Questions — AQA C4 (162 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
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)
AQA C4 2013 June Q4
12 marks Standard +0.3
4 A curve is defined by the parametric equations \(x = 8 \mathrm { e } ^ { - 2 t } - 4 , y = 2 \mathrm { e } ^ { 2 t } + 4\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. 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 \(Q\).
  3. Find the Cartesian equation of the curve in the form \(x y + 4 y - 4 x = k\), where \(k\) is an integer.
    (3 marks)
AQA C4 2013 June Q5
11 marks Standard +0.3
5 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 4 x ^ { 3 } - 11 x - 3\).
  1. Use the Factor Theorem to show that ( \(2 x + 3\) ) is a factor of \(\mathrm { f } ( x )\).
  2. Write \(\mathrm { f } ( x )\) in the form \(( 2 x + 3 ) \left( a x ^ { 2 } + b x + c \right)\), where \(a , b\) and \(c\) are integers.
    1. Show that the equation \(2 \cos 2 \theta \sin \theta + 9 \sin \theta + 3 = 0\) can be written as \(4 x ^ { 3 } - 11 x - 3 = 0\), where \(x = \sin \theta\).
    2. Hence find all solutions of the equation \(2 \cos 2 \theta \sin \theta + 9 \sin \theta + 3 = 0\) in the interval \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), giving your solutions to the nearest degree.
AQA C4 2013 June Q6
14 marks Standard +0.8
6 The points \(A , B\) and \(C\) have coordinates \(( 3 , - 2,4 ) , ( 1 , - 5,6 )\) and \(( - 4,5 , - 1 )\) respectively. The line \(l\) passes through \(A\) and has equation \(\mathbf { r } = \left[ \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right] + \lambda \left[ \begin{array} { r } 7 \\ - 7 \\ 5 \end{array} \right]\).
  1. Show that the point \(C\) lies on the line \(l\).
  2. Find a vector equation of the line that passes through points \(A\) and \(B\).
  3. The point \(D\) lies on the line through \(A\) and \(B\) such that the angle \(C D A\) is a right angle. Find the coordinates of \(D\).
  4. The point \(E\) lies on the line through \(A\) and \(B\) such that the area of triangle \(A C E\) is three times the area of triangle \(A C D\). Find the coordinates of the two possible positions of \(E\).
AQA C4 2013 June Q7
3 marks Moderate -0.8
7 The height of the tide in a certain harbour is \(h\) metres at time \(t\) hours. Successive high tides occur every 12 hours. The rate of change of the height of the tide can be modelled by a function of the form \(a \cos ( k t )\), where \(a\) and \(k\) are constants. The largest value of this rate of change is 1.3 metres per hour. Write down a differential equation in the variables \(h\) and \(t\). State the values of the constants \(a\) and \(k\).
AQA C4 2013 June Q8
10 marks Standard +0.8
8
  1. \(\quad\) Find \(\int t \cos \left( \frac { \pi } { 4 } t \right) \mathrm { d } t\).
  2. The platform of a theme park ride oscillates vertically. For the first 75 seconds of the ride, $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { t \cos \left( \frac { \pi } { 4 } t \right) } { 32 x }$$ where \(x\) metres is the height of the platform above the ground after time \(t\) seconds.
    At \(t = 0\), the height of the platform above the ground is 4 metres.
    Find the height of the platform after 45 seconds, giving your answer to the nearest centimetre.
    (6 marks)
AQA C4 2014 June Q1
5 marks Moderate -0.3
1 A curve is defined by the parametric equations \(x = \frac { t ^ { 2 } } { 2 } + 1 , y = \frac { 4 } { t } - 1\).
  1. Find the gradient at the point on the curve where \(t = 2\).
  2. Find a Cartesian equation of the curve.
    \includegraphics[max width=\textwidth, alt={}]{9f03a5f3-7fea-4fb7-b3bd-b4c0cdf662a2-02_1730_1709_977_153}
AQA C4 2014 June Q2
7 marks Moderate -0.3
2
  1. Given that \(\frac { 4 x ^ { 3 } - 2 x ^ { 2 } + 16 x - 3 } { 2 x ^ { 2 } - x + 2 }\) can be expressed as \(A x + \frac { B ( 4 x - 1 ) } { 2 x ^ { 2 } - x + 2 }\), find the values of the constants \(A\) and \(B\).
  2. The gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 x ^ { 3 } - 2 x ^ { 2 } + 16 x - 3 } { 2 x ^ { 2 } - x + 2 }$$ The point \(( - 1,2 )\) lies on the curve. Find the equation of the curve.
    [0pt] [4 marks]
AQA C4 2014 June Q3
7 marks Moderate -0.3
3
  1. Find the binomial expansion of \(( 1 - 4 x ) ^ { \frac { 1 } { 4 } }\) up to and including the term in \(x ^ { 2 }\).
    [0pt] [2 marks]
  2. Find the binomial expansion of \(( 2 + 3 x ) ^ { - 3 }\) up to and including the term in \(x ^ { 2 }\).
  3. Hence find the binomial expansion of \(\frac { ( 1 - 4 x ) ^ { \frac { 1 } { 4 } } } { ( 2 + 3 x ) ^ { 3 } }\) up to and including the term in \(x ^ { 2 }\).
    [0pt] [2 marks]