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Edexcel FP2 2003 June Q6
10 marks Standard +0.8
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
14 marks Challenging +1.2
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
16 marks Challenging +1.2
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
3 marks Moderate -0.8
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 \(r ( \cos \theta + \mathrm { i } \sin \theta ) , r > 0 , - \pi < \theta < \pi\).
Edexcel FP2 2003 June Q10
5 marks Moderate -0.3
10. (a) Sketch, on the same axes, the graphs with equation \(y = | 2 x - 3 |\), and the line with equation \(y = 5 x - 1\).
(b) Solve the inequality \(| 2 x - 3 | < 5 x - 1\).
Edexcel FP2 2003 June Q11
7 marks Standard +0.8
11. (a) Express \(\frac { 2 } { ( r + 1 ) ( r + 3 ) }\) in partial fractions.
(b) Hence prove that \(\sum _ { r = 1 } ^ { n } \frac { 2 } { ( r + 1 ) ( r + 3 ) } \equiv \frac { n ( 5 n + 13 ) } { 6 ( n + 2 ) ( n + 3 ) }\).
Edexcel FP2 2003 June Q12
10 marks Standard +0.8
12. (a) Use the substitution \(y = v x\) to transform the equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 4 x + y ) ( x + y ) } { x ^ { 2 } } , x > 0$$ into the equation $$x \frac { \mathrm {~d} v } { \mathrm {~d} x } = ( 2 + v ) ^ { 2 }$$ (b) Solve the differential equation II to find \(\boldsymbol { v }\) as a function of \(\boldsymbol { x }\) (c) Hence show that \(\quad y = - 2 x - \frac { x } { \ln x + c }\), where \(c\) is an arbitrary constant, is a general solution of the differential equation I.
Edexcel FP2 2003 June Q13
11 marks Standard +0.3
13. Given that \(z = 3 - 3 i\) express, in the form \(a + i b\), where \(a\) and \(b\) are real numbers,
  1. \(z ^ { 2 }\),
    (2)
  2. \(\frac { 1 } { z }\).
    (2)
  3. Find the exact value of each of \(| z | , \left| z ^ { 2 } \right|\) and \(\left| \frac { 1 } { z } \right|\).
    (2) The complex numbers \(z , z ^ { 2 }\) and \(\frac { 1 } { z }\) are represented by the points \(A , B\) and \(C\) respectively on an Argand diagram. The real number 1 is represented by the point \(D\), and \(O\) is the origin.
  4. Show the points \(A , B , C\) and \(D\) on an Argand diagram.
  5. Prove that \(\triangle O A B\) is similar to \(\triangle O C D\).
Edexcel FP2 2003 June Q14
14 marks Standard +0.3
14. (a) Find the value of \(\lambda\) for which \(\lambda x \cos 3 x\) is a particular integral of the differential equation $$\frac { d ^ { 2 } y } { d x ^ { 2 } } + 9 y = - 12 \sin 3 x$$ (b) Hence find the general solution of this differential equation.(4) The particular solution of the differential equation for which \(\boldsymbol { y } = \mathbf { 1 }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathbf { 2 }\) at \(\boldsymbol { x } = \mathbf { 0 }\), is \(\boldsymbol { y } = \mathbf { g } ( \boldsymbol { x } )\).
(c) Find \(\mathrm { g } ( x )\).
(d) Sketch the graph of \(y = g ( x ) , 0 \leq x \leq \pi\).
(2) \section*{15.} \section*{Figure 1} Figure 1 shows a sketch of the cardioid \(C\) with equation \(r = a ( 1 + \cos \theta ) , - \pi < \theta \leq \pi\). Also shown are the tangents to \(C\) that are parallel and perpendicular to the initial line. These tangents form a rectangle WXYZ. \includegraphics[max width=\textwidth, alt={}, center]{141c7b1b-4236-4433-84af-04fa9baa3d96-5_407_782_315_1142}
(a) Find the area of the finite region, shaded in Fig. 1, bounded by the curve \(C\).
(b) Find the polar coordinates of the points \(A\) and \(B\) where \(W Z\) touches the curve \(C\).
(c) Hence find the length of \(W X\). Given that the length of \(\boldsymbol { W } \boldsymbol { Z }\) is \(\frac { 3 \sqrt { 3 } a } { 2 }\),
(d) find the area of the rectangle \(W X Y Z\). A heart-shape is modelled by the cardioid \(C\), where \(\boldsymbol { a } = \mathbf { 1 0 ~ c m }\). The heart shape is cut from the rectangular card WXYZ, shown in Fig. 1.
