AQA C4 (Core Mathematics 4) 2007 January

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

1. 1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} t }\).
      (2 marks)
   2. Hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Find an equation of the normal to the curve at the point where \(t = 1\).
3. Find a cartesian equation of the curve.

2 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 13\).

1. Use the Remainder Theorem to find the remainder when \(\mathrm { f } ( x )\) is divided by \(( 2 x - 3 )\).
2. The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 13 + d\), where \(d\) is a constant. Given that ( \(2 x - 3\) ) is a factor of \(\mathrm { g } ( x )\), show that \(d = - 4\).
3. Express \(\mathrm { g } ( x )\) in the form \(( 2 x - 3 ) \left( x ^ { 2 } + a x + b \right)\).

3

1. Express \(\cos 2 x\) in terms of \(\sin x\).
2. 1. Hence show that \(3 \sin x - \cos 2 x = 2 \sin ^ { 2 } x + 3 \sin x - 1\) for all values of \(x\).
   2. Solve the equation \(3 \sin x - \cos 2 x = 1\) for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
3. Use your answer from part (a) to find \(\int \sin ^ { 2 } x \mathrm {~d} x\).

4

1. 1. Express \(\frac { 3 x - 5 } { x - 3 }\) in the form \(A + \frac { B } { x - 3 }\), where \(A\) and \(B\) are integers. (2 marks)
   2. Hence find \(\int \frac { 3 x - 5 } { x - 3 } \mathrm {~d} x\).
      (2 marks)
2. 1. Express \(\frac { 6 x - 5 } { 4 x ^ { 2 } - 25 }\) in the form \(\frac { P } { 2 x + 5 } + \frac { Q } { 2 x - 5 }\), where \(P\) and \(Q\) are integers.
      (3 marks)
   2. Hence find \(\int \frac { 6 x - 5 } { 4 x ^ { 2 } - 25 } \mathrm {~d} x\).

5

1. Find the binomial expansion of \(( 1 + x ) ^ { \frac { 1 } { 3 } }\) up to the term in \(x ^ { 2 }\).
2. 1. Show that \(( 8 + 3 x ) ^ { \frac { 1 } { 3 } } \approx 2 + \frac { 1 } { 4 } x - \frac { 1 } { 32 } x ^ { 2 }\) for small values of \(x\).
   2. Hence show that \(\sqrt [ 3 ] { 9 } \approx \frac { 599 } { 288 }\).

6 The points \(A , B\) and \(C\) have coordinates \(( 3 , - 2,4 ) , ( 5,4,0 )\) and \(( 11,6 , - 4 )\) respectively.

1. 1. Find the vector \(\overrightarrow { B A }\).
   2. Show that the size of angle \(A B C\) is \(\cos ^ { - 1 } \left( - \frac { 5 } { 7 } \right)\).
2. The line \(l\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 8
   - 3
   2 \end{array} \right] + \lambda \left[ \begin{array} { r } 1
   3
   - 2 \end{array} \right]\).
   1. Verify that \(C\) lies on \(l\).
   2. Show that \(A B\) is parallel to \(l\).
3. The quadrilateral \(A B C D\) is a parallelogram. Find the coordinates of \(D\).

7

1. Use the identity $$\tan ( A + B ) = \frac { \tan A + \tan B } { 1 - \tan A \tan B }$$ to express \(\tan 2 x\) in terms of \(\tan x\).
2. Show that $$2 - 2 \tan x - \frac { 2 \tan x } { \tan 2 x } = ( 1 - \tan x ) ^ { 2 }$$ for all values of \(x , \tan 2 x \neq 0\).

8

1. 1. Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} t } = y \sin t\) to obtain \(y\) in terms of \(t\).
   2. Given that \(y = 50\) when \(t = \pi\), show that \(y = 50 \mathrm { e } ^ { - ( 1 + \cos t ) }\).
2. A wave machine at a leisure pool produces waves. The height of the water, \(y \mathrm {~cm}\), above a fixed point at time \(t\) seconds is given by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} t } = y \sin t$$
   1. Given that this height is 50 cm after \(\pi\) seconds, find, to the nearest centimetre, the height of the water after 6 seconds.
   2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\) and hence verify that the water reaches a maximum height after \(\pi\) seconds.