Questions - AQA C4 - A-Level Maths

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AQA C4 2010 June Q3

1. Express $\frac { 7 x - 3 } { ( x + 1 ) ( 3 x - 2 ) }$ in the form $\frac { A } { x + 1 } + \frac { B } { 3 x - 2 }$.
2. Hence find $\int \frac { 7 x - 3 } { ( x + 1 ) ( 3 x - 2 ) } \mathrm { d } x$.
Express $\frac { 6 x ^ { 2 } + x + 2 } { 2 x ^ { 2 } - x + 1 }$ in the form $P + \frac { Q x + R } { 2 x ^ { 2 } - x + 1 }$.

AQA C4 2010 June Q4

1. Find the binomial expansion of $( 1 + x ) ^ { \frac { 3 } { 2 } }$ up to and including the term in $x ^ { 2 }$.
2. Find the binomial expansion of $( 16 + 9 x ) ^ { \frac { 3 } { 2 } }$ up to and including the term in $x ^ { 2 }$.
Use your answer to part (a)(ii) to show that $13 ^ { \frac { 3 } { 2 } } \approx 46 + \frac { a } { b }$, stating the values of the integers $a$ and $b$.

AQA C4 2010 June Q5

1. Show that the equation $3 \cos 2 x + 2 \sin x + 1 = 0$ can be written in the form $$3 \sin ^ { 2 } x - \sin x - 2 = 0$$
2. Hence, given that $3 \cos 2 x + 2 \sin x + 1 = 0$, find the possible values of $\sin x$.
1. Express $3 \cos 2 x + 2 \sin 2 x$ in the form $R \cos ( 2 x - \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, giving $\alpha$ to the nearest $0.1 ^ { \circ }$.
2. Hence solve the equation $$3 \cos 2 x + 2 \sin 2 x + 1 = 0$$ for all solutions in the interval $0 ^ { \circ } < x < 180 ^ { \circ }$, giving $x$ to the nearest $0.1 ^ { \circ }$.
  (3 marks)
  $6 \quad$ A curve has equation $x ^ { 3 } y + \cos ( \pi y ) = 7$.
Find the exact value of the $x$-coordinate at the point on the curve where $y = 1$.
Find the gradient of the curve at the point where $y = 1$.

AQA C4 2010 June Q7

7 The point $A$ has coordinates $( 4 , - 3,2 )$.
The line $l _ { 1 }$ passes through $A$ and has equation $\mathbf { r } = \left[ \begin{array} { r } 4
- 3
2 \end{array} \right] + \lambda \left[ \begin{array} { l } 2
0
1 \end{array} \right]$.
The line $l _ { 2 }$ has equation $\mathbf { r } = \left[ \begin{array} { r } - 1
3
4 \end{array} \right] + \mu \left[ \begin{array} { r } 1
- 2
- 1 \end{array} \right]$.
The point $B$ lies on $l _ { 2 }$ where $\mu = 2$.

Find the vector $\overrightarrow { A B }$.
1. Show that the lines $l _ { 1 }$ and $l _ { 2 }$ intersect.
2. The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $P$. Find the coordinates of $P$.
The point $C$ lies on a line which is parallel to $l _ { 1 }$ and which passes through the point $B$. The points $A , B , C$ and $P$ are the vertices of a parallelogram. Find the coordinates of the two possible positions of the point $C$.

AQA C4 2010 June Q8

Solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \frac { 1 } { 5 } ( x + 1 ) ^ { \frac { 1 } { 2 } }$$ given that $x = 80$ when $t = 0$. Give your answer in the form $x = \mathrm { f } ( t )$.
A fungus is spreading on the surface of a wall. The proportion of the wall that is unaffected after time $t$ hours is $x \%$. The rate of change of $x$ is modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \frac { 1 } { 5 } ( x + 1 ) ^ { \frac { 1 } { 2 } }$$ At $t = 0$, the proportion of the wall that is unaffected is $80 \%$. Find the proportion of the wall that will still be unaffected after 60 hours.
A biologist proposes an alternative model for the rate at which the fungus is spreading on the wall. The total surface area of the wall is $9 \mathrm {~m} ^ { 2 }$. The surface area that is affected at time $t$ hours is $A \mathrm {~m} ^ { 2 }$. The biologist proposes that the rate of change of $A$ is proportional to the product of the surface area that is affected and the surface area that is unaffected.
1. Write down a differential equation for this model.
  (You are not required to solve your differential equation.)
2. A solution of the differential equation for this model is given by $$A = \frac { 9 } { 1 + 4 \mathrm { e } ^ { - 0.09 t } }$$ Find the time taken for $50 \%$ of the area of the wall to be affected. Give your answer in hours to three significant figures.
  (4 marks)

