Edexcel P4 (Pure Mathematics 4) 2022 January

1. The curve \(C\) has equation

$$x y ^ { 2 } = x ^ { 2 } y + 6 \quad x \neq 0 \quad y \neq 0$$ Find an equation for the tangent to \(C\) at the point \(P ( 2,3 )\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
(6)

2. (a) Find, in ascending powers of \(x\), the first three non-zero terms of the binomial series expansion of $$\sqrt [ 3 ] { 1 + 4 x ^ { 3 } } \quad | x | < \frac { 1 } { \sqrt [ 3 ] { 4 } }$$ giving each coefficient as a simplified fraction.
(b) Use the expansion from part (a) with \(x = \frac { 1 } { 3 }\) to find a rational approximation to \(\sqrt [ 3 ] { 31 }\)
(3)

3. The curve \(C\) has parametric equations $$x = 3 + 2 \sin t \quad y = \frac { 6 } { 7 + \cos 2 t } \quad - \frac { \pi } { 2 } \leqslant t \leqslant \frac { \pi } { 2 }$$

1. Show that \(C\) has Cartesian equation $$y = \frac { 12 } { ( 7 - x ) ( 1 + x ) } \quad p \leqslant x \leqslant q$$ where \(p\) and \(q\) are constants to be found.
2. Hence, find a Cartesian equation for \(C\) in the form $$y = \frac { a } { x + b } + \frac { c } { x + d } \quad p \leqslant x \leqslant q$$ where \(a , b , c\) and \(d\) are constants.

4. \begin{figure}[h] \end{figure} A regular icosahedron of side length \(x \mathrm {~cm}\), shown in Figure 1, is expanding uniformly. The icosahedron consists of 20 congruent equilateral triangular faces of side length \(x \mathrm {~cm}\).

1. Show that the surface area, \(A \mathrm {~cm} ^ { 2 }\), of the icosahedron is given by $$A = 5 \sqrt { 3 } x ^ { 2 }$$ Given that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the icosahedron is given by $$V = \frac { 5 } { 12 } ( 3 + \sqrt { 5 } ) x ^ { 3 }$$
2. show that \(\frac { \mathrm { d } V } { \mathrm {~d} A } = \frac { ( 3 + \sqrt { 5 } ) x } { 8 \sqrt { 3 } }\) The surface area of the icosahedron is increasing at a constant rate of \(0.025 \mathrm {~cm} ^ { 2 } \mathrm {~s} ^ { - 1 }\)
3. Find the rate of change of the volume of the icosahedron when \(x = 2\), giving your answer to 2 significant figures.

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with parametric equations $$x = \sqrt { 9 - 4 t } \quad y = \frac { t ^ { 3 } } { \sqrt { 9 + 4 t } } \quad 0 \leqslant t \leqslant \frac { 9 } { 4 }$$ The curve touches the \(x\)-axis when \(t = 0\) and meets the \(y\)-axis when \(t = \frac { 9 } { 4 }\)
The region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(x\)-axis and the \(y\)-axis.

1. Show that the area of \(R\) is given by $$K \int _ { 0 } ^ { \frac { 9 } { 4 } } \frac { t ^ { 3 } } { \sqrt { 81 - 16 t ^ { 2 } } } \mathrm {~d} t$$ where \(K\) is a constant to be found.
2. Using the substitution \(u = 81 - 16 t ^ { 2 }\), or otherwise, find the exact area of \(R\).
   (Solutions relying on calculator technology are not acceptable.)

1. Three consecutive terms in a sequence of real numbers are given by

$$k , 1 + 2 k \text { and } 3 + 3 k$$ where \(k\) is a constant. Use proof by contradiction to show that this sequence is not a geometric sequence.

7. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 3 shows the design of a doorknob.
The shape of the doorknob is formed by rotating the curve shown in Figure 4 through \(360 ^ { \circ }\) about the \(x\)-axis, where the units are centimetres. The equation of the curve is given by $$\mathrm { f } ( x ) = \frac { 1 } { 4 } ( 4 - x ) \mathrm { e } ^ { x } \quad 0 \leqslant x \leqslant 4$$

1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the doorknob is given by $$V = K \int _ { 0 } ^ { 4 } \left( x ^ { 2 } - 8 x + 16 \right) \mathrm { e } ^ { 2 x } \mathrm {~d} x$$ where \(K\) is a constant to be found.
2. Hence, find the exact value of the volume of the doorknob. Give your answer in the form \(p \pi \left( \mathrm { e } ^ { q } + r \right) \mathrm { cm } ^ { 3 }\) where \(p , q\) and \(r\) are simplified rational numbers to be found.

8. With respect to a fixed origin \(O\) the points \(A\) and \(B\) have position vectors $$\left( \begin{array} { l } 6
6
2 \end{array} \right) \text { and } \left( \begin{array} { l } 6
0
7 \end{array} \right)$$ respectively. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\).

1. Write down an equation for \(l _ { 1 }\) Give your answer in the form \(\mathbf { r } = \mathbf { p } + \lambda \mathbf { q }\), where \(\lambda\) is a scalar parameter. The line \(l _ { 2 }\) has equation $$\mathbf { r } = \left( \begin{array} { l } 3
   1
   4 \end{array} \right) + \mu \left( \begin{array} { l } 1
   5

9 \end{array} \right)$$ where \(\mu\) is a scalar parameter.
(b) Show that \(l _ { 1 }\) and \(l _ { 2 }\) do not meet. The point \(C\) is on \(l _ { 2 }\) where \(\mu = - 1\)
(c) Find the acute angle between \(A C\) and \(l _ { 2 }\) Give your answer in degrees to one decimal place.

1. (a) Find the derivative with respect to \(y\) of

$$\frac { 1 } { ( 1 + 2 \ln y ) ^ { 2 } }$$ (b) Hence find a general solution to the differential equation $$3 \operatorname { cosec } ( 2 x ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y ( 1 + 2 \ln y ) ^ { 3 } \quad y > 0 \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$$ (c) Show that the particular solution of this differential equation for which \(y = 1\) at \(x = \frac { \pi } { 6 }\) is given by $$y = \mathrm { e } ^ { A \sec x - \frac { 1 } { 2 } }$$ where \(A\) is an irrational number to be found.
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