Questions FP2 - A-Level Maths

Edexcel FP2 2002 June Q9

9. (a) The point $P$ represents a complex number $z$ in an Argand diagram. Given that $$| z - 2 i | = 2 | z + i |$$

find a cartesian equation for the locus of $P$, simplifying your answer.
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. $$y \frac { d ^ { 2 } y } { d x ^ { 2 } } + \left( \frac { d y } { d x } \right) ^ { 2 } + y = 0$$

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 $x = 0$,
find the series solution for $y$, in ascending powers of $x$, up to an including the term in $x ^ { 3 }$.
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

(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.

Write down the polar corrdinates of the points $A$ and $B$ where the curves meet the initial line.(2)
Find the polar coordinates of the points $\boldsymbol { C }$ and $\boldsymbol { D }$ where the two curves meet. (4)
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. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 9 y = 4 \mathrm { e } ^ { 3 t } , \quad t \geq 0 .$$

Show that $K t ^ { 2 } e ^ { 3 t }$ is a particular integral of the differential equation, where $K$ is a constant to be found.
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 $\boldsymbol { t } = \mathbf { 0 }$,
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 $$y = \left( 1 - 3 t + 2 t ^ { 2 } \right) \mathrm { e } ^ { 3 t }$$
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 $r ( \cos \theta + \mathrm { i } \sin \theta ) , r > 0 , - \pi < \theta < \pi$.

Edexcel FP2 2003 June Q10

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

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

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

13. Given that $z = 3 - 3 i$ express, in the form $a + i b$, where $a$ and $b$ are real numbers,

$z ^ { 2 }$,
(2)
$\frac { 1 } { z }$.
(2)
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.
Show the points $A , B , C$ and $D$ on an Argand diagram.
Prove that $\triangle O A B$ is similar to $\triangle O C D$.

Edexcel FP2 2003 June Q14

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

Show that $( r + 1 ) ^ { 3 } - ( r - 1 ) ^ { 3 } \equiv A r ^ { 2 } + B$, where $A$ and $B$ are constants to be found.
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

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

4. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 65 \sin 2 x , x > 0$$

Find the general solution of the differential equation.
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

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

6. Find the complete set of values of $x$ for which $$\left| x ^ { 2 } - 2 \right| > 2 x$$

Edexcel FP2 2004 June Q7

(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

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

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 $B$.

Find, in terms of $a$, the polar coordinates of the points $A$ and $B$.
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.
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 ,
calculate the area of this badge, giving your answer to three significant figures.
(Total 16 marks)

Edexcel FP2 2004 June Q10

10. Given that $y = \tan x$,

find $\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ and $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$.
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 $\left( x - \frac { \pi } { 4 } \right) ^ { 3 }$.
Hence show that $\tan \frac { 3 \pi } { 10 } \approx 1 + \frac { \pi } { 10 } + \frac { \pi ^ { 2 } } { 200 } + \frac { \pi ^ { 3 } } { 3000 }$.

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