Questions FP2 (1157 questions)

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Edexcel FP2 2014 June Q5
  1. (a) Find the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 10 y = 27 \mathrm { e } ^ { - x }$$ (b) Find the particular solution that satisfies \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\)
Edexcel FP2 2014 June Q6
6. The transformation \(T\) from the \(z\)-plane, where \(z = x + \mathrm { i } y\), to the \(w\)-plane, where \(w = u + \mathrm { i } v\), is given by $$w = \frac { 4 ( 1 - \mathrm { i } ) z - 8 \mathrm { i } } { 2 ( - 1 + \mathrm { i } ) z - \mathrm { i } } , \quad z \neq \frac { 1 } { 4 } - \frac { 1 } { 4 } \mathrm { i }$$ The transformation \(T\) maps the points on the line \(l\) with equation \(y = x\) in the \(z\)-plane to a circle \(C\) in the \(w\)-plane.
  1. Show that $$w = \frac { a x ^ { 2 } + b x i + c } { 16 x ^ { 2 } + 1 }$$ where \(a\), \(b\) and \(c\) are real constants to be found.
  2. Hence show that the circle \(C\) has equation $$( u - 3 ) ^ { 2 } + v ^ { 2 } = k ^ { 2 }$$ where \(k\) is a constant to be found.
Edexcel FP2 2014 June Q7
7. (a) Show that the substitution \(v = y ^ { - 3 }\) transforms the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = 2 x ^ { 4 } y ^ { 4 }$$ into the differential equation $$\begin{aligned} & \frac { \mathrm { d } v } { \mathrm {~d} x } - \frac { 3 v } { x } = - 6 x ^ { 3 }
& \text { ration (II), find a general solution of differential equation (I) } \end{aligned}$$ in the form \(y ^ { 3 } = \mathrm { f } ( x )\).
Edexcel FP2 2014 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c21767d7-7331-47f7-8e59-06a0727c67c5-13_771_1036_260_593} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve \(C\) with polar equation $$r = 1 + \tan \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The tangent to the curve \(C\) at the point \(P\) is perpendicular to the initial line.
  1. Find the polar coordinates of the point \(P\). The point \(Q\) lies on the curve \(C\), where \(\theta = \frac { \pi } { 3 }\)
    The shaded region \(R\) is bounded by \(O P , O Q\) and the curve \(C\), as shown in Figure 1
  2. Find the exact area of \(R\), giving your answer in the form $$\frac { 1 } { 2 } ( \ln p + \sqrt { q } + r )$$ where \(p , q\) and \(r\) are integers to be found.
Edexcel FP2 2015 June Q1
  1. (a) Use algebra to find the set of values of \(x\) for which
$$x + 2 > \frac { 12 } { x + 3 }$$ (b) Hence, or otherwise, find the set of values of \(x\) for which $$x + 2 > \frac { 12 } { | x + 3 | }$$
Edexcel FP2 2015 June Q2
2. $$z = - 2 + ( 2 \sqrt { 3 } ) \mathrm { i }$$
  1. Find the modulus and the argument of \(z\). Using de Moivre's theorem,
  2. find \(z ^ { 6 }\), simplifying your answer,
  3. find the values of \(w\) such that \(w ^ { 4 } = z ^ { 3 }\), giving your answers in the form \(a + \mathrm { i } b\) where \(a , b \in \mathbb { R }\).
Edexcel FP2 2015 June Q3
  1. Find, in the form \(y = \mathrm { f } ( x )\), the general solution of the differential equation
$$\tan x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = 3 \cos 2 x \tan x , \quad 0 < x < \frac { \pi } { 2 }$$
Edexcel FP2 2015 June Q4
4. (a) Show that $$r ^ { 2 } ( r + 1 ) ^ { 2 } - ( r - 1 ) ^ { 2 } r ^ { 2 } \equiv 4 r ^ { 3 }$$ Given that \(\sum _ { r = 1 } ^ { n } r = \frac { 1 } { 2 } n ( n + 1 )\)
(b) use the identity in (a) and the method of differences to show that $$\left( 1 ^ { 3 } + 2 ^ { 3 } + 3 ^ { 3 } + \ldots + n ^ { 3 } \right) = ( 1 + 2 + 3 + \ldots + n ) ^ { 2 }$$
Edexcel FP2 2015 June Q5
  1. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by
$$w = \frac { z } { z + 3 \mathrm { i } } , \quad z \neq - 3 \mathrm { i }$$ The circle with equation \(| z | = 2\) is mapped by \(T\) onto the curve \(C\).
