AQA C4 (Core Mathematics 4) 2012 June

1

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)
2. 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 \(\int \frac { 4 x ^ { 3 } + 5 x - 2 } { 2 x + 1 } \mathrm {~d} x\).

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.

3

1. Find the binomial expansion of \(( 1 + 4 x ) ^ { \frac { 1 } { 2 } }\) up to and including the term in \(x ^ { 2 }\).
   (2 marks)
2. 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)

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. 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 \(T\).

5 A curve is defined by the parametric equations $$x = 2 \cos \theta , \quad y = 3 \sin 2 \theta$$

1. 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.

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.

\(\mathbf { 7 } \quad\) 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 \(B\).

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.)
2. 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)