Questions — Edexcel FP2 (291 questions)

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Edexcel FP2 Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a7ef3811-3594-4ecd-a616-36f42d26489b-06_428_803_233_577} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with polar equation $$r = a + 3 \cos \theta , \quad a > 0 , \quad 0 \leqslant \theta < 2 \pi$$ The area enclosed by the curve is \(\frac { 107 } { 2 } \pi\).
Find the value of \(a\).
Edexcel FP2 2006 January Q1
Find the set of values of \(x\) for which \(\frac { x ^ { 2 } } { x - 2 } > 2 x\).
(Total 6 marks)
Edexcel FP2 2006 January Q3
3. (a) Show that the substitution \(y = v x\) transforms the differential equation $$\frac { d y } { d x } = \frac { 3 x - 4 y } { 4 x + 3 y }$$ into the differential equation $$x \frac { \mathrm {~d} v } { \mathrm {~d} x } = - \frac { 3 v ^ { 2 } + 8 v - 3 } { 3 v + 4 }$$ (b) By solving differential equation (II), find a general solution of differential equation (I). (5)
(c) Given that \(y = 7\) at \(x = 1\), show that the particular solution of differential equation (I) can be written as $$( 3 y - x ) ( y + 3 x ) = 200$$ (5)(Total 14 marks)
Edexcel FP2 2006 January Q4
4. A curve \(C\) has polar equation \(r ^ { 2 } = a ^ { 2 } \cos 2 \theta , 0 \leq \theta \leq \frac { \pi } { 4 }\). The line \(l\) is parallel to the initial line, and \(l\) is the tangent to \(C\) at
above. above.
    1. Show that, for any point on \(C , r ^ { 2 } \sin ^ { 2 } \theta\) can be expressed in terms of \(\sin \theta\) and \(a\) only. (1)
    2. Hence, using differentiation, show that the polar coordinates of \(P\) are \(\left( \frac { a } { \sqrt { 2 } } , \frac { \pi } { 6 } \right)\).(6)
      \includegraphics[max width=\textwidth, alt={}, center]{2352f367-ddf9-4770-ace5-b561b0fbabbb-1_298_725_2163_1169} The shaded region \(R\), shown in the figure above, is bounded by \(C\), the line \(l\) and the half-line with equation
      \(\theta = \frac { \pi } { 2 }\).
  2. Show that the area of \(R\) is \(\frac { a ^ { 2 } } { 16 } ( 3 \sqrt { 3 } - 4 )\).
Edexcel FP2 2006 January Q5
5. Solve the equation \(z ^ { 5 } = \mathrm { i }\)
giving your answers in the form \(\cos \theta + \mathrm { i } \sin \theta\).
(Total 5 marks)
Edexcel FP2 2006 January Q7
7. $$( 1 + 2 x ) \frac { \mathrm { d } y } { \mathrm {~d} x } = x + 4 y ^ { 2 }$$
  1. Show that $$( 1 + 2 x ) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 1 + 2 ( 4 y - 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x }$$
  2. Differentiate equation 1 with respect to \(x\) to obtain an equation involving $$\frac { \mathrm { d } ^ { 3 } } { \mathrm {~d} x ^ { 3 } } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } , \frac { \mathrm {~d} y } { \mathrm {~d} x } , \quad x \text { and } y .$$ Given that \(y = \frac { 1 } { 2 }\) at \(x = 0\),
  3. find a series solution for \(y\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
    (6)(Total 11 marks)
Edexcel FP2 2006 January Q8
8. In the Argand diagram the point \(P\) represents the complex number \(z\). Given that arg \(\left( \frac { z - 2 \mathrm { i } } { z + 2 } \right) = \frac { \pi } { 2 }\),
  1. sketch the locus of \(P\),
  2. deduce the value of \(| \mathrm { z } + 1 - \mathrm { i } |\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is defined by $$w = \frac { 2 ( 1 + \mathrm { i } ) } { z + 2 } , \quad z \neq - 2$$
  3. Show that the locus of \(P\) in the \(z\)-plane is mapped to part of a straight line in the \(w\)-plane, and show this in an Argand diagram.
    (6)(Total 12 marks)
Edexcel FP2 2002 June Q1
  1. Find the set of values for which
$$| x - 1 | > 6 x - 1$$
Edexcel FP2 2002 June Q2
  1. Find the general solution of the differential equation \(t \frac { \mathrm {~d} v } { \mathrm {~d} t } - v = t , t > 0\) and hence show that the solution can be written in the form \(v = t ( \ln t + c )\), where \(c\) is an arbitrary cnst.
  2. This differential equation is used to model the motion of a particle which has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\). When \(t = 2\) the speed of the particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find, to 3 sf , the speed of the particle when \(t = 4\).
