 3 } - 12 x ^ { 2 } - 2 x + d$, where $d$ is a constant. When $\mathrm { f } ( x )$ is divided by ( $2 x + 1$ ), the remainder is - 2 . Use the Remainder Theorem to find the value of $d$.
The polynomial $\mathrm { g } ( x )$ is defined by $\mathrm { g } ( x ) = 8 x ^ { 3 } - 12 x ^ { 2 } - 2 x + 3$.
1. Given that $x = - \frac { 1 } { 2 }$ is a solution of the equation $\mathrm { g } ( x ) = 0$, write $\mathrm { g } ( x )$ as a product of three linear factors.
2. The function h is defined by $\mathrm { h } ( x ) = \frac { 4 x ^ { 2 } - 1 } { \mathrm {~g} ( x ) }$ for $x > 2$. Simplify $\mathrm { h } ( x )$, and hence show that h is a decreasing function.
  [0pt] [4 marks]

AQA C4 2015 June Q4

7 marks Standard +0.3

Find the binomial expansion of $( 1 + 5 x ) ^ { \frac { 1 } { 5 } }$ up to and including the term in $x ^ { 2 }$.
1. Find the binomial expansion of $( 8 + 3 x ) ^ { - \frac { 2 } { 3 } }$ up to and including the term in $x ^ { 2 }$.
2. Use your expansion from part (b)(i) to find an estimate for $\sqrt [ 3 ] { \frac { 1 } { 81 } }$, giving your answer to four decimal places.
  [0pt] [2 marks]

AQA C4 2015 June Q5

11 marks Standard +0.8

5 A curve is defined by the parametric equations $x = \cos 2 t , y = \sin t$.
The point $P$ on the curve is where $t = \frac { \pi } { 6 }$.

Find the gradient at $P$.
Find the equation of the normal to the curve at $P$ in the form $y = m x + c$.
The normal at $P$ intersects the curve again at the point $Q ( \cos 2 q , \sin q )$. Use the equation of the normal to form a quadratic equation in $\sin q$ and hence find the $x$-coordinate of $Q$.
[0pt] [5 marks]

AQA C4 2015 June Q6

12 marks Challenging +1.2

6 The points $A$ and $B$ have coordinates $( 3,2,10 )$ and $( 5 , - 2,4 )$ respectively.
The line $l$ passes through $A$ and has equation $\mathbf { r } = \left[ \begin{array} { r } 3 \\ 2 \\ 10 \end{array} \right] + \lambda \left[ \begin{array} { r } 3 \\ 1 \\ - 2 \end{array} \right]$.

Find the acute angle between $l$ and the line $A B$.
The point $C$ lies on $l$ such that angle $A B C$ is $90 ^ { \circ }$. \includegraphics[max width=\textwidth, alt={}, center]{fdd3905e-11f7-4b20-adfe-4c686018a221-12_360_339_762_852} Find the coordinates of $C$.
The point $D$ is such that $B D$ is parallel to $A C$ and angle $B C D$ is $90 ^ { \circ }$. The point $E$ lies on the line through $B$ and $D$ and is such that the length of $D E$ is half that of $A C$. Find the coordinates of the two possible positions of $E$.
[0pt] [4 marks]

AQA C4 2015 June Q7

7 marks Standard +0.3

7 A curve has equation $y ^ { 3 } + 2 \mathrm { e } ^ { - 3 x } y - x = k$, where $k$ is a constant.
The point $P \left( \ln 2 , \frac { 1 } { 2 } \right)$ lies on this curve.

Show that the exact value of $k$ is $q - \ln 2$, where $q$ is a rational number.
Find the gradient of the curve at $P$.

