Edexcel P4 (Pure Mathematics 4) 2023 January

1. $$f ( x ) = \frac { 5 x + 10 } { ( 1 - x ) ( 2 + 3 x ) }$$

1. Write \(\mathrm { f } ( x )\) in partial fraction form.
2. 1. Hence find, in ascending powers of \(x\) up to and including the terms in \(x ^ { 2 }\), the binomial series expansion of \(\mathrm { f } ( x )\). Give each coefficient as a simplified fraction.
   2. Find the range of values of \(x\) for which this expansion is valid.

1. A set of points \(P ( x , y )\) is defined by the parametric equations

$$x = \frac { t - 1 } { 2 t + 1 } \quad y = \frac { 6 } { 2 t + 1 } \quad t \neq - \frac { 1 } { 2 }$$

1. Show that all points \(P ( x , y )\) lie on a straight line.
2. Hence or otherwise, find the \(x\) coordinate of the point of intersection of this line and the line with equation \(y = x + 12\)

3. \begin{figure}[h] \end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying entirely on calculator technology are not acceptable.} Figure 1 shows a sketch of the curve with equation $$y = \sqrt { \frac { 3 x } { 3 x ^ { 2 } + 5 } } \quad x \geqslant 0$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the lines with equations \(x = \sqrt { 5 }\) and \(x = 5\) The region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
Use integration to find the exact volume of the solid generated. Give your answer in the form \(a \ln b\), where \(a\) is an irrational number and \(b\) is a prime number.

1. (a) Using the substitution \(u = \sqrt { 2 x + 1 }\), show that

$$\int _ { 4 } ^ { 12 } \sqrt { 8 x + 4 } \mathrm { e } ^ { \sqrt { 2 x + 1 } } \mathrm {~d} x$$ may be expressed in the form $$\int _ { a } ^ { b } k u ^ { 2 } \mathrm { e } ^ { u } \mathrm {~d} u$$ where \(a\), \(b\) and \(k\) are constants to be found.
(b) Hence find, by algebraic integration, the exact value of $$\int _ { 4 } ^ { 12 } \sqrt { 8 x + 4 } e ^ { \sqrt { 2 x + 1 } } d x$$ giving your answer in simplest form.

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation $$y ^ { 2 } = 2 x ^ { 2 } + 15 x + 10 y$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The curve is not defined for values of \(x\) in the interval ( \(p , q\) ), as shown in Figure 2.
2. Using your answer to part (a) or otherwise, find the value of \(p\) and the value of \(q\).
   (Solutions relying entirely on calculator technology are not acceptable.)

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

• the point \(A\) has position vector \(2 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k }\)
• the point \(B\) has position vector \(8 \mathbf { i } + 3 \mathbf { j } - 7 \mathbf { k }\)

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

1. 1. Find \(\overrightarrow { A B }\)
   2. Find a vector equation for the line \(l\) The point \(C\) has position vector \(3 \mathbf { i } + 5 \mathbf { j } + 2 \mathbf { k }\)
      The point \(P\) lies on \(l\)
      Given that \(\overrightarrow { C P }\) is perpendicular to \(l\)
2. find the position vector of the point \(P\)

1. The volume \(V \mathrm {~cm} ^ { 3 }\) of a spherical balloon with radius \(r \mathrm {~cm}\) is given by the formula

$$V = \frac { 4 } { 3 } \pi r ^ { 3 }$$

1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} r }\) giving your answer in simplest form. At time \(t\) seconds, the volume of the balloon is increasing according to the differential equation $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 900 } { ( 2 t + 3 ) ^ { 2 } } \quad t \geqslant 0$$ Given that \(V = 0\) when \(t = 0\)
2. 1. solve this differential equation to show that $$V = \frac { 300 t } { 2 t + 3 }$$
   2. Hence find the upper limit to the volume of the balloon.
3. Find the radius of the balloon at \(t = 3\), giving your answer in cm to 3 significant figures.
4. Find the rate of increase of the radius of the balloon at \(t = 3\), giving your answer to 2 significant figures. Show your working and state the units of your answer.

8. \begin{figure}[h] \end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying entirely on calculator technology are not acceptable.} A curve \(C\) has parametric equations $$x = \sin ^ { 2 } t \quad y = 2 \tan t \quad 0 \leqslant t < \frac { \pi } { 2 }$$ The point \(P\) with parameter \(t = \frac { \pi } { 4 }\) lies on \(C\).
The line \(l\) is the normal to \(C\) at \(P\), as shown in Figure 3.

1. Show, using calculus, that an equation for \(l\) is $$8 y + 2 x = 17$$ The region \(S\), shown shaded in Figure 3, is bounded by \(C , l\) and the \(x\)-axis.
2. Find, using calculus, the exact area of \(S\).

1. A student was asked to prove, for \(p \in \mathbb { N }\), that
   "if \(p ^ { 3 }\) is a multiple of 3 , then \(p\) must be a multiple of 3 "

The start of the student's proof by contradiction is shown in the box below. Assumption:
There exists a number \(p , p \in \mathbb { N }\), such that \(p ^ { 3 }\) is a multiple of 3 , and \(p\) is NOT a multiple of 3 Let \(p = 3 k + 1 , k \in \mathbb { N }\). $$\text { Consider } \begin{aligned} p ^ { 3 } = ( 3 k + 1 ) ^ { 3 } & = 27 k ^ { 3 } + 27 k ^ { 2 } + 9 k + 1
& = 3 \left( 9 k ^ { 3 } + 9 k ^ { 2 } + 3 k \right) + 1 \quad \text { which is not a multiple of } 3 \end{aligned}$$

1. Show the calculations and statements that are required to complete the proof.
2. Hence prove, by contradiction, that \(\sqrt [ 3 ] { 3 }\) is an irrational number.