Edexcel FP2 2002 June Q4
4. The curve \(C\) has polar equation \(r = 3 a \cos \theta , - \frac { \pi } { 2 } \leq \frac { \pi } { 2 }\). The curve \(D\) has polar equation \(r = a ( 1 + \cos \theta ) , - \pi \leq \theta < \pi\). Given that \(a\) is a positive constant, (a) sketch, on the same diagram, the graphs of \(C\) and \(D\), indicating where each curve cuts the initial line. The graphs of \(C\) intersect at the pole \(O\) and at the points \(P\) and \(Q\).
(b) Find the polar coordinates of \(P\) and \(Q\).
(c) Use integration to find the exact area enclosed by the curve \(D\) and the lines \(\theta = 0\) and \(\theta = \frac { \pi } { 3 }\) The region \(R\) contains all points which lie outside \(D\) and inside \(C\).
Given that the value of the smaller area enclosed by the curve \(C\) and the line \(\theta = \frac { \pi } { 3 }\) is $$\frac { 3 a ^ { 2 } } { 16 } ( 2 \pi - 3 \sqrt { } 3 )$$ (d) show that the area of \(R\) is \(\pi a ^ { 2 }\).
Edexcel FP2 2002 June Q5
5. Using algebra, find the set of values of \(x\) for which \(2 x - 5 > \frac { 3 } { x }\).
Edexcel FP2 2002 June Q6
6. (a) Find the general solution of the differential equation $$\cos x \frac { \mathrm {~d} y } { \mathrm {~d} x } + ( \sin x ) y = \cos ^ { 3 } x$$ (b) Show that, for \(0 \leq x \leq 2 \pi\), there are two points on the \(x\)-axis through which all the solution curves for this differential equation pass.
(c) Sketch the graph, for \(0 \leq x \leq 2 \pi\), of the particular solution for which \(y = 0\) at \(x = 0\).
Edexcel FP2 2002 June Q7
7. (a) Find the general solution of the differential equation $$2 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 7 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 3 y = 3 t ^ { 2 } + 11 t$$ (b) Find the particular solution of this differential equation for which \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} t } = 1\) when \(t = 0\).
(c) For this particular solution, calculate the value of \(y\) when \(t = 1\).
Edexcel FP2 2002 June Q8
8. \section*{Figure 1} The curve \(C\) shown in Fig. 1 has polar equation $$r = a ( 3 + \sqrt { 5 } \cos \theta ) , \quad - \pi \leq \theta < \pi .$$ \includegraphics[max width=\textwidth, alt={}, center]{6d92bf8a-df0d-421c-8246-8160f5921ee6-2_460_792_1503_970}
  1. Find the polar coordinates of the points \(P\) and \(Q\) where the tangents to \(C\) are parallel to the initial line. (6) The curve \(C\) represents the perimeter of the surface of a swimming pool. The direct distance from \(P\) to \(Q\) is 20 m.
  2. Calculate the value of \(a\).
  3. Find the area of the surface of the pool. (6)
Edexcel FP2 2002 June Q9
9. (a) The point \(P\) represents a complex number \(z\) in an Argand diagram. Given that $$| z - 2 i | = 2 | z + i |$$
  1. find a cartesian equation for the locus of \(P\), simplifying your answer.
  2. sketch the locus of \(P\).
    (b) A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is a translation \(- 7 + 11\) i followed by an enlargement with centre the origin and scale factor 3 . Write down the transformation \(T\) in the form $$w = a z + b , \quad a , b \in \mathbb { C }$$
Edexcel FP2 2002 June Q10
10. $$y \frac { d ^ { 2 } y } { d x ^ { 2 } } + \left( \frac { d y } { d x } \right) ^ { 2 } + y = 0$$
  1. Find an expression for \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\). Given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) at \(x = 0\),
  2. find the series solution for \(y\), in ascending powers of \(x\), up to an including the term in \(x ^ { 3 }\).
  3. Comment on whether it would be sensible to use your series solution to give estimates for \(y\) at \(x = 0.2\) and at \(x = 50\).
Edexcel FP2 2003 June Q1
  1. (i) (a) On the same Argand diagram sketch the loci given by the following equations.
$$| z - 1 | = 1 , \quad , , \arg ( z + 1 ) = \frac { \pi } { 12 } , \quad , \arg ( z + 1 ) = \frac { \pi } { 2 }$$ (b) Shade on your diagram the region for which $$| z - 1 | \leq 1 \quad \text { and } \quad \frac { \pi } { 12 } \leq \arg ( z + 1 ) \leq \frac { \pi } { 2 }$$ (ii) (a) Show that the transformation \(\quad w = \frac { z - 1 } { z } , \quad z \neq 0\), $$\text { maps } | z - 1 | = 1 \text { in the } \boldsymbol { z } \text {-plane onto } | w | = | w - 1 | \text { in the } \boldsymbol { w } \text {-plane. }$$ The region \(| z - 1 | \leq 1\) in the \(z\)-plane is mapped onto the region \(T\) in the \(w\)-plane.
