Edexcel P4 (Pure Mathematics 4) 2022 June

1. The binomial expansion of

$$( 3 + k x ) ^ { - 2 } \quad | k x | < 3$$ where \(k\) is a non-zero constant, may be written in the form $$A + B x + C x ^ { 2 } + D x ^ { 3 } + \ldots$$ where \(A\), \(B\), \(C\) and \(D\) are constants.

1. Find the value of \(A\) Given that \(C = 3 B\)
2. show that $$k ^ { 2 } + 6 k = 0$$
3. Hence (i) find the value of \(k\)
   (ii) find the value of \(D\)

1. (a) Express \(\frac { 1 } { ( 1 + 3 x ) ( 1 - x ) }\) in partial fractions.
   (b) Hence find the solution of the differential equation

$$( 1 + 3 x ) ( 1 - x ) \frac { \mathrm { d } y } { \mathrm {~d} x } = \tan y \quad - \frac { 1 } { 3 } < x \leqslant \frac { 1 } { 2 }$$ for which \(x = \frac { 1 } { 2 }\) when \(y = \frac { \pi } { 2 }\)
Give your answer in the form \(\sin ^ { n } y = \mathrm { f } ( x )\) where \(n\) is an integer to be found.

3. \begin{figure}[h] \end{figure} A tablet is dissolving in water.
The tablet is modelled as a cylinder, shown in Figure 1.
At \(t\) seconds after the tablet is dropped into the water, the radius of the tablet is \(x \mathrm {~mm}\) and the length of the tablet is \(3 x \mathrm {~mm}\). The cross-sectional area of the tablet is decreasing at a constant rate of \(0.5 \mathrm {~mm} ^ { 2 } \mathrm {~s} ^ { - 1 }\)

1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) when \(x = 7\)
2. Find, according to the model, the rate of decrease of the volume of the tablet when \(x = 4\)

1. In this question you must show all stages of your working.

\section*{Solutions relying on calculator technology are not acceptable.} A curve has equation $$16 x ^ { 3 } - 9 k x ^ { 2 } y + 8 y ^ { 3 } = 875$$ where \(k\) is a constant.

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 k x y - 16 x ^ { 2 } } { 8 y ^ { 2 } - 3 k x ^ { 2 } }$$ Given that the curve has a turning point at \(x = \frac { 5 } { 2 }\)
2. find the value of \(k\)

1. In this question you must show all stages of your working.

\section*{Solutions relying on calculator technology are not acceptable.}

1. Use the substitution \(x = 2 \sin u\) to show that $$\int _ { 0 } ^ { 1 } \frac { 3 x + 2 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } d x = \int _ { 0 } ^ { p } \left( \frac { 3 } { 2 } \operatorname { secutanu } + \frac { 1 } { 2 } \sec ^ { 2 } u \right) d u$$ where \(p\) is a constant to be found.
2. Hence find the exact value of $$\int _ { 0 } ^ { 1 } \frac { 3 x + 2 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } d x$$

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

• the point \(A\) has position vector \(\quad \mathbf { i } - 4 \mathbf { j } + 3 \mathbf { k }\)
• the point \(B\) has position vector \(5 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k }\)
• the point \(C\) has position vector \(3 \mathbf { i } + p \mathbf { j } - \mathbf { k }\)
  where \(p\) is a constant.
  The line \(l\) passes through \(A\) and \(B\).
  1. Find a vector equation for the line \(l\)

Given that \(\overrightarrow { A C }\) is perpendicular to \(l\)

find the value of \(p\)

Hence find the area of triangle \(A B C\), giving your answer as a surd in simplest form.

1. In this question you must show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.} The curve \(C\) has parametric equations $$x = \sin t - 3 \cos ^ { 2 } t \quad y = 3 \sin t + 2 \cos t \quad 0 \leqslant t \leqslant 5$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\) where \(t = \pi\) The point \(P\) lies on \(C\) where \(t = \pi\)
2. Find the equation of the tangent to the curve at \(P\) in the form \(y = m x + c\) where \(m\) and \(c\) are constants to be found. Given that the tangent to the curve at \(P\) cuts \(C\) at the point \(Q\)
3. show that the value of \(t\) at point \(Q\) satisfies the equation $$9 \cos ^ { 2 } t + 2 \cos t - 7 = 0$$
4. Hence find the exact value of the \(y\) coordinate of \(Q\)

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 2 shows the curve with equation $$y = 10 x \mathrm { e } ^ { - \frac { 1 } { 2 } x } \quad 0 \leqslant x \leqslant 10$$ The finite region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(x\)-axis and the line with equation \(x = 10\) The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.

1. Show that the volume, \(V\), of this solid is given by $$V = k \int _ { 0 } ^ { 10 } x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x$$ where \(k\) is a constant to be found.
2. Find \(\int x ^ { 2 } e ^ { - x } d x\) Figure 3 represents an exercise weight formed by joining two of these solids together.
   The exercise weight has mass 5 kg and is 20 cm long.
   Given that $$\text { density } = \frac { \text { mass } } { \text { volume } }$$ and using your answers to part (a) and part (b),
3. find the density of this exercise weight. Give your answer in grams per \(\mathrm { cm } ^ { 3 }\) to 3 significant figures.

1. Use proof by contradiction to show that, when \(n\) is an integer,

$$n ^ { 2 } - 2$$ is never divisible by 4