Edexcel P4 (Pure Mathematics 4) 2021 June

1. Given that \(k\) is a constant and the binomial expansion of

$$\sqrt { 1 + k x } \quad | k x | < 1$$ in ascending powers of \(x\) up to the term in \(x ^ { 3 }\) is $$1 + \frac { 1 } { 8 } x + A x ^ { 2 } + B x ^ { 3 }$$

1. 1. find the value of \(k\),
   2. find the value of the constant \(A\) and the constant \(B\).
2. Use the expansion to find an approximate value to \(\sqrt { 1.15 }\) Show your working and give your answer to 6 decimal places.

2.
\includegraphics[max width=\textwidth, alt={}, center]{960fe82f-c180-422c-b409-a5cdc5fae924-06_974_1088_116_548} \section*{Figure 1} Figure 1 shows a sketch of part of the curve with equation $$y = \frac { 9 } { ( 2 x - 3 ) ^ { 1.25 } } \quad x > \frac { 3 } { 2 }$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the line with equation \(y = 9\) and the line with equation \(x = 6\) This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution. Find, by algebraic integration, the exact volume of the solid generated.

3. \begin{figure}[h] \end{figure} A bowl with circular cross section and height 20 cm is shown in Figure 2.
The bowl is initially empty and water starts flowing into the bowl.
When the depth of water is \(h \mathrm {~cm}\), the volume of water in the bowl, \(V \mathrm {~cm} ^ { 3 }\), is modelled by the equation $$V = \frac { 1 } { 3 } h ^ { 2 } ( h + 4 ) \quad 0 \leqslant h \leqslant 20$$ Given that the water flows into the bowl at a constant rate of \(160 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\), find, according to the model,

1. the time taken to fill the bowl,
2. the rate of change of the depth of the water, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), when \(h = 5\)

4. Use algebraic integration and the substitution \(u = \sqrt { x }\) to find the exact value of $$\int _ { 1 } ^ { 4 } \frac { 10 } { 5 x + 2 x \sqrt { x } } \mathrm {~d} x$$ Write your answer in the form \(4 \ln \left( \frac { a } { b } \right)\), where \(a\) and \(b\) are integers to be found.
(Solutions relying entirely on calculator technology are not acceptable.)

5. A curve has equation $$y ^ { 2 } = y \mathrm { e } ^ { - 2 x } - 3 x$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 y \mathrm { e } ^ { - 2 x } + 3 } { \mathrm { e } ^ { - 2 x } - 2 y }$$ The curve crosses the \(y\)-axis at the origin and at the point \(P\).
   The tangent to the curve at the origin and the tangent to the curve at \(P\) meet at the point \(R\).
2. Find the coordinates of \(R\).
   \includegraphics[max width=\textwidth, alt={}, center]{960fe82f-c180-422c-b409-a5cdc5fae924-17_2644_1838_121_116}

6. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C\) with parametric equations $$x = 2 \cos 2 t \quad y = 4 \sin t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the \(y\)-axis.

1. 1. Show, making your working clear, that the area of \(R = \int _ { 0 } ^ { \frac { \pi } { 4 } } 32 \sin ^ { 2 } t \cos t d t\)
   2. Hence find, by algebraic integration, the exact value of the area of \(R\).
2. Show that all points on \(C\) satisfy \(y = \sqrt { a x + b }\), where \(a\) and \(b\) are constants to be found. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where f is the function $$f ( x ) = \sqrt { a x + b } \quad - 2 \leqslant x \leqslant 2$$ and \(a\) and \(b\) are the constants found in part (b).
3. State the range of f.

1. Relative to a fixed origin \(O\), the line \(l\) has equation

$$\mathbf { r } = \left( \begin{array} { r } 1
- 10
- 9 \end{array} \right) + \lambda \left( \begin{array} { l } 4
4
2 \end{array} \right) \quad \text { where } \lambda \text { is a scalar parameter }$$ Given that \(\overrightarrow { O A }\) is a unit vector parallel to \(l\),

1. find \(\overrightarrow { O A }\) The point \(X\) lies on \(l\).
   Given that \(X\) is the point on \(l\) that is closest to the origin,
2. find the coordinates of \(X\). The points \(O , X\) and \(A\) form the triangle \(O X A\).
3. Find the exact area of triangle \(O X A\).

8. (a) Given that \(y = 1\) at \(x = 0\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 x y ^ { \frac { 1 } { 3 } } } { \mathrm { e } ^ { 2 x } } \quad y \geqslant 0$$ giving your answer in the form \(y ^ { 2 } = \mathrm { g } ( x )\).
(b) Hence find the equation of the horizontal asymptote to the curve with equation \(y ^ { 2 } = \mathrm { g } ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{960fe82f-c180-422c-b409-a5cdc5fae924-27_2644_1840_118_111} \includegraphics[max width=\textwidth, alt={}, center]{960fe82f-c180-422c-b409-a5cdc5fae924-29_2646_1838_121_116}

9. (i) Relative to a fixed origin \(O\), the points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) respectively. Points \(A , B\) and \(C\) lie in a straight line, with \(B\) lying between \(A\) and \(C\).
Given \(A B : A C = 1 : 3\) show that $$\mathbf { c } = 3 \mathbf { b } - 2 \mathbf { a }$$ (ii) Given that \(n \in \mathbb { N }\), prove by contradiction that if \(n ^ { 2 }\) is a multiple of 3 then \(n\) is a multiple of 3