Edexcel P4 (Pure Mathematics 4) 2024 January

1. Find, in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), the binomial expansion of

$$( 1 - 4 x ) ^ { - 3 } \quad | x | < \frac { 1 } { 4 }$$ fully simplifying each term.

1. Given that

$$\frac { 3 x + 4 } { ( x - 2 ) ( 2 x + 1 ) ^ { 2 } } \equiv \frac { A } { x - 2 } + \frac { B } { 2 x + 1 } + \frac { C } { ( 2 x + 1 ) ^ { 2 } }$$

1. find the values of the constants \(A , B\) and \(C\).
2. Hence find the exact value of $$\int _ { 7 } ^ { 12 } \frac { 3 x + 4 } { ( x - 2 ) ( 2 x + 1 ) ^ { 2 } } \mathrm {~d} x$$ giving your answer in the form \(p \ln q + r\) where \(p\), \(q\) and \(r\) are rational numbers.

3. \begin{figure}[h] \end{figure} The curve \(C\), shown in Figure 1, has equation $$y ^ { 2 } x + 3 y = 4 x ^ { 2 } + k \quad y > 0$$ where \(k\) is a constant.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\) The point \(P ( p , 2 )\), where \(p\) is a constant, lies on \(C\).
   Given that \(P\) is the minimum turning point on \(C\),
2. find
   1. the value of \(p\)
   2. the value of \(k\)

4. \begin{figure}[h] \end{figure} A cone, shown in Figure 2, has

• fixed height 5 cm
• base radius \(r \mathrm {~cm}\)
• slant height \(l \mathrm {~cm}\)
  1. Find an expression for \(l\) in terms of \(r\)

Given that the base radius is increasing at a constant rate of 3 cm per minute,

find the rate at which the total surface area of the cone is changing when the radius of the cone is 1.5 cm . Give your answer in \(\mathrm { cm } ^ { 2 }\) per minute to one decimal place.
[0pt] [The total surface area, \(S\), of a cone is given by the formula \(S = \pi r ^ { 2 } + \pi r l\) ]

1. (a) Find \(\int x ^ { 2 } \cos 2 x d x\)
   (b) Hence solve the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} t } = \left( \frac { t \cos t } { y } \right) ^ { 2 }$$ giving your answer in the form \(y ^ { n } = \mathrm { f } ( t )\) where \(n\) is an integer.

1. Relative to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations

$$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( 3 \mathbf { i } + p \mathbf { j } + 7 \mathbf { k } ) + \lambda ( 2 \mathbf { i } - 5 \mathbf { j } + 4 \mathbf { k } )
& l _ { 2 } : \mathbf { r } = ( 8 \mathbf { i } - 2 \mathbf { j } + 5 \mathbf { k } ) + \mu ( 4 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } ) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(p\) is a constant.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect,

1. find the value of \(p\),
2. find the position vector of the point of intersection.
3. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) Give your answer in degrees to one decimal place. The point \(A\) lies on \(l _ { 1 }\) with parameter \(\lambda = 2\)
   The point \(B\) lies on \(l _ { 2 }\) with \(\overrightarrow { A B }\) perpendicular to \(l _ { 2 }\)
4. Find the coordinates of \(B\)

1. (a) Using the substitution \(u = 4 x + 2 \sin 2 x\), or otherwise, show that

$$\int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { e } ^ { 4 x + 2 \sin 2 x } \cos ^ { 2 } x \mathrm {~d} x = \frac { 1 } { 8 } \left( \mathrm { e } ^ { 2 \pi } - 1 \right)$$ Figure 3 The curve shown in Figure 3, has equation $$y = 6 \mathrm { e } ^ { 2 x + \sin 2 x } \cos x$$ The region \(R\), shown shaded in Figure 3, is bounded by the positive \(x\)-axis, the positive \(y\)-axis and the curve. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid.
(b) Use the answer to part (a) to find the volume of the solid formed, giving the answer in simplest form.

1. Use proof by contradiction to prove that the curve with equation

$$y = 2 x + x ^ { 3 } + \cos x$$ has no stationary points.

9. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve \(C\) with parametric equations $$x = \sec t \quad y = \sqrt { 3 } \tan \left( t + \frac { \pi } { 3 } \right) \quad \frac { \pi } { 6 } < t < \frac { \pi } { 2 }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\)
2. Find an equation for the tangent to \(C\) at the point where \(t = \frac { \pi } { 3 }\) Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
3. Show that all points on \(C\) satisfy the equation $$y = \frac { A x ^ { 2 } + B \sqrt { 3 x ^ { 2 } - 3 } } { 4 - 3 x ^ { 2 } }$$ where \(A\) and \(B\) are constants to be found.