Edexcel C4 (Core Mathematics 4) 2013 June

1. Find \(\int x ^ { 2 } e ^ { x } d x\).
2. Hence find the exact value of \(\int _ { 0 } ^ { 1 } x ^ { 2 } \mathrm { e } ^ { x } \mathrm {~d} x\).

1. Use the binomial expansion to show that $$\left. \sqrt { ( } \frac { 1 + x } { 1 - x } \right) \approx 1 + x + \frac { 1 } { 2 } x ^ { 2 } , \quad | x | < 1$$
2. Substitute \(x = \frac { 1 } { 26 }\) into $$\sqrt { \left( \frac { 1 + x } { 1 - x } \right) = 1 + x + \frac { 1 } { 2 } x ^ { 2 } }$$ to obtain an approximation to \(\sqrt { } 3\)
   Give your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers.

3. \begin{figure}[h] \end{figure} Figure 1 shows the finite region \(R\) bounded by the \(x\)-axis, the \(y\)-axis, the line \(x = \frac { \pi } { 2 }\) and the curve with equation $$y = \sec \left( \frac { 1 } { 2 } x \right) , \quad 0 \leqslant x \leqslant \frac { \pi } { 2 }$$ The table shows corresponding values of \(x\) and \(y\) for \(y = \sec \left( \frac { 1 } { 2 } x \right)\).

1. Complete the table above giving the missing value of \(y\) to 6 decimal places.
2. Using the trapezium rule, with all of the values of \(y\) from the completed table, find an approximation for the area of \(R\), giving your answer to 4 decimal places. Region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
3. Use calculus to find the exact volume of the solid formed.

1. A curve \(C\) has parametric equations

$$x = 2 \sin t , \quad y = 1 - \cos 2 t , \quad - \frac { \pi } { 2 } \leqslant t \leqslant \frac { \pi } { 2 }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point where \(t = \frac { \pi } { 6 }\)
2. Find a cartesian equation for \(C\) in the form $$y = \mathrm { f } ( x ) , \quad - k \leqslant x \leqslant k$$ stating the value of the constant \(k\).
3. Write down the range of \(\mathrm { f } ( x )\).

1. (a) Use the substitution \(x = u ^ { 2 } , u > 0\), to show that

$$\int \frac { 1 } { x ( 2 \sqrt { x } - 1 ) } \mathrm { d } x = \int \frac { 2 } { u ( 2 u - 1 ) } \mathrm { d } u$$ (b) Hence show that $$\int _ { 1 } ^ { 9 } \frac { 1 } { x ( 2 \sqrt { x } - 1 ) } \mathrm { d } x = 2 \ln \left( \frac { a } { b } \right)$$ where \(a\) and \(b\) are integers to be determined.

6. Water is being heated in a kettle. At time \(t\) seconds, the temperature of the water is \(\theta ^ { \circ } \mathrm { C }\). The rate of increase of the temperature of the water at any time \(t\) is modelled by the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = \lambda ( 120 - \theta ) , \quad \theta \leqslant 100$$ where \(\lambda\) is a positive constant. Given that \(\theta = 20\) when \(t = 0\),

1. solve this differential equation to show that $$\theta = 120 - 100 \mathrm { e } ^ { - \lambda t }$$ When the temperature of the water reaches \(100 ^ { \circ } \mathrm { C }\), the kettle switches off.
2. Given that \(\lambda = 0.01\), find the time, to the nearest second, when the kettle switches off.

7. A curve is described by the equation $$x ^ { 2 } + 4 x y + y ^ { 2 } + 27 = 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). A point \(Q\) lies on the curve.
   The tangent to the curve at \(Q\) is parallel to the \(y\)-axis.
   Given that the \(x\) coordinate of \(Q\) is negative,
2. use your answer to part (a) to find the coordinates of \(Q\).

1. With respect to a fixed origin \(O\), the line \(l\) has equation

$$\mathbf { r } = \left( \begin{array} { c }

13
8
1 \end{array} \right) + \lambda \left( \begin{array} { r } 2
2
- 1 \end{array} \right) \text {, where } \lambda \text { is a scalar parameter. }$$ The point \(A\) lies on \(l\) and has coordinates ( \(3 , - 2,6\) ).
The point \(P\) has position vector ( \(- p \mathbf { i } + 2 p \mathbf { k }\) ) relative to \(O\), where \(p\) is a constant.
Given that vector \(\overrightarrow { P A }\) is perpendicular to \(l\),

1. find the value of \(p\). Given also that \(B\) is a point on \(l\) such that \(\angle B P A = 45 ^ { \circ }\),
2. find the coordinates of the two possible positions of \(B\).