Questions — AQA C4 (162 questions)

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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 \(\int \frac { 3 x - 5 } { ( x + 3 ) ( 2 x - 1 ) } \mathrm { d } x\).
AQA C4 2005 June Q3
6 marks Moderate -0.8
3
  1. Find the remainder when \(2 x ^ { 3 } - x ^ { 2 } + 2 x - 2\) is divided by \(2 x - 1\).
  2. Given that \(\frac { 2 x ^ { 3 } - x ^ { 2 } + 2 x - 2 } { 2 x - 1 } = x ^ { 2 } + a + \frac { b } { 2 x - 1 }\), find the values of \(a\) and \(b\).
AQA C4 2005 June Q4
8 marks Moderate -0.3
4
  1. Find the binomial expansion of \(( 1 + x ) ^ { - \frac { 1 } { 2 } }\) up to the term in \(x ^ { 2 }\).
  2. Hence, or otherwise, obtain the binomial expansion of \(\frac { 1 } { \sqrt { 1 + 2 x } }\) up to the term in \(x ^ { 2 }\), in simplified form.
  3. Use your answer to part (b) with \(x = - 0.1\) to show that \(\sqrt { 5 } \approx 2.23\).
AQA C4 2005 June Q5
10 marks Moderate -0.8
5 A curve is defined by the parametric equations $$x = 2 t + \frac { 1 } { t } , \quad y = \frac { 1 } { t } , \quad t \neq 0$$
  1. Find the coordinates of the point on the curve where \(t = \frac { 1 } { 2 }\).
  2. Show that the cartesian equation of the curve can be written as $$x y - y ^ { 2 } = 2$$
  3. Show that the gradient of the curve at the point \(( 3,2 )\) is 2 .
AQA C4 2005 June Q6
12 marks Moderate -0.3
6
  1. Express \(\sin 2 x\) in terms of \(\sin x\) and \(\cos x\).
  2. Using the identity \(\cos ( A + B ) = \cos A \cos B - \sin A \sin B\) :
    1. express \(\cos 2 x\) in terms of \(\sin x\) and \(\cos x\);
    2. show, by writing \(3 x\) as \(( 2 x + x )\), that $$\cos 3 x = 4 \cos ^ { 3 } x - 3 \cos x$$
  3. Show that \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \cos ^ { 3 } x \mathrm {~d} x = \frac { 2 } { 3 }\).
AQA C4 2005 June Q7
12 marks Standard +0.3
7 The points \(A\) and \(B\) have coordinates \(( 1,4,2 )\) and \(( 2 , - 1,3 )\) respectively.
The line \(l\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 2 \\ - 1 \\ 3 \end{array} \right] + \lambda \left[ \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right]\).
  1. Show that the distance between the points \(A\) and \(B\) is \(3 \sqrt { 3 }\).
  2. The line \(A B\) makes an acute angle \(\theta\) with \(l\). Show that \(\cos \theta = \frac { 7 } { 9 }\).
  3. The point \(P\) on the line \(l\) is where \(\lambda = p\).
    1. Show that $$\overrightarrow { A P } \cdot \left[ \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right] = 7 + 3 p$$
    2. Hence find the coordinates of the foot of the perpendicular from the point \(A\) to the line \(l\).
AQA C4 2005 June Q8
14 marks Moderate -0.3
8
  1. A cup of coffee is cooling down in a room. At time \(t\) minutes after the coffee is made, its temperature is \(x ^ { \circ } \mathrm { C }\), where $$x = 15 + 70 \mathrm { e } ^ { - \frac { t } { 40 } }$$
    1. Find the temperature of the coffee when it is made.
    2. Find the temperature of the coffee 30 minutes after it is made.
    3. Find how long it will take for the coffee to cool down to \(60 ^ { \circ } \mathrm { C }\).
    1. Use integration to solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \frac { 1 } { 40 } ( x - 15 ) , \quad x > 15$$ given that \(x = 85\) when \(t = 0\), expressing \(t\) in terms of \(x\).
    2. Hence show that \(x = 15 + 70 \mathrm { e } ^ { - \frac { t } { 40 } }\).
AQA C4 2006 June Q1
8 marks Moderate -0.3
1
  1. The polynomial \(\mathrm { p } ( x )\) is defined by \(\mathrm { p } ( x ) = 6 x ^ { 3 } - 19 x ^ { 2 } + 9 x + 10\).
    1. Find \(\mathrm { p } ( 2 )\).
    2. Use the Factor Theorem to show that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\).
    3. Write \(\mathrm { p } ( x )\) as the product of three linear factors.
  2. Hence simplify \(\frac { 3 x ^ { 2 } - 6 x } { 6 x ^ { 3 } - 19 x ^ { 2 } + 9 x + 10 }\).
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.