(e) Find a numerical value for the area of card wasted in making this heart shape.
8. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is defined by $$w = \frac { z + 1 } { i z - 1 } , \quad z \neq - i$$ where \(z = x + \mathrm { i } y , w = u + \mathrm { i } v\) and \(x , y , u\) and \(v\) are real. \(T\) transforms the circle \(| z | = 1\) in the \(z\)-plane onto a straight line \(L\) in the \(w\)-plane.
(a) Find an equation of \(L\) giving your answer in terms of \(u\) and \(v\).
(b) Show that \(T\) transforms the line \(\operatorname { Im } z = 0\) in the \(z\)-plane onto a circle \(C\) in the \(w\)-plane, giving the centre and radius of this circle.
(c) On a single Argand diagram sketch \(L\) and \(C\). Question: Solve $$x ^ { 5 } = - ( 9 \sqrt { 3 } ) i$$
Edexcel FP2 2004 June Q1
8 marks Standard +0.3
  1. Show that \(( r + 1 ) ^ { 3 } - ( r - 1 ) ^ { 3 } \equiv A r ^ { 2 } + B\), where \(A\) and \(B\) are constants to be found.
  2. Prove by the method of differences that \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 ) , n > 1\).
    (6)(Total 8 marks)
Edexcel FP2 2004 June Q3
11 marks Standard +0.3
3. (a) Sketch, on the same axes, the graph of \(y = | ( x - 2 ) ( x - 4 ) |\), and the line with equation \(y = 6 - 2 x\).
(b) Find the exact values of \(x\) for which \(| ( x - 2 ) ( x - 4 ) | = 6 - 2 x\).
(c) Hence solve the inequality \(| ( x - 2 ) ( x - 4 ) | < 6 - 2 x\).
(2)(Total 11 marks)
Edexcel FP2 2004 June Q4
12 marks Challenging +1.2
4. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 65 \sin 2 x , x > 0$$
  1. Find the general solution of the differential equation.
  2. Show that for large values of \(x\) this general solution may be approximated by a sine function and find this sine function.
    (3)(Total 12 marks)
Edexcel FP2 2004 June Q5
16 marks Challenging +1.8
5. (a) Sketch the curve with polar equation \(\quad r = 3 \cos 2 \theta , \quad - \frac { \pi } { 4 } \leq \theta < \frac { \pi } { 4 }\) (b) Find the area of the smaller finite region enclosed between the curve and the half-line $$\theta = \frac { \pi } { 6 }$$ (c) Find the exact distance between the two tangents which are parallel to the initial line.
(8)(Total 16 marks)
Edexcel FP2 2004 June Q6
7 marks Standard +0.8
6. Find the complete set of values of \(x\) for which $$\left| x ^ { 2 } - 2 \right| > 2 x$$
Edexcel FP2 2004 June Q7
11 marks Standard +0.3
  1. (a) Find the general solution of the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y = x$$ Given that \(y = 1\) at \(x = 0\),
(b) find the exact values of the coordinates of the minimum point of the particular solution curve,
(c) draw a sketch of this particular solution curve.
Edexcel FP2 2004 June Q8
12 marks Standard +0.8
8. (a) Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 2 y = 2 \mathrm { e } ^ { - t }$$ (b) Find the particular solution that satisfies \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} t } = 1\) at \(t = 0\).
(6)(Total 12 marks)
Edexcel FP2 2004 June Q9
16 marks Challenging +1.3
9. The diagram is a sketch of the two curves \(C _ { 1 }\) and \(C _ { 2 }\) with polar equations \(C _ { 1 } : r = 3 a ( 1 - \cos \theta ) , - \pi \leq \theta < \pi\) \(\mathrm { C } _ { 2 } : r = a ( 1 + \cos \theta ) , - \pi \leq \theta < \pi\). \includegraphics[max width=\textwidth, alt={}, center]{8646b60a-3822-4d41-8978-1ccad1e216d6-2_318_776_1567_1082} The curves meet at the pole \(O\), and at the points \(A\) and \(B\).
  1. Find, in terms of \(a\), the polar coordinates of the points \(A\) and \(B\).
  2. Show that the length of the line \(A B\) is \(\frac { 3 \sqrt { } 3 } { 2 } a\). The region inside \(C _ { 2 }\) and outside \(C _ { 1 }\) is shown shaded in the diagram above.