    1. Show that \(C\) is a circle.
    2. Find the centre and radius of \(C\). The region \(| z | \leqslant 2\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.
  2. Shade the region \(R\) on an Argand diagram.
Edexcel FP2 2015 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{49da3c56-ccd1-4599-95d8-d1395461bcca-11_451_1063_237_438} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve \(C\), shown in Figure 1, has polar equation $$r = 3 a ( 1 + \cos \theta ) , \quad 0 \leqslant \theta < \pi$$ The tangent to \(C\) at the point \(A\) is parallel to the initial line.
  1. Find the polar coordinates of \(A\). The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\), the initial line and the line \(O A\).
  2. Use calculus to find the area of the shaded region \(R\), giving your answer in the form \(a ^ { 2 } ( p \pi + q \sqrt { 3 } )\), where \(p\) and \(q\) are rational numbers.
Edexcel FP2 2015 June Q7
7. $$y = \tan ^ { 2 } x , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$$
  1. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 \sec ^ { 4 } x - 4 \sec ^ { 2 } x\)
  2. Hence show that \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = 8 \sec ^ { 2 } x \tan x \left( A \sec ^ { 2 } x + B \right)\), where \(A\) and \(B\) are constants to be found.
  3. Find the Taylor series expansion of \(\tan ^ { 2 } x\), in ascending powers of \(\left( x - \frac { \pi } { 3 } \right)\), up to and including the term in \(\left( x - \frac { \pi } { 3 } \right) ^ { 3 }\)
Edexcel FP2 2015 June Q8
  1. (a) Show that the transformation \(x = \mathrm { e } ^ { u }\) transforms the differential equation
$$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 7 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 16 y = 2 \ln x , \quad x > 0$$ into the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} u ^ { 2 } } - 8 \frac { \mathrm {~d} y } { \mathrm {~d} u } + 16 y = 2 u$$ (b) Find the general solution of the differential equation (II), expressing \(y\) as a function of \(u\).
(c) Hence obtain the general solution of the differential equation (I).
Edexcel FP2 2016 June Q1
  1. Use algebra to find the set of values of \(x\) for which
$$\frac { x } { x + 1 } < \frac { 2 } { x + 2 }$$
Edexcel FP2 2016 June Q2
2. (a) Show that, for \(r > 0\) $$r - 3 + \frac { 1 } { r + 1 } - \frac { 1 } { r + 2 } = \frac { r ^ { 3 } - 7 r - 5 } { ( r + 1 ) ( r + 2 ) }$$ (b) Hence prove, using the method of differences, that $$\sum _ { r = 1 } ^ { n } \frac { r ^ { 3 } - 7 r - 5 } { ( r + 1 ) ( r + 2 ) } = \frac { n \left( n ^ { 2 } + a n + b \right) } { 2 ( n + 2 ) }$$ where \(a\) and \(b\) are constants to be found.
Edexcel FP2 2016 June Q3
3. (a) Find the four roots of the equation \(z ^ { 4 } = 8 ( \sqrt { 3 } + \mathrm { i } )\) in the form \(z = r \mathrm { e } ^ { \mathrm { i } \theta }\)
(b) Show these roots on an Argand diagram.
Edexcel FP2 2016 June Q4
4. (i) $$p \frac { \mathrm {~d} x } { \mathrm {~d} t } + q x = r \quad \text { where } p , q \text { and } r \text { are constants }$$ Given that \(x = 0\) when \(t = 0\)
  1. find \(x\) in terms of \(t\)
  2. find the limiting value of \(x\) as \(t \rightarrow \infty\)
    (ii) $$\frac { \mathrm { d } y } { \mathrm {~d} \theta } + 2 y = \sin \theta$$ Given that \(y = 0\) when \(\theta = 0\), find \(y\) in terms of \(\theta\)
Edexcel FP2 2016 June Q5
5. (a) Use de Moivre's theorem to show that $$\sin ^ { 5 } \theta \equiv a \sin 5 \theta + b \sin 3 \theta + c \sin \theta$$ where \(a\), \(b\) and \(c\) are constants to be found.
