OCR C4 (Core Mathematics 4) 2009 June

1 Find the quotient and the remainder when \(3 x ^ { 4 } - x ^ { 3 } - 3 x ^ { 2 } - 14 x - 8\) is divided by \(x ^ { 2 } + x + 2\).

2 Use the substitution \(x = \tan \theta\) to find the exact value of $$\int _ { 1 } ^ { \sqrt { 3 } } \frac { 1 - x ^ { 2 } } { 1 + x ^ { 2 } } \mathrm {~d} x$$

3

1. Expand \(( a + x ) ^ { - 2 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\).
2. When \(( 1 - x ) ( a + x ) ^ { - 2 }\) is expanded, the coefficient of \(x ^ { 2 }\) is 0 . Find the value of \(a\).

4

1. Differentiate \(\mathrm { e } ^ { x } ( \sin 2 x - 2 \cos 2 x )\), simplifying your answer.
2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \mathrm { e } ^ { x } \sin 2 x \mathrm {~d} x\).

5 A curve has parametric equations $$x = 2 t + t ^ { 2 } , \quad y = 2 t ^ { 2 } + t ^ { 3 }$$

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\) and find the gradient of the curve at the point \(( 3 , - 9 )\).
2. By considering \(\frac { y } { x }\), find a cartesian equation of the curve, giving your answer in a form not involving fractions.

6 The expression \(\frac { 4 x } { ( x - 5 ) ( x - 3 ) ^ { 2 } }\) is denoted by \(\mathrm { f } ( x )\).

1. Express f \(( x )\) in the form \(\frac { A } { x - 5 } + \frac { B } { x - 3 } + \frac { C } { ( x - 3 ) ^ { 2 } }\), where \(A , B\) and \(C\) are constants.
2. Hence find the exact value of \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\).

1. The vector \(\mathbf { u } = \frac { 3 } { 13 } \mathbf { i } + b \mathbf { j } + c \mathbf { k }\) is perpendicular to the vector \(4 \mathbf { i } + \mathbf { k }\) and to the vector \(4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\). Find the values of \(b\) and \(c\), and show that \(\mathbf { u }\) is a unit vector.
2. Calculate, to the nearest degree, the angle between the vectors \(4 \mathbf { i } + \mathbf { k }\) and \(4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\).

8

1. Given that \(14 x ^ { 2 } - 7 x y + y ^ { 2 } = 2\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 28 x - 7 y } { 7 x - 2 y }\).
2. The points \(L\) and \(M\) on the curve \(14 x ^ { 2 } - 7 x y + y ^ { 2 } = 2\) each have \(x\)-coordinate 1 . The tangents to the curve at \(L\) and \(M\) meet at \(N\). Find the coordinates of \(N\).

9 A tank contains water which is heated by an electric water heater working under the action of a thermostat. The temperature of the water, \(\theta ^ { \circ } \mathrm { C }\), may be modelled as follows. When the water heater is first switched on, \(\theta = 40\). The heater causes the temperature to increase at a rate \(k _ { 1 } { } ^ { \circ } \mathrm { C }\) per second, where \(k _ { 1 }\) is a constant, until \(\theta = 60\). The heater then switches off.

1. Write down, in terms of \(k _ { 1 }\), how long it takes for the temperature to increase from \(40 ^ { \circ } \mathrm { C }\) to \(60 ^ { \circ } \mathrm { C }\). The temperature of the water then immediately starts to decrease at a variable rate \(k _ { 2 } ( \theta - 20 ) ^ { \circ } \mathrm { C }\) per second, where \(k _ { 2 }\) is a constant, until \(\theta = 40\).
2. Write down a differential equation to represent the situation as the temperature is decreasing.
3. Find the total length of time for the temperature to increase from \(40 ^ { \circ } \mathrm { C }\) to \(60 ^ { \circ } \mathrm { C }\) and then decrease to \(40 ^ { \circ } \mathrm { C }\). Give your answer in terms of \(k _ { 1 }\) and \(k _ { 2 }\). 4