Questions C4 (1219 questions)

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AQA C4 2013 January Q2
11 marks Standard +0.3
2 It is given that \(\mathrm { f } ( x ) = \frac { 7 x - 1 } { ( 1 + 3 x ) ( 3 - x ) }\).
  1. Express \(\mathrm { f } ( x )\) in the form \(\frac { A } { 3 - x } + \frac { B } { 1 + 3 x }\), where \(A\) and \(B\) are integers.
    (3 marks)
    1. Find the first three terms of the binomial expansion of \(\mathrm { f } ( x )\) in the form \(a + b x + c x ^ { 2 }\), where \(a\), \(b\) and \(c\) are rational numbers.
      (7 marks)
    2. State why the binomial expansion cannot be expected to give a good approximation to \(\mathrm { f } ( x )\) at \(x = 0.4\).
      (1 mark)
AQA C4 2013 January Q3
12 marks Standard +0.3
3
    1. Express \(3 \cos x + 2 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
      (3 marks)
    2. Hence find the minimum value of \(3 \cos x + 2 \sin x\) and the value of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\) where the minimum occurs. Give your value of \(x\) to the nearest \(0.1 ^ { \circ }\).
    1. Show that \(\cot x - \sin 2 x = \cot x \cos 2 x\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
    2. Hence, or otherwise, solve the equation $$\cot x - \sin 2 x = 0$$ in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).
AQA C4 2013 January Q4
8 marks Standard +0.3
4
  1. A curve is defined by the equation \(x ^ { 2 } - y ^ { 2 } = 8\).
    1. Show that at any point \(( p , q )\) on the curve, where \(q \neq 0\), the gradient of the curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { p } { q }\).
      (2 marks)
    2. Show that the tangents at the points \(( p , q )\) and \(( p , - q )\) intersect on the \(x\)-axis.
      (4 marks)
  2. Show that \(x = t + \frac { 2 } { t } , y = t - \frac { 2 } { t }\) are parametric equations of the curve \(x ^ { 2 } - y ^ { 2 } = 8\).
    (2 marks)
AQA C4 2013 January Q5
9 marks Standard +0.3
5
  1. Find \(\int x \sqrt { x ^ { 2 } + 3 } \mathrm {~d} x\).
    (2 marks)
  2. Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x \sqrt { x ^ { 2 } + 3 } } { \mathrm { e } ^ { 2 y } }$$ given that \(y = 0\) when \(x = 1\). Give your answer in the form \(y = \mathrm { f } ( x )\).
AQA C4 2013 January Q6
15 marks Standard +0.3
6
  1. The points \(A , B\) and \(C\) have coordinates \(( 3,1 , - 6 ) , ( 5 , - 2,0 )\) and \(( 8 , - 4 , - 6 )\) respectively.
    1. Show that the vector \(\overrightarrow { A C }\) is given by \(\overrightarrow { A C } = n \left[ \begin{array} { r } 1 \\ - 1 \\ 0 \end{array} \right]\), where \(n\) is an integer.
    2. Show that the acute angle \(A C B\) is given by \(\cos ^ { - 1 } \left( \frac { 5 \sqrt { 2 } } { 14 } \right)\).
  2. Find a vector equation of the line \(A C\).
  3. The point \(D\) has coordinates \(( 6 , - 1 , p )\). It is given that the lines \(A C\) and \(B D\) intersect.
    1. Find the value of \(p\).
    2. Show that \(A B C D\) is a rhombus, and state the length of each of its sides.
AQA C4 2013 January Q7
13 marks Standard +0.3
7 A biologist is investigating the growth of a population of a species of rodent. The biologist proposes the model $$N = \frac { 500 } { 1 + 9 \mathrm { e } ^ { - \frac { t } { 8 } } }$$ for the number of rodents, \(N\), in the population \(t\) weeks after the start of the investigation. Use this model to answer the following questions.
    1. Find the size of the population at the start of the investigation.
    2. Find the size of the population 24 weeks after the start of the investigation. your answer to the nearest whole number.
    3. Find the number of weeks that it will take the population to reach 400 . Give your answer in the form \(t = r \ln s\), where \(r\) and \(s\) are integers.
    1. Show that the rate of growth, \(\frac { \mathrm { d } N } { \mathrm {~d} t }\), is given by $$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { N } { 4000 } ( 500 - N )$$
    2. The maximum rate of growth occurs after \(T\) weeks. Find the value of \(T\).
AQA C4 2011 June Q1
7 marks Moderate -0.8
1 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 4 x ^ { 3 } - 13 x + 6\).
  1. Find \(\mathrm { f } ( - 2 )\).
  2. Use the Factor Theorem to show that \(2 x - 3\) is a factor of \(\mathrm { f } ( x )\).
  3. Simplify \(\frac { 2 x ^ { 2 } + x - 6 } { \mathrm { f } ( x ) }\).
AQA C4 2011 June Q2
6 marks Moderate -0.8
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.
  2. 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
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. 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 \(n\).
  2. 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
13 marks Standard +0.2
4
  1. 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 \(\theta = \frac { \pi } { 3 }\).
  2. Find the exact value of \(\int _ { - \frac { \pi } { 4 } } ^ { \frac { \pi } { 4 } } \sin ^ { 2 } x \mathrm {~d} x\).
AQA C4 2011 June Q5
12 marks Standard +0.3
5 The points \(A\) and \(B\) have coordinates \(( 5,1 , - 2 )\) and \(( 4 , - 1,3 )\) respectively.
The line \(l\) has equation \(\mathbf { r } = \left[ \begin{array} { r } - 8 \\ 5 \\ - 6 \end{array} \right] + \mu \left[ \begin{array} { r } 5 \\ 0 \\ - 2 \end{array} \right]\).
  1. 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
10 marks Standard +0.3
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.
  1. Find the exact value of \(C\).
  2. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  3. Verify that \(P \left( 1 , \frac { 1 } { \mathrm { e } } \right)\) is a stationary point on the curve.
AQA C4 2011 June Q7
7 marks Moderate -0.3
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.
  1. 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
13 marks Standard +0.8
8
  1. 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)
  2. 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
11 marks Moderate -0.3
1
    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
7 marks Standard +0.3
2
  1. 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 \(0.1 ^ { \circ }\).
  2. 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
8 marks Standard +0.3
3
  1. 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.
  3. 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
8 marks Easy -1.2
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 \(N\).
  2. 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
9 marks Standard +0.3
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 \(\theta = \frac { \pi } { 6 }\).
  2. 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
8 marks Standard +0.8
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
12 marks Standard +0.3
\(\mathbf { 7 } \quad\) 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 \(P\).
  1. Show that \(q = 4\) and find the coordinates of \(P\).
  2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular.
  3. 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
12 marks Standard +0.3
8
  1. 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
10 marks Standard +0.3
1
    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
8 marks Moderate -0.3
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 \(\sin 2 \alpha\).
  2. 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
7 marks Standard +0.3
3
  1. 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)