  3. Find, in terms of \(a\), the area of this region. A badge is designed which has the shape of the shaded region.
    Given that the length of the line \(A B\) is 4.5 cm ,
  4. calculate the area of this badge, giving your answer to three significant figures.
    (Total 16 marks)
Edexcel FP2 2004 June Q10
8 marks Standard +0.8
10. Given that \(y = \tan x\),
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\).
  2. Find the Taylor series expansion of \(\tan x\) in ascending powers of \(\left( x - \frac { \pi } { 4 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 4 } \right) ^ { 3 }\).
  3. Hence show that \(\tan \frac { 3 \pi } { 10 } \approx 1 + \frac { \pi } { 10 } + \frac { \pi ^ { 2 } } { 200 } + \frac { \pi ^ { 3 } } { 3000 }\).
Edexcel FP2 2004 June Q11
11 marks Standard +0.3
11. (b) Hence find the Maclaurin series expansion of \(\mathrm { e } ^ { x } \cos x\), in ascending powers of \(x\), up to and including the term in \(x ^ { 4 }\).
(Total 11 marks)
Edexcel FP2 2004 June Q12
14 marks Challenging +1.2
12. The transformation \(T\) from the complex \(z\)-plane to the complex \(w\)-plane is given by $$w = \frac { z + 1 } { z + \mathrm { i } } , \quad z \neq - \mathrm { i }$$
  1. Show that \(T\) maps points on the half-line \(\arg ( z ) = \frac { \pi } { 4 }\) in the \(z\)-plane into points on the circle \(| w | = 1\) in the \(w\)-plane.
  2. Find the image under \(T\) in the \(w\)-plane of the circle \(| Z | = 1\) in the \(z\)-plane.
  3. Sketch on separate diagrams the circle \(| \mathbf { Z } | = 1\) in the \(z\)-plane and its image under \(T\) in the \(w\)-plane.
  4. Mark on your sketches the point \(P\), where \(z = \mathrm { i }\), and its image \(Q\) under \(T\) in the \(w\)-plane.
Edexcel FP2 2005 June Q1
5 marks Standard +0.8
  1. Sketch the graph of \(y = | x - 2 a |\), given that \(a > 0\).
  2. Solve \(| x - 2 a | > 2 x + a\), where \(a > 0\).
    (3)(Total 5 marks)
Edexcel FP2 2005 June Q3
12 marks Challenging +1.2
3. (a) Show that the transformation \(y = x v\) transforms the equation $$\begin{array} { l l } x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + \left( 2 + 9 x ^ { 2 } \right) y = x ^ { 5 } , \\ \text { into the equation } & \frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 9 v = x ^ { 2 } . \end{array}$$I (b) Solve the differential equation II to find \(v\) as a function of \(x\).
(c) Hence state the general solution of the differential equation I.
(1)(Total 12 marks)
Edexcel FP2 2005 June Q4
13 marks Standard +0.8
4. The curve \(C\) has polar equation \(\quad r = 6 \cos \theta , \quad - \frac { \pi } { 2 } \leq \theta < \frac { \pi } { 2 }\), and the line \(D\) has polar equation \(\quad r = 3 \sec \left( \frac { \pi } { 3 } - \theta \right) , \quad - \frac { \pi } { 6 } < \theta < \frac { 5 \pi } { 6 }\).
  1. Find a cartesian equation of \(C\) and a cartesian equation of \(D\).
  2. Sketch on the same diagram the graphs of \(C\) and \(D\), indicating where each cuts the initial line. The graphs of \(C\) and \(D\) intersect at the points \(P\) and \(Q\).
  3. Find the polar coordinates of \(P\) and \(Q\).
    (5)(Total 13 marks)
Edexcel FP2 2005 June Q5
7 marks Standard +0.8
5. Find the general solution of the differential equation $$( x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y = \frac { 1 } { x } , \quad x > 0 .$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
(7)(Total 7 marks)
Edexcel FP2 2005 June Q6
12 marks Standard +0.3
6. (a) On the same diagram, sketch the graphs of \(y = \left| x ^ { 2 } - 4 \right|\) and \(y = | 2 x - 1 |\), showing the coordinates of the points where the graphs meet the axes.
(b) Solve \(\left| x ^ { 2 } - 4 \right| = | 2 x - 1 |\), giving your answers in surd form where appropriate.
(c) Hence, or otherwise, find the set of values of \(x\) for which \(\left| x ^ { 2 } - 4 \right| > | 2 x - 1 |\).
(3)(Total 12 marks)