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AQA C4 2008 January Q4
9 marks Moderate -0.3
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 \(\pounds 1000\).
  2. 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
10 marks Standard +0.3
5 A curve is defined by the parametric equations \(x = 2 t + \frac { 1 } { t ^ { 2 } } , \quad y = 2 t - \frac { 1 } { t ^ { 2 } }\).
  1. 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 \(P\).
  2. 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
5 marks Moderate -0.5
6 A curve has equation \(3 x y - 2 y ^ { 2 } = 4\).
Find the gradient of the curve at the point \(( 2,1 )\).
AQA C4 2008 January Q7
14 marks Standard +0.3
7
    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
5 marks Moderate -0.8
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
11 marks Standard +0.3
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 \(\lambda\).
  2. 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
8 marks Moderate -0.3
1
  1. 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 \(\frac { 4 x ^ { 3 } - 7 x - 3 } { 2 x ^ { 2 } + 3 x + 1 }\).
  2. 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
6 marks Standard +0.3
2
  1. 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.
  2. 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
13 marks Standard +0.3
3
    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
7 marks Moderate -0.3
4
    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)
  2. 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
9 marks Standard +0.3
5
  1. Express \(\sin 2 x\) in terms of \(\sin x\) and \(\cos x\).
  2. 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.
  3. 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
10 marks Standard +0.3
6 A curve is defined by the equation \(x ^ { 2 } y + y ^ { 3 } = 2 x + 1\).
  1. Find the gradient of the curve at the point \(( 2,1 )\).
  2. 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
10 marks Standard +0.3
7
  1. 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)
  2. 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
12 marks Standard +0.3
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 \(\cos \theta = \frac { 5 } { 2 \sqrt { 7 } }\).
  2. 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 \(l\).
  3. The point \(D\) lies on \(l\) such that angle \(O D C = 90 ^ { \circ }\). Find the coordinates of \(D\).
AQA C4 2010 January Q1
8 marks Moderate -0.3
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 \(\mathrm { f } ( x )\).
  2. Simplify $$\frac { 15 x ^ { 2 } - 6 x } { f ( x ) }$$ giving your answer in a fully factorised form.
AQA C4 2010 January Q2
10 marks Standard +0.3
2
  1. 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.
  3. 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
7 marks Standard +0.3
3
    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 \(x ^ { 2 }\).
  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
8 marks Standard +0.3
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.
  1. Find the values of \(A\) and \(B\).
  2. Hence find \(\int \frac { 10 x ^ { 2 } + 8 } { ( x + 1 ) ( 5 x - 1 ) } \mathrm { d } x\).
AQA C4 2010 January Q5
5 marks Moderate -0.3
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
10 marks Moderate -0.3
6
    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 \(\cos 2 \theta\).
  2. 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
6 marks Moderate -0.3
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
11 marks Standard +0.3
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 \(\mathbf { r } = \left[ \begin{array} { r } 2 \\ - 1 \\ - 5 \end{array} \right] + \lambda \left[ \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right]\).
  1. Verify that the point \(B\) lies on the line \(l\).
  2. Find the vector \(\overrightarrow { B C }\).
  3. 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
10 marks Moderate -0.3
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.
  1. Use this model to:
    1. find the height of the toadstool when \(t = 0\);
    2. show that \(A = 60\), correct to two significant figures.
  2. 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
7 marks Moderate -0.3
1
  1. 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 \(0.1 ^ { \circ }\).
  2. Solve the equation \(2 \sin x + \cos x = 1\) for \(0 ^ { \circ } \leqslant x < 360 ^ { \circ }\).
AQA C4 2005 June Q2
6 marks Moderate -0.8
2
  1. Express \(\frac { 3 x - 5 } { ( x + 3 ) ( 2 x - 1 ) }\) in the form \(\frac { A } { x + 3 } + \frac { B } { 2 x - 1 }\).
  2. Hence find \(\int \frac { 3 x - 5 } { ( x + 3 ) ( 2 x - 1 ) } \mathrm { d } x\).