Questions FP2 - A-Level Maths

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.
Find the image under $T$ in the $w$-plane of the circle $| Z | = 1$ in the $z$-plane.
Sketch on separate diagrams the circle $| \mathbf { Z } | = 1$ in the $z$-plane and its image under $T$ in the $w$-plane.
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

Sketch the graph of $y = | x - 2 a |$, given that $a > 0$.
Solve $| x - 2 a | > 2 x + a$, where $a > 0$.
(3)(Total 5 marks)

Edexcel FP2 2005 June Q3

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

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 $\quad r = 3 \sec \left( \frac { \pi } { 3 } - \theta \right) , \quad - \frac { \pi } { 6 } < \theta < \frac { 5 \pi } { 6 }$.

Find a cartesian equation of $C$ and a cartesian equation of $D$.
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 $Q$.
Find the polar coordinates of $P$ and $Q$.
(5)(Total 13 marks)

Edexcel FP2 2005 June Q5

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

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)

Edexcel FP2 2005 June Q7

7. (a) Find the general solution of the differential equation $$2 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 5 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 x = 2 t + 9$$ (b) Find the particular solution of this differential equation for which $x = 3$ and $\frac { \mathrm { d } x } { \mathrm {~d} t } = - 1$ when $t = 0$. The particular solution in part (b) is used to model the motion of a particle $P$ on the $x$-axis. At time $t$ seconds $( t \geq 0 ) , P$ is $x$ metres from the origin $O$.
(c) Show that the minimum distance between $O$ and $P$ is $\frac { 1 } { 2 } ( 5 + \ln 2 ) \mathrm { m }$ and justify that the distance is a minimum.
(4)(Total 14 marks)

Edexcel FP2 2005 June Q8

8. The curve $C$ which passes through $O$ has polar equation $$r = 4 a ( 1 + \cos \theta ) , \quad - \pi < \theta \leq \pi .$$ The line $l$ has polar equation $$r = 3 a \sec \theta , \quad - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 } .$$ The line $l$ cuts $C$ at the points $P$ and $Q$, as shown in the diagram.

Prove that $P Q = 6 \sqrt { } 3 a$. The region $R$, shown shaded in the diagram, is bounded by $l$ and $C$.
Use calculus to find the exact area of $R$.
\includegraphics[max width=\textwidth, alt={}, center]{d9aa1f75-ef35-4bf0-85c2-dff36872ca46-2_714_778_1959_1153}

Edexcel FP2 2005 June Q9

9. A complex number $z$ is represented by the point $P$ in the Argand diagram. Given that $$| z - 3 \mathrm { i } | = 3$$

sketch the locus of $P$.
Find the complex number $z$ which satisfies both $| z - 3 i | = 3$ and $\arg ( z - 3 i ) = \frac { 3 } { 4 } \pi$. The transformation $T$ from the $z$-plane to the $w$-plane is given by $$w = \frac { 2 \mathrm { i } } { z }$$
Show that $T$ maps $| z - 3 i | = 3$ to a line in the $w$-plane, and give the cartesian equation of this line.
(5)(Total 11 marks)

Edexcel FP2 2005 June Q10

10. (a) Given that $z = e ^ { \mathrm { i } \theta }$, show that $$z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$ where $n$ is a positive integer.
(b) Show that $$\sin ^ { 5 } \theta = \frac { 1 } { 16 } ( \sin 5 \theta - 5 \sin 3 \theta + 10 \sin \theta )$$ (c) Hence solve, in the interval $0 \leq \theta < 2 \pi$, $$\sin 5 \theta - 5 \sin 3 \theta + 6 \sin \theta = 0$$ (5)(Total 12 marks)

Edexcel FP2 2005 June Q11

11. The variable $y$ satisfies the differential equation $$4 \left( 1 + x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = y$$ At $x = 0 , y = 1$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 }$.

