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AQA C4 2006 June Q2
8 marks Moderate -0.3
2
  1. Obtain the binomial expansion of \(( 1 - x ) ^ { - 3 }\) up to and including the term in \(x ^ { 2 }\).
  2. Hence obtain the binomial expansion of \(\left( 1 - \frac { 5 } { 2 } x \right) ^ { - 3 }\) up to and including the term in \(x ^ { 2 }\).
  3. Find the range of values of \(x\) for which the binomial expansion of \(\left( 1 - \frac { 5 } { 2 } x \right) ^ { - 3 }\) would be valid.
  4. Given that \(x\) is small, show that \(\left( \frac { 4 } { 2 - 5 x } \right) ^ { 3 } \approx a + b x + c x ^ { 2 }\), where \(a , b\) and \(c\) are integers.
AQA C4 2006 June Q3
8 marks Moderate -0.3
3
  1. Given that \(\frac { 9 x ^ { 2 } - 6 x + 5 } { ( 3 x - 1 ) ( x - 1 ) }\) can be written in the form \(3 + \frac { A } { 3 x - 1 } + \frac { B } { x - 1 }\), where \(A\) and \(B\) are integers, find the values of \(A\) and \(B\).
  2. Hence, or otherwise, find \(\int \frac { 9 x ^ { 2 } - 6 x + 5 } { ( 3 x - 1 ) ( x - 1 ) } \mathrm { d } x\).
AQA C4 2006 June Q4
9 marks Moderate -0.8
4
    1. Express \(\sin 2 x\) in terms of \(\sin x\) and \(\cos x\).
    2. Express \(\cos 2 x\) in terms of \(\cos x\).
  2. Show that $$\sin 2 x - \tan x = \tan x \cos 2 x$$ for all values of \(x\).
  3. Solve the equation \(\sin 2 x - \tan x = 0\), giving all solutions in degrees in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
AQA C4 2006 June Q5
14 marks Standard +0.3
5 A curve is defined by the equation $$y ^ { 2 } - x y + 3 x ^ { 2 } - 5 = 0$$
  1. Find the \(y\)-coordinates of the two points on the curve where \(x = 1\).
    1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - 6 x } { 2 y - x }\).
    2. Find the gradient of the curve at each of the points where \(x = 1\).
    3. Show that, at the two stationary points on the curve, \(33 x ^ { 2 } - 5 = 0\).
AQA C4 2006 June Q6
12 marks Moderate -0.3
6 The points \(A\) and \(B\) have coordinates \(( 2,4,1 )\) and \(( 3,2 , - 1 )\) respectively. The point \(C\) is such that \(\overrightarrow { O C } = 2 \overrightarrow { O B }\), where \(O\) is the origin.
  1. Find the vectors:
    1. \(\overrightarrow { O C }\);
    2. \(\overrightarrow { A B }\).
    1. Show that the distance between the points \(A\) and \(C\) is 5 .
    2. Find the size of angle \(B A C\), giving your answer to the nearest degree.
  3. The point \(P ( \alpha , \beta , \gamma )\) is such that \(B P\) is perpendicular to \(A C\). Show that \(4 \alpha - 3 \gamma = 15\).
AQA C4 2006 June Q7
6 marks Moderate -0.8
7 Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x y ^ { 2 }$$ given that \(y = 1\) when \(x = 2\). Give your answer in the form \(y = \mathrm { f } ( x )\).
AQA C4 2006 June Q8
10 marks Standard +0.3
8 A disease is spreading through a colony of rabbits. There are 5000 rabbits in the colony. At time \(t\) hours, \(x\) is the number of rabbits infected. The rate of increase of the number of rabbits infected is proportional to the product of the number of rabbits infected and the number not yet infected.
    1. Formulate a differential equation for \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) in terms of the variables \(x\) and \(t\) and a constant of proportionality \(k\).
    2. Initially, 1000 rabbits are infected and the disease is spreading at a rate of 200 rabbits per hour. Find the value of the constant \(k\).
      (You are not required to solve your differential equation.)
  2. The solution of the differential equation in this model is $$t = 4 \ln \left( \frac { 4 x } { 5000 - x } \right)$$
    1. Find the time after which 2500 rabbits will be infected, giving your answer in hours to one decimal place.
    2. Find, according to this model, the number of rabbits infected after 30 hours.
AQA C4 2007 June Q1
5 marks Moderate -0.8
1
  1. Find the remainder when \(2 x ^ { 2 } + x - 3\) is divided by \(2 x + 1\).
    (2 marks)
  2. Simplify the algebraic fraction \(\frac { 2 x ^ { 2 } + x - 3 } { x ^ { 2 } - 1 }\).
    (3 marks)
AQA C4 2007 June Q2
12 marks Moderate -0.3
2
    1. Find the binomial expansion of \(( 1 + x ) ^ { - 1 }\) up to the term in \(x ^ { 3 }\).
    2. Hence, or otherwise, obtain the binomial expansion of \(\frac { 1 } { 1 + 3 x }\) up to the term in \(x ^ { 3 }\).
  2. Express \(\frac { 1 + 4 x } { ( 1 + x ) ( 1 + 3 x ) }\) in partial fractions.
    1. Find the binomial expansion of \(\frac { 1 + 4 x } { ( 1 + x ) ( 1 + 3 x ) }\) up to the term in \(x ^ { 3 }\).
    2. Find the range of values of \(x\) for which the binomial expansion of \(\frac { 1 + 4 x } { ( 1 + x ) ( 1 + 3 x ) }\) is valid.