AQA C4 2011 June Q1

1 The polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = 4 x ^ { 3 } - 13 x + 6$.

Find $\mathrm { f } ( - 2 )$.
Use the Factor Theorem to show that $2 x - 3$ is a factor of $\mathrm { f } ( x )$.
Simplify $\frac { 2 x ^ { 2 } + x - 6 } { \mathrm { f } ( x ) }$.

AQA C4 2011 June Q2

2 The average weekly pay of a footballer at a certain club was $\pounds 80$ on 1 August 1960. By 1 August 1985, this had risen to $\pounds 2000$. The average weekly pay of a footballer at this club can be modelled by the equation $$P = A k ^ { t }$$ where $\pounds P$ is the average weekly pay $t$ years after 1 August 1960, and $A$ and $k$ are constants.

1. Write down the value of $A$.
2. Show that the value of $k$ is 1.137411 , correct to six decimal places.
Use this model to predict the year in which, on 1 August, the average weekly pay of a footballer at this club will first exceed $\pounds 100000$.

AQA C4 2011 June Q3

1. Find the binomial expansion of $( 1 - x ) ^ { \frac { 1 } { 3 } }$ up to and including the term in $x ^ { 2 }$.
2. Hence, or otherwise, show that $$( 125 - 27 x ) ^ { \frac { 1 } { 3 } } \approx 5 + \frac { m } { 25 } x + \frac { n } { 3125 } x ^ { 2 }$$ for small values of $x$, stating the values of the integers $m$ and $n$.
Use your result from part (a)(ii) to find an approximate value of $\sqrt [ 3 ] { 119 }$, giving your answer to five decimal places.
(2 marks)

AQA C4 2011 June Q4

A curve is defined by the parametric equations $x = 3 \cos 2 \theta , y = 2 \cos \theta$.
1. Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { k \cos \theta }$, where $k$ is an integer.
2. Find an equation of the normal to the curve at the point where $\theta = \frac { \pi } { 3 }$.
Find the exact value of $\int _ { - \frac { \pi } { 4 } } ^ { \frac { \pi } { 4 } } \sin ^ { 2 } x \mathrm {~d} x$.

AQA C4 2011 June Q5

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

Find a vector equation of the line that passes through $A$ and $B$.
1. Show that the line that passes through $A$ and $B$ intersects the line $l$, and find the coordinates of the point of intersection, $P$.
2. The point $C$ lies on $l$ such that triangle $P B C$ has a right angle at $B$. Find the coordinates of $C$.

AQA C4 2011 June Q6

6 A curve is defined by the equation $2 y + \mathrm { e } ^ { 2 x } y ^ { 2 } = x ^ { 2 } + C$, where $C$ is a constant. The point $P \left( 1 , \frac { 1 } { \mathrm { e } } \right)$ lies on the curve.

Find the exact value of $C$.
Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.
Verify that $P \left( 1 , \frac { 1 } { \mathrm { e } } \right)$ is a stationary point on the curve.

AQA C4 2011 June Q7

7 A giant snowball is melting. The snowball can be modelled as a sphere whose surface area is decreasing at a constant rate with respect to time. The surface area of the sphere is $A \mathrm {~cm} ^ { 2 }$ at time $t$ days after it begins to melt.

Write down a differential equation in terms of the variables $A$ and $t$ and a constant $k$, where $k > 0$, to model the melting snowball.
1. Initially, the radius of the snowball is 60 cm , and 9 days later, the radius has halved. Show that $A = 1200 \pi ( 12 - t )$.
  (You may assume that the surface area of a sphere is given by $A = 4 \pi r ^ { 2 }$, where $r$ is the radius.)
2. Use this model to find the number of days that it takes the snowball to melt completely.

