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Edexcel P4 2018 Specimen Q6
4 marks Standard +0.3
6. Prove by contradiction that, if \(a , b\) are positive real numbers, then \(a + b \geqslant 2 \sqrt { a b }\) \includegraphics[max width=\textwidth, alt={}, center]{4de08317-5fb9-4789-8d57-ccf463224c78-20_2655_1943_114_118}
Edexcel P4 2018 Specimen Q7
5 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4de08317-5fb9-4789-8d57-ccf463224c78-21_664_1244_301_351} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the curve \(C\) with parametric equations $$x = 4 \cos \left( t + \frac { \pi } { 6 } \right) \quad y = 2 \sin t \quad 0 \leqslant t \leqslant 2 \pi$$
  1. Show that $$x + y = 2 \sqrt { 3 } \cos t$$
  2. Show that a cartesian equation of \(C\) is $$( x + y ) ^ { 2 } + a y ^ { 2 } = b$$ where \(a\) and \(b\) are integers to be found. \includegraphics[max width=\textwidth, alt={}, center]{4de08317-5fb9-4789-8d57-ccf463224c78-22_2673_1948_107_118}
Edexcel P4 2018 Specimen Q8
11 marks Standard +0.3
8. Water is being heated in a kettle. At time \(t\) seconds, the temperature of the water is \(\theta ^ { \circ } \mathrm { C }\). The rate of increase of the temperature of the water at time \(t\) is modelled by the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = \lambda ( 120 - \theta ) \quad \theta \leqslant 100$$ where \(\lambda\) is a positive constant.
Given that \(\theta = 20\) when \(t = 0\)
  1. solve this differential equation to show that $$\theta = 120 - 100 \mathrm { e } ^ { - \lambda t }$$ When the temperature of the water reaches \(100 ^ { \circ } \mathrm { C }\), the kettle switches off.
  2. Given that \(\lambda = 0.01\), find the time, to the nearest second, when the kettle switches off.
    \includegraphics[max width=\textwidth, alt={}]{4de08317-5fb9-4789-8d57-ccf463224c78-26_2642_1833_118_118}
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Edexcel P4 2018 Specimen Q9
15 marks Standard +0.3
  1. With respect to a fixed origin \(O\), the line \(l _ { 1 }\) is given by the equation
$$\mathbf { r } = \left( \begin{array} { r } 8 \\ 1 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { r } - 5 \\ 4 \\ 3 \end{array} \right)$$ where \(\mu\) is a scalar parameter.
The point \(A\) lies on \(l _ { 1 }\) where \(\mu = 1\)
  1. Find the coordinates of \(A\). The point \(P\) has position vector \(\left( \begin{array} { l } 1 \\ 5 \\ 2 \end{array} \right)\) The line \(l _ { 2 }\) passes through the point \(P\) and is parallel to the line \(l _ { 1 }\)
  2. Write down a vector equation for the line \(l _ { 2 }\)
  3. Find the exact value of the distance \(A P\). Give your answer in the form \(k \sqrt { 2 }\), where \(k\) is a constant to be found. The acute angle between \(A P\) and \(l _ { 2 }\) is \(\theta\)
  4. Find the value of \(\cos \theta\) A point \(E\) lies on the line \(l _ { 2 }\) Given that \(A P = P E\),
  5. find the area of triangle \(A P E\),
  6. find the coordinates of the two possible positions of \(E\).
Edexcel FP2 Q4
Standard +0.8
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
6 marks Moderate -0.3
Find the set of values of \(x\) for which \(\frac { x ^ { 2 } } { x - 2 } > 2 x\).
(Total 6 marks)
Edexcel FP2 2006 January Q3
14 marks Challenging +1.2
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
15 marks Challenging +1.8
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 marks Standard +0.3
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
11 marks Challenging +1.2
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
12 marks Challenging +1.2
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
5 marks Moderate -0.5
  1. Find the set of values for which
$$| x - 1 | > 6 x - 1$$
Edexcel FP2 2002 June Q2
10 marks Standard +0.3
  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
18 marks Challenging +1.8
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
7 marks Moderate -0.3
5. Using algebra, find the set of values of \(x\) for which \(2 x - 5 > \frac { 3 } { x }\).
Edexcel FP2 2002 June Q6
11 marks Standard +0.8
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
14 marks Standard +0.3
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
15 marks Challenging +1.2
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
7 marks Standard +0.3
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
12 marks Challenging +1.2
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
10 marks Standard +0.8
  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
10 marks Challenging +1.2
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
8 marks Challenging +1.2
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
5 marks Standard +0.8
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
6 marks Standard +0.3
5. Solve the inequality \(\frac { 1 } { 2 x + 1 } > \frac { x } { 3 x - 2 }\).