Questions P4 (127 questions)

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Edexcel P4 2024 June Q6
  1. show that \end{itemize} $$\frac { \mathrm { d } A } { \mathrm {~d} \theta } = K ( 1 - \cos \theta )$$ where \(K\) is a constant to be found.
  2. Find, in \(\mathrm { cm } ^ { 2 } \mathrm {~s} ^ { - 1 }\), the rate of increase of the area of the segment when \(\theta = \frac { \pi } { 3 }\) 5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e583bf92-d6a9-4f1a-b3c8-372afa6e0a0e-10_803_1086_248_493} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of the curve defined by the parametric equations $$x = t ^ { 2 } + 2 t \quad y = \frac { 2 } { t ( 3 - t ) } \quad a \leqslant t \leqslant b$$ where \(a\) and \(b\) are constants.
    The ends of the curve lie on the line with equation \(y = 1\)
  3. Find the value of \(a\) and the value of \(b\) The region \(R\), shown shaded in Figure 2, is bounded by the curve and the line with equation \(y = 1\)
  4. Show that the area of region \(R\) is given by $$M - k \int _ { a } ^ { b } \frac { t + 1 } { t ( 3 - t ) } \mathrm { d } t$$ where \(M\) and \(k\) are constants to be found.
    1. Write \(\frac { t + 1 } { t ( 3 - t ) }\) in partial fractions.
    2. Use algebraic integration to find the exact area of \(R\), giving your answer in simplest form.
Edexcel P4 2021 October Q8
  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.}