Questions — AQA C4 (160 questions)

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AQA C4 2005 June Q6
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
  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
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
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
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
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
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
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 \(y ^ { 3 } = \frac { 2 ( x - 1 ) } { x + 1 }\).