(b) Shade the region \(T\) on an Argand diagram.
Edexcel FP2 2003 June Q2
2. (a) Use de Moivre's theorem to show that $$\cos 5 \theta = 16 \cos ^ { 5 } \theta - 20 \cos ^ { 3 } \theta + 5 \cos \theta$$ (b) Hence find 3 distinct solutions of the equation \(16 x ^ { 5 } - 20 x ^ { 3 } + 5 x + 1 = 0\), giving your answers to 3 decimal places where appropriate.
Edexcel FP2 2003 June Q3
3. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - y ^ { 2 } , \quad y = 1 \text { at } x = 0 \text {. (I) }$$ (b) By differentiating (I) twice with respect to \(x\), show that $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } + 2 y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } - 2 = 0$$ (c) Hence, for (I), find the series solution for \(\boldsymbol { y }\) in ascending powers of \(\boldsymbol { x }\) up to and including the term in \(\boldsymbol { x } ^ { \mathbf { 3 } }\). (4)
Edexcel FP2 2003 June Q4
4. (a) Express as a simplified single fraction \(\frac { 1 } { ( r - 1 ) ^ { 2 } } - \frac { 1 } { r ^ { 2 } }\).
(b) Hence prove, by the method of differences, that \(\quad \sum _ { r = 2 } ^ { n } \frac { 2 r - 1 } { r ^ { 2 } ( r - 1 ) ^ { 2 } } = 1 - \frac { 1 } { n ^ { 2 } }\).
Edexcel FP2 2003 June Q5
5. Solve the inequality \(\frac { 1 } { 2 x + 1 } > \frac { x } { 3 x - 2 }\).
Edexcel FP2 2003 June Q6
6. (a) Using the substitution \(t = x ^ { 2 }\), or otherwise, find $$\int x ^ { 3 } \mathrm { e } ^ { - x ^ { 2 } } \mathrm {~d} x$$ (b) Find the general solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = x \mathrm { e } ^ { - x ^ { 2 } } , \quad x > 0$$
Edexcel FP2 2003 June Q7
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{141c7b1b-4236-4433-84af-04fa9baa3d96-2_568_1431_1637_258}
\end{figure} A logo is designed which consists of two overlapping closed curves. The polar equations of these curves are \(r = \boldsymbol { a } ( \mathbf { 3 } + \mathbf { 2 } \cos \boldsymbol { \theta } )\) and $$r = a ( 5 - 2 \cos \theta ) , \quad 0 \leq \theta < 2 \pi .$$ Figure 1 is a sketch (not to scale) of these two curves.
  1. Write down the polar corrdinates of the points \(A\) and \(B\) where the curves meet the initial line.(2)
  2. Find the polar coordinates of the points \(\boldsymbol { C }\) and \(\boldsymbol { D }\) where the two curves meet. (4)
  3. Show that the area of the overlapping region, which is shaded in the figure, is $$\frac { a ^ { 2 } } { 3 } ( 49 \pi - 48 \sqrt { } 3 )$$
Edexcel FP2 2003 June Q8
8. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 9 y = 4 \mathrm { e } ^ { 3 t } , \quad t \geq 0 .$$
  1. Show that \(K t ^ { 2 } e ^ { 3 t }\) is a particular integral of the differential equation, where \(K\) is a constant to be found.
  2. Find the general solution of the differential equation. (3) Given that a particular solution satisfies \(\boldsymbol { y } = 3\) and \(\frac { \mathrm { d } y } { \mathrm {~d} t } = 1\) when \(\boldsymbol { t } = \mathbf { 0 }\),
  3. find this solution.(4) Another particular solution which satisfies \(\boldsymbol { y } = \mathbf { 1 }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} t } = \mathbf { 0 }\) when \(\boldsymbol { t } = \mathbf { 0 }\), has equation $$y = \left( 1 - 3 t + 2 t ^ { 2 } \right) \mathrm { e } ^ { 3 t }$$
  4. For this particular solution draw a sketch graph of \(y\) against \(t\), showing where the graph crosses the \(t\)-axis. Determine also the coordinates of the minimum of the point on the sketch graph.
Edexcel FP2 2003 June Q9
9. $$z = 4 \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right) , \text { and } \boldsymbol { w } = 3 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$$ Express zw in the form \(r ( \cos \theta + \mathrm { i } \sin \theta ) , r > 0 , - \pi < \theta < \pi\).