Edexcel P4 (Pure Mathematics 4) 2020 October

1. Given that \(n\) is an integer, use algebra, to prove by contradiction, that if \(n ^ { 3 }\) is even then \(n\) is even.
   

1. (a) Use the binomial expansion to expand

$$( 4 - 5 x ) ^ { - \frac { 1 } { 2 } } \quad | x | < \frac { 4 } { 5 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) giving each coefficient as a fully simplified fraction. $$f ( x ) = \frac { 2 + k x } { \sqrt { 4 - 5 x } } \quad \text { where } k \text { is a constant and } | x | < \frac { 4 } { 5 }$$ Given that the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), is $$1 + \frac { 3 } { 10 } x + m x ^ { 2 } + \ldots \quad \text { where } m \text { is a constant }$$ (b) find the value of \(k\),
(c) find the value of \(m\).

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { e } ^ { 0.5 x } - 2\)
The region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the \(y\)-axis. The region \(R\) is rotated \(360 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution.
Show that the volume of this solid can be written in the form \(a \ln 2 + b\), where \(a\) and \(b\) are constants to be found.

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with parametric equations $$x = 2 t ^ { 2 } - 6 t , \quad y = t ^ { 3 } - 4 t , \quad t \in \mathbb { R }$$ The curve cuts the \(x\)-axis at the origin and at the points \(A\) and \(B\), as shown in Figure 2.

1. Find the coordinates of \(A\) and show that \(B\) has coordinates (20, 0).
2. Show that the equation of the tangent to the curve at \(B\) is $$7 y + 4 x - 80 = 0$$ The tangent to the curve at \(B\) cuts the curve again at the point \(P\).
3. Find, using algebra, the \(x\) coordinate of \(P\).

5. \begin{figure}[h] \end{figure}

1. Find \(\int \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x\) Figure 3 shows a sketch of part of the curve with equation $$y = \frac { 3 + 2 x + \ln x } { x ^ { 2 } } \quad x > 0.5$$ The finite region \(R\), shown shaded in Figure 3, is bounded by the curve, the line with equation \(x = 2\), the \(x\)-axis and the line with equation \(x = 4\)
2. Use the answer to part (a) to find the exact area of \(R\), writing your answer in simplest form.

6. A curve \(C\) has equation $$y = x ^ { \sin x } \quad x > 0 \quad y > 0$$

1. Find, by firstly taking natural logarithms, an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Hence show that the \(x\) coordinates of the stationary points of \(C\) are solutions of the equation $$\tan x + x \ln x = 0$$

7. (i) Using a suitable substitution, find, using calculus, the value of $$\int _ { 1 } ^ { 5 } \frac { 3 x } { \sqrt { 2 x - 1 } } \mathrm {~d} x$$ (Solutions relying entirely on calculator technology are not acceptable.)
(ii) Find $$\int \frac { 6 x ^ { 2 } - 16 } { ( x + 1 ) ( 2 x - 3 ) } d x$$

8. Relative to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{aligned} & l _ { 1 } : \quad \mathbf { r } = \left( \begin{array} { r } 4
- 3
2 \end{array} \right) + \lambda \left( \begin{array} { r } 3
- 2
- 1 \end{array} \right) \quad \text { where } \lambda \text { is a scalar parameter }
& l _ { 2 } : \quad \mathbf { r } = \left( \begin{array} { r } 2
0
- 9 \end{array} \right) + \mu \left( \begin{array} { r } 2
- 1
- 3 \end{array} \right) \quad \text { where } \mu \text { is a scalar parameter } \end{aligned}$$ Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(X\),

1. find the position vector of \(X\). The point \(P ( 10 , - 7,0 )\) lies on \(l _ { 1 }\)
   The point \(Q\) lies on \(l _ { 2 }\)
   Given that \(\overrightarrow { P Q }\) is perpendicular to \(l _ { 2 }\)
2. calculate the coordinates of \(Q\).
   

9. Bacteria are growing on the surface of a dish in a laboratory. The area of the dish, \(A \mathrm {~cm} ^ { 2 }\), covered by the bacteria, \(t\) days after the bacteria start to grow, is modelled by the differential equation $$\frac { \mathrm { d } A } { \mathrm {~d} t } = \frac { A ^ { \frac { 3 } { 2 } } } { 5 t ^ { 2 } } \quad t > 0$$ Given that \(A = 2.25\) when \(t = 3\)

1. show that $$A = \left( \frac { p t } { q t + r } \right) ^ { 2 }$$ where \(p , q\) and \(r\) are integers to be found. According to the model, there is a limit to the area that will be covered by the bacteria.
2. Find the value of this limit.
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