Edexcel P4 (Pure Mathematics 4) 2021 October

1. The curve \(C\) has equation

$$2 x - 4 y ^ { 2 } + 3 x ^ { 2 } y = 4 x ^ { 2 } + 8$$ The point \(P ( 3,2 )\) lies on \(C\).
Find the equation of the normal to \(C\) at the point \(P\), writing your answer in the form \(a x + b y + c = 0\) where \(a\), \(b\) and \(c\) are integers to be found.

2. Find the particular solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 y ^ { 2 } } { \sqrt { 4 x + 5 } } \quad x > - \frac { 5 } { 4 }$$ for which \(y = \frac { 1 } { 3 }\) at \(x = - \frac { 1 } { 4 }\) giving your answer in the form \(y = \mathrm { f } ( x )\)
(6)

3. $$\mathrm { g } ( x ) = \frac { 3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 6 } { x ( x + 3 ) } \equiv A x + B + \frac { C } { x } + \frac { D } { x + 3 }$$

1. Find the values of the constants \(A , B , C\) and \(D\). A curve has equation $$y = g ( x ) \quad x > 0$$ Using the answer to part (a),
2. find \(\mathrm { g } ^ { \prime } ( x )\).
3. Hence, explain why \(\mathrm { g } ^ { \prime } ( x ) > 3\) for all values of \(x\) in the domain of g .

4. $$\mathrm { f } ( x ) = \sqrt { 1 - 4 x ^ { 2 } } \quad | x | < \frac { 1 } { 2 }$$

1. Find, in ascending powers of \(x\), the first four non-zero terms of the binomial expansion of \(\mathrm { f } ( x )\). Give each coefficient in simplest form.
2. By substituting \(x = \frac { 1 } { 4 }\) into the binomial expansion of \(\mathrm { f } ( x )\), obtain an approximation for \(\sqrt { 3 }\) Give your answer to 4 decimal places.
   \includegraphics[max width=\textwidth, alt={}, center]{08756c4b-6619-42da-ac8a-2bf065c01de8-13_42_63_2606_1852}

5. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with parametric equations $$x = 5 + 2 \tan t \quad y = 8 \sec ^ { 2 } t \quad - \frac { \pi } { 3 } \leqslant t \leqslant \frac { \pi } { 4 }$$

1. Use parametric differentiation to find the gradient of \(C\) at \(x = 3\) The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where f is a quadratic function.
2. Find \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a\), \(b\) and \(c\) are constants to be found.
3. Find the range of f.

6. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation $$y = \frac { 16 \sin 2 x } { ( 3 + 4 \sin x ) ^ { 2 } } \quad 0 \leqslant x \leqslant \frac { \pi } { 2 }$$ The region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(x\)-axis and the line with equation \(x = \frac { \pi } { 6 }\) Using the substitution \(u = 3 + 4 \sin x\), show that the area of \(R\) can be written in the form \(a + \ln b\), where \(a\) and \(b\) are rational constants to be found.

7. With respect to a fixed origin \(O\),

• the line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { r } 4
  2
  - 3 \end{array} \right) + \lambda \left( \begin{array} { r } - 4
  - 3
  5 \end{array} \right)\) where \(\lambda\) is a scalar constant
• the point \(A\) has position vector \(9 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }\)

Given that \(X\) is the point on \(l\) nearest to \(A\),

1. find
   1. the coordinates of \(X\)
   2. the shortest distance from \(A\) to \(l\). Give your answer in the form \(\sqrt { d }\), where \(d\) is an integer. The point \(B\) is the image of \(A\) after reflection in \(l\).
2. Find the position vector of \(B\). Solutions relying on calculator technology are not acceptable. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{08756c4b-6619-42da-ac8a-2bf065c01de8-26_668_661_408_644} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure}

1. Find \(\int x ^ { 2 } \ln x \mathrm {~d} x\) Figure 3 shows a sketch of part of the curve with equation $$y = x \ln x \quad x > 0$$ The region \(R\), shown shaded in Figure 3, lies entirely above the \(x\)-axis and is bounded by the curve, the \(x\)-axis and the line with equation \(x = \mathrm { e }\). This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
2. Find the exact volume of the solid formed, giving your answer in simplest form. \section*{8. In this question you must show all stages of your working.
   In this question you must show all stages of your working.}

9. \begin{figure}[h] \end{figure} Figure 4 shows a cylindrical tank that contains some water. The tank has an internal diameter of 8 m and an internal height of 4.2 m .
Water is flowing into the tank at a constant rate of \(( 0.6 \pi ) \mathrm { m } ^ { 3 }\) per minute. There is a tap at point \(T\) at the bottom of the tank. At time \(t\) minutes after the tap has been opened,

• the depth of the water is \(h\) metres
• the water is leaving the tank at a rate of \(( 0.15 \pi h ) \mathrm { m } ^ { 3 }\) per minute
  1. Show that

$$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 12 - 3 h } { 320 }$$ Given that the depth of the water in the tank is 0.5 m when the tap is opened,

find the time taken for the depth of water in the tank to reach 3.5 m .

10. (a) A student's attempt to answer the question
"Prove by contradiction that if \(n ^ { 3 }\) is even, then \(n\) is even" is shown below. Line 5 of the proof is missing. Assume that there exists a number \(n\) such that \(n ^ { 3 }\) is even, but \(n\) is odd. If \(n\) is odd then \(n = 2 p + 1\) where \(p \in \mathbb { Z }\)
So \(n ^ { 3 } = ( 2 p + 1 ) ^ { 3 }\) $$\begin{aligned} & = 8 p ^ { 3 } + 12 p ^ { 2 } + 6 p + 1
& = \end{aligned}$$ This contradicts our initial assumption, so if \(n ^ { 3 }\) is even, then \(n\) is even. Complete this proof by filling in line 5.
(b) Hence, prove by contradiction that \(\sqrt [ 3 ] { 2 }\) is irrational.