AQA C4 2007 June Q3
10 marks Moderate -0.3
3
  1. Express \(4 \cos x + 3 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 360 ^ { \circ }\), giving your value for \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
  2. Hence solve the equation \(4 \cos x + 3 \sin x = 2\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\), giving all solutions to the nearest \(0.1 ^ { \circ }\).
  3. Write down the minimum value of \(4 \cos x + 3 \sin x\) and find the value of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\) at which this minimum value occurs.
AQA C4 2007 June Q4
11 marks Moderate -0.8
4 A biologist is researching the growth of a certain species of hamster. She proposes that the length, \(x \mathrm {~cm}\), of a hamster \(t\) days after its birth is given by $$x = 15 - 12 \mathrm { e } ^ { - \frac { t } { 14 } }$$
  1. Use this model to find:
    1. the length of a hamster when it is born;
    2. the length of a hamster after 14 days, giving your answer to three significant figures.
    1. Show that the time for a hamster to grow to 10 cm in length is given by \(t = 14 \ln \left( \frac { a } { b } \right)\), where \(a\) and \(b\) are integers.
    2. Find this time to the nearest day.
    1. Show that $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { 14 } ( 15 - x )$$
    2. Find the rate of growth of the hamster, in cm per day, when its length is 8 cm .
      (1 mark)
AQA C4 2007 June Q5
10 marks Moderate -0.3
5 The point \(P ( 1 , a )\), where \(a > 0\), lies on the curve \(y + 4 x = 5 x ^ { 2 } y ^ { 2 }\).
  1. Show that \(a = 1\).
  2. Find the gradient of the curve at \(P\).
  3. Find an equation of the tangent to the curve at \(P\).
AQA C4 2007 June Q6
8 marks Moderate -0.3
6 A curve is given by the parametric equations $$x = \cos \theta \quad y = \sin 2 \theta$$
    1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} \theta }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} \theta }\).
      (2 marks)
    2. Find the gradient of the curve at the point where \(\theta = \frac { \pi } { 6 }\).
  2. Show that the cartesian equation of the curve can be written as $$y ^ { 2 } = k x ^ { 2 } \left( 1 - x ^ { 2 } \right)$$ where \(k\) is an integer.
AQA C4 2007 June Q7
11 marks Standard +0.3
7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = \left[ \begin{array} { r } 8 \\ 6 \\ - 9 \end{array} \right] + \lambda \left[ \begin{array} { r } 3 \\ - 3 \\ - 1 \end{array} \right]\) and \(\mathbf { r } = \left[ \begin{array} { r } - 4 \\ 0 \\ 11 \end{array} \right] + \mu \left[ \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right]\) respectively.
  1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular.
  2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of the point of intersection, \(P\).
  3. The point \(A ( - 4,0,11 )\) lies on \(l _ { 2 }\). The point \(B\) on \(l _ { 1 }\) is such that \(A P = B P\). Find the length of \(A B\).
AQA C4 2007 June Q8
8 marks Moderate -0.3
8
  1. Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sqrt { 1 + 2 y } } { x ^ { 2 } }$$ given that \(y = 4\) when \(x = 1\).
  2. Show that the solution can be written as \(y = \frac { 1 } { 2 } \left( 15 - \frac { 8 } { x } + \frac { 1 } { x ^ { 2 } } \right)\).
AQA C4 2008 June Q1
9 marks Moderate -0.3
1 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 27 x ^ { 3 } - 9 x + 2\).
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(3 x + 1\).
    1. Show that f \(\left( - \frac { 2 } { 3 } \right) = 0\).
    2. Express \(\mathrm { f } ( x )\) as a product of three linear factors.
    3. Simplify $$\frac { 27 x ^ { 3 } - 9 x + 2 } { 9 x ^ { 2 } + 3 x - 2 }$$
AQA C4 2008 June Q2
10 marks Moderate -0.3
2 A curve is defined, for \(t \neq 0\), by the parametric equations $$x = 4 t + 3 , \quad y = \frac { 1 } { 2 t } - 1$$ At the point \(P\) on the curve, \(t = \frac { 1 } { 2 }\).
  1. Find the gradient of the curve at the point \(P\).
  2. Find an equation of the normal to the curve at the point \(P\).
  3. Find a cartesian equation of the curve.
AQA C4 2008 June Q3
8 marks Standard +0.3
3
  1. By writing \(\sin 3 x\) as \(\sin ( x + 2 x )\), show that \(\sin 3 x = 3 \sin x - 4 \sin ^ { 3 } x\) for all values of \(x\).
