AQA C4 (Core Mathematics 4) 2006 January

1

1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + 2\).
   1. Find f(1).
   2. Show that \(\mathrm { f } ( - 2 ) = 0\).
   3. Hence, or otherwise, show that $$\frac { ( x - 1 ) ( x + 2 ) } { 3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + 2 } = \frac { 1 } { a x + b }$$ where \(a\) and \(b\) are integers.
2. The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + d\). When \(\mathrm { g } ( x )\) is divided by \(( 3 x - 1 )\), the remainder is 2 . Find the value of \(d\).

2 A curve is defined by the parametric equations $$x = 3 - 4 t \quad y = 1 + \frac { 2 } { t }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Find the equation of the tangent to the curve at the point where \(t = 2\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
3. Verify that the cartesian equation of the curve can be written as $$( x - 3 ) ( y - 1 ) + 8 = 0$$

3 It is given that \(3 \cos \theta - 2 \sin \theta = R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).

1. Find the value of \(R\).
2. Show that \(\alpha \approx 33.7 ^ { \circ }\).
3. Hence write down the maximum value of \(3 \cos \theta - 2 \sin \theta\) and find a positive value of \(\theta\) at which this maximum value occurs.

4 On 1 January 1900, a sculpture was valued at \(\pounds 80\).
When the sculpture was sold on 1 January 1956, its value was \(\pounds 5000\).
The value, \(\pounds V\), of the sculpture is modelled by the formula \(V = A k ^ { t }\), where \(t\) is the time in years since 1 January 1900 and \(A\) and \(k\) are constants.

1. Write down the value of \(A\).
2. Show that \(k \approx 1.07664\).
3. Use this model to:
   1. show that the value of the sculpture on 1 January 2006 will be greater than £200 000;
   2. find the year in which the value of the sculpture will first exceed \(\pounds 800000\).

5

1. 1. Obtain the binomial expansion of \(( 1 - x ) ^ { - 1 }\) up to and including the term in \(x ^ { 2 }\).
      (2 marks)
   2. Hence, or otherwise, show that $$\frac { 1 } { 3 - 2 x } \approx \frac { 1 } { 3 } + \frac { 2 } { 9 } x + \frac { 4 } { 27 } x ^ { 2 }$$ for small values of \(x\).
2. Obtain the binomial expansion of \(\frac { 1 } { ( 1 - x ) ^ { 2 } }\) up to and including the term in \(x ^ { 2 }\).
3. Given that \(\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }\) can be written in the form \(\frac { A } { ( 3 - 2 x ) } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 1 - x ) ^ { 2 } }\), find the values of \(A , B\) and \(C\).
4. Hence find the binomial expansion of \(\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }\) up to and including the term in \(x ^ { 2 }\).

6

1. Express \(\cos 2 x\) in the form \(a \cos ^ { 2 } x + b\), where \(a\) and \(b\) are constants.
2. Hence show that \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \cos ^ { 2 } x \mathrm {~d} x = \frac { \pi } { a }\), where \(a\) is an integer.

7 The quadrilateral \(A B C D\) has vertices \(A ( 2,1,3 ) , B ( 6,5,3 ) , C ( 6,1 , - 1 )\) and \(D ( 2 , - 3 , - 1 )\).
The line \(l _ { 1 }\) has vector equation \(\mathbf { r } = \left[ \begin{array} { r } 6
1
- 1 \end{array} \right] + \lambda \left[ \begin{array} { l } 1
1
0 \end{array} \right]\).

1. 1. Find the vector \(\overrightarrow { A B }\).
   2. Show that the line \(A B\) is parallel to \(l _ { 1 }\).
   3. Verify that \(D\) lies on \(l _ { 1 }\).
2. The line \(l _ { 2 }\) passes through \(D ( 2 , - 3 , - 1 )\) and \(M ( 4,1,1 )\).
   1. Find the vector equation of \(l _ { 2 }\).
   2. Find the angle between \(l _ { 2 }\) and \(A C\).

8

1. Solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - 2 ( x - 6 ) ^ { \frac { 1 } { 2 } }$$ to find \(t\) in terms of \(x\), given that \(x = 70\) when \(t = 0\).
2. Liquid fuel is stored in a tank. At time \(t\) minutes, the depth of fuel in the tank is \(x \mathrm {~cm}\). Initially there is a depth of 70 cm of fuel in the tank. There is a tap 6 cm above the bottom of the tank. The flow of fuel out of the tank is modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - 2 ( x - 6 ) ^ { \frac { 1 } { 2 } }$$
   1. Explain what happens when \(x = 6\).
   2. Find how long it will take for the depth of fuel to fall from 70 cm to 22 cm .