AQA C4 2015 June Q8

12 marks Standard +0.3

A pond is initially empty and is then filled gradually with water. After $t$ minutes, the depth of the water, $x$ metres, satisfies the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { \sqrt { 4 + 5 x } } { 5 ( 1 + t ) ^ { 2 } }$$ Solve this differential equation to find $x$ in terms of $t$.
Another pond is gradually filling with water. After $t$ minutes, the surface of the water forms a circle of radius $r$ metres. The rate of change of the radius is inversely proportional to the area of the surface of the water.
1. Write down a differential equation, in the variables $r$ and $t$ and a constant of proportionality, which represents how the radius of the surface of the water is changing with time.
  (You are not required to solve your differential equation.)
2. When the radius of the pond is 1 metre, the radius is increasing at a rate of 4.5 metres per second. Find the radius of the pond when the radius is increasing at a rate of 0.5 metres per second.
  [0pt] [2 marks]
  \includegraphics[max width=\textwidth, alt={}]{fdd3905e-11f7-4b20-adfe-4c686018a221-18_1277_1709_1430_153}
  
  \includegraphics[max width=\textwidth, alt={}]{fdd3905e-11f7-4b20-adfe-4c686018a221-20_2288_1707_221_153}

AQA C4 2014 June Q7

9 marks Moderate -0.3

1. Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
2. Hence find the exact value of the gradient of the curve at $A$.
The normal at $A$ crosses the $y$-axis at the point $B$. Find the exact value of the $y$-coordinate of $B$.
[0pt] [2 marks]

AQA C4 Q5

10 marks Standard +0.3

1. Obtain the binomial expansion of $( 1 - x ) ^ { - 1 }$ up to and including the term in $x ^ { 2 }$.
2. Hence, or otherwise, show that $$\frac { 1 } { 3 - 2 x } \approx \frac { 1 } { 3 } + \frac { 2 } { 9 } x + \frac { 4 } { 27 } x ^ { 2 }$$ for small values of $x$.
Obtain the binomial expansion of $\frac { 1 } { ( 1 - x ) ^ { 2 } }$ up to and including the term in $x ^ { 2 }$.
Given that $\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }$ can be written in the form $\frac { A } { ( 3 - 2 x ) } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 1 - x ) ^ { 2 } }$, find the values of $A , B$ and $C$.
Hence find the binomial expansion of $\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }$ up to and including the term in $x ^ { 2 }$.

AQA C4 2006 January Q1

8 marks Moderate -0.8

The polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = 3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + 2$.
1. Find f(1).
2. Show that $\mathrm { f } ( - 2 ) = 0$.
3. Hence, or otherwise, show that $$\frac { ( x - 1 ) ( x + 2 ) } { 3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + 2 } = \frac { 1 } { a x + b }$$ where $a$ and $b$ are integers.
The polynomial $\mathrm { g } ( x )$ is defined by $\mathrm { g } ( x ) = 3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + d$. When $\mathrm { g } ( x )$ is divided by $( 3 x - 1 )$, the remainder is 2 . Find the value of $d$.

AQA C4 2006 January Q2

11 marks Moderate -0.3

2 A curve is defined by the parametric equations $$x = 3 - 4 t \quad y = 1 + \frac { 2 } { t }$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$.
Find the equation of the tangent to the curve at the point where $t = 2$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
Verify that the cartesian equation of the curve can be written as $$( x - 3 ) ( y - 1 ) + 8 = 0$$

AQA C4 2006 January Q3

6 marks Moderate -0.3

3 It is given that $3 \cos \theta - 2 \sin \theta = R \cos ( \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.

Find the value of $R$.
Show that $\alpha \approx 33.7 ^ { \circ }$.
Hence write down the maximum value of $3 \cos \theta - 2 \sin \theta$ and find a positive value of $\theta$ at which this maximum value occurs.

AQA C4 2006 January Q4

9 marks Moderate -0.8

4 On 1 January 1900, a sculpture was valued at $\pounds 80$.
When the sculpture was sold on 1 January 1956, its value was $\pounds 5000$.
The value, $\pounds V$, of the sculpture is modelled by the formula $V = A k ^ { t }$, where $t$ is the time in years since 1 January 1900 and $A$ and $k$ are constants.

Write down the value of $A$.
Show that $k \approx 1.07664$.
Use this model to:
1. show that the value of the sculpture on 1 January 2006 will be greater than £200 000;
2. find the year in which the value of the sculpture will first exceed $\pounds 800000$.