AQA C4 2011 June Q8

Express $\frac { 1 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }$ in the form $\frac { A } { 3 - 2 x } + \frac { B } { 1 - x } + \frac { C } { ( 1 - x ) ^ { 2 } }$.
(4 marks)
Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 \sqrt { y } } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }$$ where $y = 0$ when $x = 0$, expressing your answer in the form $$y ^ { p } = q \ln [ \mathrm { f } ( x ) ] + \frac { x } { 1 - x }$$ where $p$ and $q$ are constants.

AQA C4 2012 June Q1

1. Express $\frac { 5 x - 6 } { x ( x - 3 ) }$ in the form $\frac { A } { x } + \frac { B } { x - 3 }$.
  (2 marks)
2. Find $\int \frac { 5 x - 6 } { x ( x - 3 ) } \mathrm { d } x$.
  (2 marks)
1. Given that $$4 x ^ { 3 } + 5 x - 2 = ( 2 x + 1 ) \left( 2 x ^ { 2 } + p x + q \right) + r$$ find the values of the constants $p , q$ and $r$.
2. Find $\int \frac { 4 x ^ { 3 } + 5 x - 2 } { 2 x + 1 } \mathrm {~d} x$.

AQA C4 2012 June Q2

Express $\sin x - 3 \cos x$ in the form $R \sin ( x - \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, giving your value of $\alpha$ to the nearest $0.1 ^ { \circ }$.
Hence find the values of $x$ in the interval $0 ^ { \circ } < x < 360 ^ { \circ }$ for which $$\sin x - 3 \cos x + 2 = 0$$ giving your values of $x$ to the nearest degree.

AQA C4 2012 June Q3

Find the binomial expansion of $( 1 + 4 x ) ^ { \frac { 1 } { 2 } }$ up to and including the term in $x ^ { 2 }$.
(2 marks)
1. Find the binomial expansion of $( 4 - x ) ^ { - \frac { 1 } { 2 } }$ up to and including the term in $x ^ { 2 }$.
2. State the range of values of $x$ for which the expansion in part (b)(i) is valid.
Find the binomial expansion of $\sqrt { \frac { 1 + 4 x } { 4 - x } }$ up to and including the term in $x ^ { 2 }$.
(2 marks)

AQA C4 2012 June Q4

4 The value, $\pounds V$, of an initial investment, $\pounds P$, at the end of $n$ years is given by the formula $$V = P \left( 1 + \frac { r } { 100 } \right) ^ { n }$$ where $r \%$ per year is the fixed interest rate.
Mr Brown invests $\pounds 1000$ in Barcelona Bank at a fixed interest rate of $3 \%$ per year.

1. Find the value of Mr Brown's investment at the end of 5 years. Give your value to the nearest $\pounds 10$.
2. The value of Mr Brown's investment will first exceed $\pounds 2000$ after $N$ complete years. Find the value of $N$.
Mrs White invests $\pounds 1500$ in Bilbao Bank at a fixed interest rate of $1.5 \%$ per year. Mr Brown and Mrs White invest their money at the same time. The value of Mr Brown's investment will first exceed the value of Mrs White's investment after $T$ complete years. Find the value of $T$.

AQA C4 2012 June Q5

5 A curve is defined by the parametric equations $$x = 2 \cos \theta , \quad y = 3 \sin 2 \theta$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = a \sin \theta + b \operatorname { cosec } \theta$$ where $a$ and $b$ are integers.
2. Find the gradient of the normal to the curve at the point where $\theta = \frac { \pi } { 6 }$.
Show that the cartesian equation of the curve can be expressed as $$y ^ { 2 } = p x ^ { 2 } \left( 4 - x ^ { 2 } \right)$$ where $p$ is a rational number.

AQA C4 2012 June Q6

6 A curve is defined by the equation $9 x ^ { 2 } - 6 x y + 4 y ^ { 2 } = 3$. Find the coordinates of the two stationary points of this curve.

AQA C4 2012 June Q7

$\mathbf { 7 } \quad$ The line $l _ { 1 }$ has equation $\mathbf { r } = \left[ \begin{array} { r } 0
- 2
q \end{array} \right] + \lambda \left[ \begin{array} { r } 2
0
- 1 \end{array} \right]$, where $q$ is an integer. The line $l _ { 2 }$ has equation $\mathbf { r } = \left[ \begin{array} { l } 8
3
5 \end{array} \right] + \mu \left[ \begin{array} { l } 2
5
4 \end{array} \right]$. The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $P$.