Find the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at $x = 0$.
(1) (c) Find the value of $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$ at $x = 0$
Express $y$ as a series, in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.
Find the value that the series gives for $y$ at $x = 0.1$, giving your answer to 5 decimal places.
(1)(Total 14 marks)

Edexcel FP2 2006 June Q1

Given that $3 x \sin 2 x$ is a particular integral of the differential equation

$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = k \cos 2 x$$ where $k$ is a constant,

calculate the value of $k$,
find the particular solution of the differential equation for which at $x = 0 , y = 2$, and for which at $x = \frac { \pi } { 4 } , y = \frac { \pi } { 2 }$.
(4)(Total 8 marks)

Edexcel FP2 2006 June Q2

2. Given that for all real values of $r , \quad ( 2 r + 1 ) ^ { 3 } - ( 2 r - 1 ) ^ { 3 } = A r ^ { 2 } + B$, where $A$ and $B$ are constants,

find the value of $A$ and the value of $B$.
Hence, or otherwise, prove that $\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$.
Calculate $\sum _ { r = 1 } ^ { 40 } ( 3 r - 1 ) ^ { 2 }$.
(3)(Total 10 marks)

Edexcel FP2 2006 June Q3

3. (a) Use algebra to find the exact solutions of the equation $$\left| 2 x ^ { 2 } + x - 6 \right| = 6 - 3 x$$ (b) On the same diagram, sketch the curve with equation $y = \left| 2 x ^ { 2 } + x - 6 \right|$ and the line with equation $y = 6 - 3 x$.
(c) Find the set of values of $x$ for which $$\left| 2 x ^ { 2 } + x - 6 \right| > 6 - 3 x$$ (3)(Total 12 marks)

Edexcel FP2 2006 June Q4

4. During an industrial process, the mass of salt, $S \mathrm {~kg}$, dissolved in a liquid $t$ minutes after the process begins is modelled by the differential equation $$\frac { \mathrm { d } S } { \mathrm {~d} t } + \frac { 2 S } { 120 - t } = \frac { 1 } { 4 } , \quad 0 \leq t < 120$$ Given that $S = 6$ when $t = 0$,

find $S$ in terms of $t$,
calculate the maximum mass of salt that the model predicts will be dissolved in the liquid at any one time during the process.
(4)(Total 12 marks)

Edexcel FP2 2006 June Q5

5. (a) Find the Taylor expansion of $\cos 2 x$ in ascending powers of $\left( x - \frac { \pi } { 4 } \right)$ up to and including the term in $\left( x - \frac { \pi } { 4 } \right) ^ { 5 }$.
(b) Use your answer to (a) to obtain an estimate of $\cos 2$, giving your answer to 6 decimal places.
(3)(Total 8 marks)

Edexcel FP2 2006 June Q6

6. (a) Use de Moivre's theorem to show that $\boldsymbol { \operatorname { s i n } } 5 \boldsymbol { \theta } = \boldsymbol { \operatorname { s i n } } \boldsymbol { \theta } \left( \mathbf { 1 6 } \mathbf { c o s } ^ { 4 } \boldsymbol { \theta } - \mathbf { 1 2 } \boldsymbol { \operatorname { c o s } } ^ { 2 } \boldsymbol { \theta } + \mathbf { 1 } \right)$.
(b) Hence, or otherwise, solve, for $0 \leq \theta < \pi$ $$\sin 5 \theta + \cos \theta \sin 2 \theta = 0$$ (6)(Total 11 marks)

Edexcel FP2 2006 June Q7

7. $$\frac { \mathrm { d } ^ { 2 x } } { \mathrm {~d} t ^ { 2 } } + 3 \sin x = 0 . \quad \text { At } t = 0 , \quad x = 0 \quad \text { and } \quad \frac { \mathrm { d } x } { \mathrm {~d} t } = 0.4$$ (b) Find a series solution for $x$, in ascending powers of $t$, up to and including the term in $t ^ { 3 }$.
(c) Use your answer to (b) to obtain an estimate of $x$ at $t = 0.3$.
(2)(Total 11 marks)