AQA C4 2006 January Q5

15 marks Standard +0.3

1. Obtain the binomial expansion of $( 1 - x ) ^ { - 1 }$ up to and including the term in $x ^ { 2 }$.
  (2 marks)
2. Hence, or otherwise, show that $$\frac { 1 } { 3 - 2 x } \approx \frac { 1 } { 3 } + \frac { 2 } { 9 } x + \frac { 4 } { 27 } x ^ { 2 }$$ for small values of $x$.
Obtain the binomial expansion of $\frac { 1 } { ( 1 - x ) ^ { 2 } }$ up to and including the term in $x ^ { 2 }$.
Given that $\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }$ can be written in the form $\frac { A } { ( 3 - 2 x ) } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 1 - x ) ^ { 2 } }$, find the values of $A , B$ and $C$.
Hence find the binomial expansion of $\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }$ up to and including the term in $x ^ { 2 }$.

AQA C4 2006 January Q6

7 marks Moderate -0.8

Express $\cos 2 x$ in the form $a \cos ^ { 2 } x + b$, where $a$ and $b$ are constants.
Hence show that $\int _ { 0 } ^ { \frac { \pi } { 2 } } \cos ^ { 2 } x \mathrm {~d} x = \frac { \pi } { a }$, where $a$ is an integer.

AQA C4 2006 January Q7

10 marks Moderate -0.3

7 The quadrilateral $A B C D$ has vertices $A ( 2,1,3 ) , B ( 6,5,3 ) , C ( 6,1 , - 1 )$ and $D ( 2 , - 3 , - 1 )$.
The line $l _ { 1 }$ has vector equation $\mathbf { r } = \left[ \begin{array} { r } 6 \\ 1 \\ - 1 \end{array} \right] + \lambda \left[ \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right]$.

1. Find the vector $\overrightarrow { A B }$.
2. Show that the line $A B$ is parallel to $l _ { 1 }$.
3. Verify that $D$ lies on $l _ { 1 }$.
The line $l _ { 2 }$ passes through $D ( 2 , - 3 , - 1 )$ and $M ( 4,1,1 )$.
1. Find the vector equation of $l _ { 2 }$.
2. Find the angle between $l _ { 2 }$ and $A C$.

AQA C4 2006 January Q8

9 marks Moderate -0.3

Solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - 2 ( x - 6 ) ^ { \frac { 1 } { 2 } }$$ to find $t$ in terms of $x$, given that $x = 70$ when $t = 0$.
Liquid fuel is stored in a tank. At time $t$ minutes, the depth of fuel in the tank is $x \mathrm {~cm}$. Initially there is a depth of 70 cm of fuel in the tank. There is a tap 6 cm above the bottom of the tank. The flow of fuel out of the tank is modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - 2 ( x - 6 ) ^ { \frac { 1 } { 2 } }$$
1. Explain what happens when $x = 6$.
2. Find how long it will take for the depth of fuel to fall from 70 cm to 22 cm .

AQA C4 2007 January Q1

11 marks Moderate -0.8

1 A curve is defined by the parametric equations $$x = 1 + 2 t , \quad y = 1 - 4 t ^ { 2 }$$

1. Find $\frac { \mathrm { d } x } { \mathrm {~d} t }$ and $\frac { \mathrm { d } y } { \mathrm {~d} t }$.
  (2 marks)
2. Hence 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.

AQA C4 2007 January Q2

6 marks Moderate -0.8

2 The polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 13$.

Use the Remainder Theorem to find the remainder when $\mathrm { f } ( x )$ is divided by $( 2 x - 3 )$.
The polynomial $\mathrm { g } ( x )$ is defined by $\mathrm { g } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 13 + d$, where $d$ is a constant. Given that ( $2 x - 3$ ) is a factor of $\mathrm { g } ( x )$, show that $d = - 4$.
Express $\mathrm { g } ( x )$ in the form $( 2 x - 3 ) \left( x ^ { 2 } + a x + b \right)$.

AQA C4 2007 January Q3

9 marks Standard +0.3

Express $\cos 2 x$ in terms of $\sin x$.
1. Hence show that $3 \sin x - \cos 2 x = 2 \sin ^ { 2 } x + 3 \sin x - 1$ for all values of $x$.
2. Solve the equation $3 \sin x - \cos 2 x = 1$ for $0 ^ { \circ } < x < 360 ^ { \circ }$.
Use your answer from part (a) to find $\int \sin ^ { 2 } x \mathrm {~d} x$.

Questions — AQA C4 (162 questions)

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AQA C4 2014 June Q4

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