Questions - AQA C4 - A-Level Maths

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AQA C4 2014 June Q7

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 2016 June Q7

Find the exact value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at the point on $C$ where $t = \frac { 2 } { 3 }$.
Show that $x = \frac { 4 - \mathrm { e } ^ { 2 - 6 t } } { 4 }$ can be rearranged into the form $\mathrm { e } ^ { 3 t } = \frac { \mathrm { e } } { 2 \sqrt { ( 1 - x ) } }$.
Hence find the Cartesian equation of $C$, giving your answer in the form $$y = \frac { \mathrm { e } } { \mathrm { f } ( x ) [ 1 - \ln ( \mathrm { f } ( x ) ) ] }$$

AQA C4 Q5

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

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

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

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

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

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

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

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

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

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

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

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 $\int \sin ^ { 2 } x \mathrm {~d} x$.

AQA C4 2007 January Q4

1. Express $\frac { 3 x - 5 } { x - 3 }$ in the form $A + \frac { B } { x - 3 }$, where $A$ and $B$ are integers. (2 marks)
2. Hence find $\int \frac { 3 x - 5 } { x - 3 } \mathrm {~d} x$.
  (2 marks)
1. Express $\frac { 6 x - 5 } { 4 x ^ { 2 } - 25 }$ in the form $\frac { P } { 2 x + 5 } + \frac { Q } { 2 x - 5 }$, where $P$ and $Q$ are integers.
  (3 marks)
2. Hence find $\int \frac { 6 x - 5 } { 4 x ^ { 2 } - 25 } \mathrm {~d} x$.

AQA C4 2007 January Q5

Find the binomial expansion of $( 1 + x ) ^ { \frac { 1 } { 3 } }$ up to the term in $x ^ { 2 }$.
1. Show that $( 8 + 3 x ) ^ { \frac { 1 } { 3 } } \approx 2 + \frac { 1 } { 4 } x - \frac { 1 } { 32 } x ^ { 2 }$ for small values of $x$.
2. Hence show that $\sqrt [ 3 ] { 9 } \approx \frac { 599 } { 288 }$.

AQA C4 2007 January Q6

6 The points $A , B$ and $C$ have coordinates $( 3 , - 2,4 ) , ( 5,4,0 )$ and $( 11,6 , - 4 )$ respectively.

1. Find the vector $\overrightarrow { B A }$.
2. Show that the size of angle $A B C$ is $\cos ^ { - 1 } \left( - \frac { 5 } { 7 } \right)$.
The line $l$ has equation $\mathbf { r } = \left[ \begin{array} { r } 8
- 3
2 \end{array} \right] + \lambda \left[ \begin{array} { r } 1
3
- 2 \end{array} \right]$.
1. Verify that $C$ lies on $l$.
2. Show that $A B$ is parallel to $l$.
The quadrilateral $A B C D$ is a parallelogram. Find the coordinates of $D$.

AQA C4 2007 January Q7

Use the identity $$\tan ( A + B ) = \frac { \tan A + \tan B } { 1 - \tan A \tan B }$$ to express $\tan 2 x$ in terms of $\tan x$.
Show that $$2 - 2 \tan x - \frac { 2 \tan x } { \tan 2 x } = ( 1 - \tan x ) ^ { 2 }$$ for all values of $x , \tan 2 x \neq 0$.

AQA C4 2007 January Q8

1. Solve the differential equation $\frac { \mathrm { d } y } { \mathrm {~d} t } = y \sin t$ to obtain $y$ in terms of $t$.
2. Given that $y = 50$ when $t = \pi$, show that $y = 50 \mathrm { e } ^ { - ( 1 + \cos t ) }$.
A wave machine at a leisure pool produces waves. The height of the water, $y \mathrm {~cm}$, above a fixed point at time $t$ seconds is given by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} t } = y \sin t$$
1. Given that this height is 50 cm after $\pi$ seconds, find, to the nearest centimetre, the height of the water after 6 seconds.
2. Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }$ and hence verify that the water reaches a maximum height after $\pi$ seconds.

AQA C4 2008 January Q1

Given that $\frac { 3 } { 9 - x ^ { 2 } }$ can be expressed in the form $k \left( \frac { 1 } { 3 + x } + \frac { 1 } { 3 - x } \right)$, find the value of the rational number $k$.
Show that $\int _ { 1 } ^ { 2 } \frac { 3 } { 9 - x ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln \left( \frac { a } { b } \right)$, where $a$ and $b$ are integers.

AQA C4 2008 January Q2

The polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 18 x + 8$.
1. Use the Factor Theorem to show that $( 2 x - 1 )$ is a factor of $\mathrm { f } ( x )$.
2. Write $\mathrm { f } ( x )$ in the form $( 2 x - 1 ) \left( x ^ { 2 } + p x + q \right)$, where $p$ and $q$ are integers.
3. Simplify the algebraic fraction $\frac { 4 x ^ { 2 } + 16 x } { 2 x ^ { 3 } + 3 x ^ { 2 } - 18 x + 8 }$.
Express the algebraic fraction $\frac { 2 x ^ { 2 } } { ( x + 5 ) ( x - 3 ) }$ in the form $A + \frac { B + C x } { ( x + 5 ) ( x - 3 ) }$, where $A , B$ and $C$ are integers.

AQA C4 2008 January Q3

Obtain the binomial expansion of $( 1 + x ) ^ { \frac { 1 } { 2 } }$ up to and including the term in $x ^ { 2 }$.
Hence obtain the binomial expansion of $\sqrt { 1 + \frac { 3 } { 2 } x }$ up to and including the term in $x ^ { 2 }$.
Hence show that $\sqrt { \frac { 2 + 3 x } { 8 } } \approx a + b x + c x ^ { 2 }$ for small values of $x$, where $a , b$ and $c$ are constants to be found.

AQA C4 2008 January Q4

4 David is researching changes in the selling price of houses. One particular house was sold on 1 January 1885 for $\pounds 20$. Sixty years later, on 1 January 1945, it was sold for $\pounds 2000$. David proposes a model $$P = A k ^ { t }$$ for the selling price, $\pounds P$, of this house, where $t$ is the time in years after 1 January 1885 and $A$ and $k$ are constants.

1. Write down the value of $A$.
2. Show that, to six decimal places, $k = 1.079775$.
3. Use the model, with this value of $k$, to estimate the selling price of this house on 1 January 2008. Give your answer to the nearest $\pounds 1000$.
For another house, which was sold for $\pounds 15$ on 1 January 1885, David proposes the model $$Q = 15 \times 1.082709 ^ { t }$$ for the selling price, $\pounds Q$, of this house $t$ years after 1 January 1885. Calculate the year in which, according to these models, these two houses would have had the same selling price.

AQA C4 2008 January Q5

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

At the point $P$ on the curve, $t = \frac { 1 } { 2 }$.
1. Find the coordinates of $P$.
2. Find an equation of the tangent to the curve at $P$.
Show that the cartesian equation of the curve can be written as $$( x - y ) ( x + y ) ^ { 2 } = k$$ where $k$ is an integer.

AQA C4 2008 January Q6

6 A curve has equation $3 x y - 2 y ^ { 2 } = 4$.
Find the gradient of the curve at the point $( 2,1 )$.

Questions — AQA C4 (160 questions)

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