Show that $q = 4$ and find the coordinates of $P$.
Show that $l _ { 1 }$ and $l _ { 2 }$ are perpendicular.
The point $A$ lies on the line $l _ { 1 }$ where $\lambda = 1$.
1. Find $A P ^ { 2 }$.
2. The point $B$ lies on the line $l _ { 2 }$ so that the right-angled triangle $A P B$ is isosceles. Find the coordinates of the two possible positions of $B$.

AQA C4 2012 June Q8

A water tank has a height of 2 metres. The depth of the water in the tank is $h$ metres at time $t$ minutes after water begins to enter the tank. The rate at which the depth of the water in the tank increases is proportional to the difference between the height of the tank and the depth of the water. Write down a differential equation in the variables $h$ and $t$ and a positive constant $k$.
(You are not required to solve your differential equation.)
1. Another water tank is filling in such a way that $t$ minutes after the water is turned on, the depth of the water, $x$ metres, increases according to the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { 15 x \sqrt { 2 x - 1 } }$$ The depth of the water is 1 metre when the water is first turned on.
  Solve this differential equation to find $t$ as a function of $x$.
2. Calculate the time taken for the depth of the water in the tank to reach 2 metres, giving your answer to the nearest 0.1 of a minute.
  (l mark)

AQA C4 2013 June Q1

1. Express $\frac { 5 - 8 x } { ( 2 + x ) ( 1 - 3 x ) }$ in the form $\frac { A } { 2 + x } + \frac { B } { 1 - 3 x }$, where $A$ and $B$ are integers.
  (3 marks)
2. Hence show that $\int _ { - 1 } ^ { 0 } \frac { 5 - 8 x } { ( 2 + x ) ( 1 - 3 x ) } \mathrm { d } x = p \ln 2$, where $p$ is rational.
  (4 marks)
1. Given that $\frac { 9 - 18 x - 6 x ^ { 2 } } { 2 - 5 x - 3 x ^ { 2 } }$ can be written as $C + \frac { 5 - 8 x } { 2 - 5 x - 3 x ^ { 2 } }$, find the value of $C$.
  (1 mark)
2. Hence find the exact value of the area of the region bounded by the curve $y = \frac { 9 - 18 x - 6 x ^ { 2 } } { 2 - 5 x - 3 x ^ { 2 } }$, the $x$-axis and the lines $x = - 1$ and $x = 0$. You may assume that $y > 0$ when $- 1 \leqslant x \leqslant 0$.

AQA C4 2013 June Q2

2 The acute angles $\alpha$ and $\beta$ are given by $\tan \alpha = \frac { 2 } { \sqrt { 5 } }$ and $\tan \beta = \frac { 1 } { 2 }$.

1. Show that $\sin \alpha = \frac { 2 } { 3 }$, and find the exact value of $\cos \alpha$.
2. Hence find the exact value of $\sin 2 \alpha$.
Show that the exact value of $\cos ( \alpha - \beta )$ can be expressed as $\frac { 2 } { 15 } ( k + \sqrt { 5 } )$, where $k$ is an integer.

AQA C4 2013 June Q3

Find the binomial expansion of $( 1 + 6 x ) ^ { - \frac { 1 } { 3 } }$ up to and including the term in $x ^ { 2 }$.
1. Find the binomial expansion of $( 27 + 6 x ) ^ { - \frac { 1 } { 3 } }$ up to and including the term in $x ^ { 2 }$, simplifying the coefficients.
2. Given that $\sqrt [ 3 ] { \frac { 2 } { 7 } } = \frac { 2 } { \sqrt [ 3 ] { 28 } }$, use your binomial expansion from part (b)(i) to obtain an approximation to $\sqrt [ 3 ] { \frac { 2 } { 7 } }$, giving your answer to six decimal places.
  (2 marks)

AQA C4 2013 June Q4

4 A curve is defined by the parametric equations $x = 8 \mathrm { e } ^ { - 2 t } - 4 , y = 2 \mathrm { e } ^ { 2 t } + 4$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$.
The point $P$, where $t = \ln 2$, lies on the curve.
1. Find the gradient of the curve at $P$.
2. Find the coordinates of $P$.
3. The normal at $P$ crosses the $x$-axis at the point $Q$. Find the coordinates of $Q$.
Find the Cartesian equation of the curve in the form $x y + 4 y - 4 x = k$, where $k$ is an integer.
(3 marks)

Questions — AQA C4 (160 questions)

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