Edexcel FP2 2006 June Q8

8. The point $P$ represents a complex number $z$ on an Argand diagram, where $$| z - 6 + 3 i | = 3 | z + 2 - i |$$

Show that the locus of $P$ is a circle, giving the coordinates of the centre and the radius of this circle. The point $Q$ represents a complex number $z$ on an Argand diagram, where $$\tan [ \arg ( z + 6 ) ] = \frac { 1 } { 2 }$$
On the same Argand diagram, sketch the locus of $P$ and the locus of $Q$.
On your diagram, shade the region which satisfies both $$| z - 6 + 3 \mathrm { i } | > 3 | z + 2 - \mathrm { i } | \text { and } \tan [ \arg ( z + 6 ) ] > \frac { 1 } { 2 }$$ (2)(Total 14 marks)

Edexcel FP2 2007 June Q1

Obtain the general solution of the differential equation

$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = \cos x , \quad x > 0$$ giving your answer in the form $y = \mathrm { f } ( x )$.
(Total 8 marks)

Edexcel FP2 2007 June Q2

2.
\includegraphics[max width=\textwidth, alt={}, center]{d6befd60-de40-41b6-8ae5-48656dbca40c-1_734_1228_888_479} The diagram above shows a sketch of the curve with equation $$y = \frac { x ^ { 2 } - 1 } { | x + 2 | } , \quad x \neq - 2$$ The curve crosses the $x$-axis at $x = 1$ and $x = - 1$ and the line $x = - 2$ is an asymptote of the curve.

Use algebra to solve the equation $\frac { x ^ { 2 } - 1 } { | x + 2 | } = 3 ( 1 - x )$.
Hence, or otherwise, find the set of values of $x$ for which $$\frac { x ^ { 2 } - 1 } { | x + 2 | } < 3 ( 1 - x )$$ (Total 9 marks)

Edexcel FP2 2007 June Q3

3. A scientist is modelling the amount of a chemical in the human bloodstream. The amount $x$ of the chemical, measured in $\mathrm { mg } l ^ { - 1 }$, at time $t$ hours satisfies the differential equation $$2 x \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 6 \left( \frac { \mathrm { dx } } { \mathrm { dt } } \right) ^ { 2 } = x ^ { 2 } - 3 x ^ { 4 } , \quad x > 0$$

Show that the substitution $\mathrm { y } = \frac { 1 } { x ^ { 2 } }$ transforms this differential equation into $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + y = 3$$
Find the general solution of differential equation $I$. Given that at time $t = 0 , x = \frac { 1 } { 2 }$ and $\frac { \mathrm { d } x } { \mathrm {~d} t } = 0$,
find an expression for $x$ in terms of $t$,
write down the maximum value of $x$ as $t$ varies.

Edexcel FP2 2007 June Q4

4.
\includegraphics[max width=\textwidth, alt={}, center]{d6befd60-de40-41b6-8ae5-48656dbca40c-3_535_1027_276_577} The diagram above shows a sketch of the curve $C$ with polar equation $$r = 4 \sin \theta \cos ^ { 2 } \theta , \quad 0 \leq \theta < \frac { \pi } { 2 }$$ The tangent to $C$ at the point $P$ is perpendicular to the initial line.

Show that $P$ has polar coordinates $\left( \frac { 3 } { 2 } , \frac { \pi } { 6 } \right)$. The point $Q$ on $C$ has polar coordinates $\left( \sqrt { 2 } , \frac { \pi } { 4 } \right)$.
The shaded region $R$ is bounded by $O P , O Q$ and $C$, as shown in the diagram above.
Show that the area of $R$ is given by $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } \left( \sin ^ { 2 } 2 \theta \cos 2 \theta + \frac { 1 } { 2 } - \frac { 1 } { 2 } \cos 4 \theta \right) \mathrm { d } \theta$$
Hence, or otherwise, find the area of $R$, giving your answer in the form $a + b \pi$, where $a$ and $b$ are rational numbers.
(Total 14 marks)

Edexcel FP2 2007 June Q5

5. Find the set of values of $x$ for which $$\frac { x + 1 } { 2 x - 3 } < \frac { 1 } { x - 3 }$$

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