  2. Hence, or otherwise, find \(\int \sin ^ { 3 } x \mathrm {~d} x\).
AQA C4 2008 June Q4
7 marks Standard +0.3
4
    1. Obtain the binomial expansion of \(( 1 - x ) ^ { \frac { 1 } { 4 } }\) up to and including the term in \(x ^ { 2 }\).
    2. Hence show that \(( 81 - 16 x ) ^ { \frac { 1 } { 4 } } \approx 3 - \frac { 4 } { 27 } x - \frac { 8 } { 729 } x ^ { 2 }\) for small values of \(x\).
  2. Use the result from part (a)(ii) to find an approximation for \(\sqrt [ 4 ] { 80 }\), giving your answer to seven decimal places.
AQA C4 2008 June Q5
10 marks Moderate -0.3
5
  1. The angle \(\alpha\) is acute and \(\sin \alpha = \frac { 4 } { 5 }\).
    1. Find the value of \(\cos \alpha\).
    2. Express \(\cos ( \alpha - \beta )\) in terms of \(\sin \beta\) and \(\cos \beta\).
    3. Given also that the angle \(\beta\) is acute and \(\cos \beta = \frac { 5 } { 13 }\), find the exact value of \(\cos ( \alpha - \beta )\).
    1. Given that \(\tan 2 x = 1\), show that \(\tan ^ { 2 } x + 2 \tan x - 1 = 0\).
    2. Hence, given that \(\tan 45 ^ { \circ } = 1\), show that \(\tan 22 \frac { 1 } { 2 } ^ { \circ } = \sqrt { 2 } - 1\).
AQA C4 2008 June Q6
10 marks Moderate -0.3
6
  1. Express \(\frac { 2 } { x ^ { 2 } - 1 }\) in the form \(\frac { A } { x - 1 } + \frac { B } { x + 1 }\).
  2. Hence find \(\int \frac { 2 } { x ^ { 2 } - 1 } \mathrm {~d} x\).
  3. Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 y } { 3 \left( x ^ { 2 } - 1 \right) }\), given that \(y = 1\) when \(x = 3\). Show that the solution can be written as \(y ^ { 3 } = \frac { 2 ( x - 1 ) } { x + 1 }\).
AQA C4 2008 June Q7
12 marks Standard +0.8
7 The coordinates of the points \(A\) and \(B\) are ( \(3 , - 2,1\) ) and ( \(5,3,0\) ) respectively. The line \(l\) has equation \(\mathbf { r } = \left[ \begin{array} { l } 5 \\ 3 \\ 0 \end{array} \right] + \lambda \left[ \begin{array} { r } 1 \\ 0 \\ - 3 \end{array} \right]\).
  1. Find the distance between \(A\) and \(B\).
  2. Find the acute angle between the lines \(A B\) and \(l\). Give your answer to the nearest degree.
  3. The points \(B\) and \(C\) lie on \(l\) such that the distance \(A C\) is equal to the distance \(A B\). Find the coordinates of \(C\).
AQA C4 2008 June Q8
9 marks Moderate -0.5
8
  1. The number of fish in a lake is decreasing. After \(t\) years, there are \(x\) fish in the lake. The rate of decrease of the number of fish is proportional to the number of fish currently in the lake.
    1. Formulate a differential equation, in the variables \(x\) and \(t\) and a constant of proportionality \(k\), where \(k > 0\), to model the rate at which the number of fish in the lake is decreasing.
    2. At a certain time, there were 20000 fish in the lake and the rate of decrease was 500 fish per year. Find the value of \(k\).
  2. The equation $$P = 2000 - A \mathrm { e } ^ { - 0.05 t }$$ is proposed as a model for the number of fish, \(P\), in another lake, where \(t\) is the time in years and \(A\) is a positive constant. On 1 January 2008, a biologist estimated that there were 700 fish in this lake.
    1. Taking 1 January 2008 as \(t = 0\), find the value of \(A\).
    2. Hence find the year during which, according to this model, the number of fish in this lake will first exceed 1900.
AQA C4 2009 June Q1
5 marks Moderate -0.8
1
  1. Use the Remainder Theorem to find the remainder when \(3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 5\) is divided by \(3 x - 1\).
  2. Express \(\frac { 3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 5 } { 3 x - 1 }\) in the form \(a x ^ { 2 } + b x + \frac { c } { 3 x - 1 }\), where \(a , b\) and \(c\) are integers.
AQA C4 2009 June Q2
11 marks Standard +0.3
2 A curve is defined by the parametric equations $$x = \frac { 1 } { t } , \quad y = t + \frac { 1 } { 2 t }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find an equation of the normal to the curve at the point where \(t = 1\).
  3. Show that the cartesian equation of the curve can be written in the form $$x ^ { 2 } - 2 x y + k = 0$$ where \(k\) is an integer.