(b) Hence show that \(\int _ { 0 } ^ { \frac { \pi } { 3 } } \sin ^ { 5 } \theta \mathrm {~d} \theta = \frac { 53 } { 480 }\)
VILM SIHI NITIIIUMI ON OC
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VALV SIHI NI JIIIM ION OO
Edexcel FP2 2016 June Q6
6. (a) Find the Taylor series expansion about \(\frac { \pi } { 4 }\) of \(\tan x\) in ascending powers of \(\left( x - \frac { \pi } { 4 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 4 } \right) ^ { 3 }\).
(b) Deduce that an approximation for \(\tan \frac { 5 \pi } { 12 }\) is \(1 + \frac { \pi } { 3 } + \frac { \pi ^ { 2 } } { 18 } + \frac { \pi ^ { 3 } } { 81 }\)
Edexcel FP2 2016 June Q7
7. (a) Show that the substitution \(x = e ^ { u }\) transforms the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = - x ^ { - 2 } , \quad x > 0$$ into the equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} u ^ { 2 } } - 3 \frac { \mathrm {~d} y } { \mathrm {~d} u } + 2 y = - \mathrm { e } ^ { - 2 u }$$ (b) Find the general solution of the differential equation (II).
(c) Hence obtain the general solution of the differential equation (I) giving your answer in the form \(y = \mathrm { f } ( x )\)
Edexcel FP2 2016 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{18620cc5-2377-480b-b815-63bfc6a9760a-15_618_942_255_584} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve \(C _ { 1 }\) with equation $$r = 7 \cos \theta , \quad - \frac { \pi } { 2 } < \theta \leqslant \frac { \pi } { 2 }$$ and the curve \(C _ { 2 }\) with equation $$r = 3 ( 1 + \cos \theta ) , \quad - \pi < \theta \leqslant \pi$$ are shown on Figure 1.
The curves \(C _ { 1 }\) and \(C _ { 2 }\) both pass through the pole and intersect at the point \(P\) and the point \(Q\).
  1. Find the polar coordinates of \(P\) and the polar coordinates of \(Q\). The regions enclosed by the curve \(C _ { 1 }\) and the curve \(C _ { 2 }\) overlap, and the common region \(R\) is shaded in Figure 1.
  2. Find the area of \(R\).
Edexcel FP2 2017 June Q1
  1. (a) Show that, for \(r > 0\)
$$\frac { 1 } { r ^ { 2 } } - \frac { 1 } { ( r + 1 ) ^ { 2 } } \equiv \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }$$ (b) Hence prove that, for \(n \in \mathbb { N }\) $$\sum _ { r = 1 } ^ { n } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } } = \frac { n ( n + 2 ) } { ( n + 1 ) ^ { 2 } }$$ (c) Show that, for \(n \in \mathbb { N } , n > 1\) $$\sum _ { r = n } ^ { 3 n } \frac { 6 r + 3 } { r ^ { 2 } ( r + 1 ) ^ { 2 } } = \frac { a n ^ { 2 } + b n + c } { n ^ { 2 } ( 3 n + 1 ) ^ { 2 } }$$ where \(a , b\) and \(c\) are constants to be found.
Edexcel FP2 2017 June Q2
2. Use algebra to find the set of values of \(x\) for which $$\frac { x - 2 } { 2 ( x + 2 ) } \leqslant \frac { 12 } { x ( x + 2 ) }$$ "
Edexcel FP2 2017 June Q3
3. Solve the equation $$z ^ { 3 } + 32 + 32 i \sqrt { 3 } = 0$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(r > 0\) and \(- \pi < \theta \leqslant \pi\)
Edexcel FP2 2017 June Q4
4. $$y = \ln \left( \frac { 1 } { 1 - 2 x } \right) , \quad | x | < \frac { 1 } { 2 }$$
  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. Hence, or otherwise, find the series expansion of \(\ln \left( \frac { 1 } { 1 - 2 x } \right)\) about \(x = 0\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient in its simplest form.
  3. Use your expansion to find an approximate value for \(\ln \left( \frac { 3 } { 2 } \right)\), giving your answer
    to 3 decimal places.
Edexcel FP2 2017 June Q5
5. (a) Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } = 26 \sin 3 x$$ (b) Find the particular solution of this differential equation for which \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)
when \(x = 0\) when \(x = 0\)