Edexcel P4 (Pure Mathematics 4) 2023 June

1. (a) Find the first 4 terms of the binomial expansion, in ascending powers of \(x\), of

$$\left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { - \frac { 3 } { 2 } } \quad | x | < \frac { 1 } { 2 }$$ giving each term in simplest form. Given that $$\left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { n } \left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { - \frac { 3 } { 2 } } = \left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { \frac { 1 } { 2 } }$$ (b) write down the value of \(n\).
(c) Hence, or otherwise, find the first 3 terms of the binomial expansion, in ascending powers of \(x\), of $$\left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { \frac { 1 } { 2 } } \quad | x | < \frac { 1 } { 2 }$$ giving each term in simplest form.

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$2 ^ { x } - 4 x y + y ^ { 2 } = 13 \quad y \geqslant 0$$ The point \(P\) lies on \(C\) and has \(x\) coordinate 2

1. Find the \(y\) coordinate of \(P\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The tangent to \(C\) at \(P\) crosses the \(x\)-axis at the point \(Q\).
3. Find the \(x\) coordinate of \(Q\), giving your answer in the form \(\frac { a \ln 2 + b } { c \ln 2 + d }\) where \(a , b , c\) and \(d\) are integers to be found.

3. $$\mathrm { f } ( x ) = \frac { 8 x - 5 } { ( 2 x - 1 ) ( 4 x - 3 ) } \quad x > 1$$

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence find \(\int \mathrm { f } ( x ) \mathrm { d } x\)
3. Use the answer to part (b) to find the value of \(k\) for which $$\int _ { k } ^ { 3 k } \mathrm { f } ( x ) \mathrm { d } x = \frac { 1 } { 2 } \ln 20$$

1. Relative to a fixed origin \(O\),

• the point \(A\) has position vector \(4 \mathbf { i } + 8 \mathbf { j } + \mathbf { k }\)
• the point \(B\) has position vector \(5 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k }\)
• the point \(P\) has position vector \(2 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\)

The straight line \(l\) passes through \(A\) and \(B\).

1. Find a vector equation for \(l\). The point \(C\) lies on \(l\) so that \(P C\) is perpendicular to \(l\).
2. Find the coordinates of \(C\). The point \(P ^ { \prime }\) is the reflection of \(P\) in the line \(l\).
3. Find the coordinates of \(P ^ { \prime }\)
4. Hence find \(\left| \overrightarrow { P P ^ { \prime } } \right|\), giving your answer as a simplified surd.

1. (i) Find

$$\int x ^ { 2 } \mathrm { e } ^ { x } \mathrm {~d} x$$ (4)
(ii) Use the substitution \(u = \sqrt { 1 - 3 x }\) to show that $$\int \frac { 27 x } { \sqrt { 1 - 3 x } } \mathrm {~d} x = - 2 ( 1 - 3 x ) ^ { \frac { 1 } { 2 } } ( A x + B ) + k$$ where \(A\) and \(B\) are integers to be found and \(k\) is an arbitrary constant.

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The temperature, \(\theta ^ { \circ } \mathrm { C }\), of a car engine, \(t\) minutes after the engine is turned off, is modelled by the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k ( \theta - 15 ) ^ { 2 }$$ where \(k\) is a constant.
Given that the temperature of the car engine

• is \(85 ^ { \circ } \mathrm { C }\) at the instant the engine is turned off
• is \(40 ^ { \circ } \mathrm { C }\) exactly 10 minutes after the engine is turned off
  1. solve the differential equation to show that, according to the model

$$\theta = \frac { a t + b } { c t + d }$$ where \(a , b , c\) and \(d\) are integers to be found.

Hence find, according to the model, the time taken for the temperature of the car engine to reach \(20 ^ { \circ } \mathrm { C }\). Give your answer to the nearest minute.

1. Use proof by contradiction to prove that \(\sqrt { 7 }\) is irrational.
   (You may assume that if \(k\) is an integer and \(k ^ { 2 }\) is a multiple of 7 then \(k\) is a multiple of 7 )

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with parametric equations $$x = t + \frac { 1 } { t } \quad y = t - \frac { 1 } { t } \quad t > 0.7$$ The curve \(C\) intersects the \(x\)-axis at the point \(Q\).

1. Find the \(x\) coordinate of \(Q\). The line \(l\) is the normal to \(C\) at the point \(P\) as shown in Figure 2.
   Given that \(t = 2\) at \(P\)
2. write down the coordinates of \(P\)
3. Using calculus, show that an equation of \(l\) is $$3 x + 5 y = 15$$ The region, \(R\), shown shaded in Figure 2 is bounded by the curve \(C\), the line \(l\) and the \(x\)-axis.
4. Using algebraic integration, find the exact volume of the solid of revolution formed when the region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.