Questions FP2 - A-Level Maths

tat t d at t cll $$\begin{array} { c c } \pi & \theta
\hline \pi & \theta \end{array}$$
G tat $t$ magtud $t$ mul xcd by dug $t$ cll - $d t$ alu

CAIE FP2 2014 June Q2

2 Aatcl ma kgmaacaccltct adadu mt At tm tatcl at tt At tm cd agl tat tadal cmt $t$ acclat at tm cd a magtud $\quad \pi \quad \theta m$ Fd

talu ta cmt t acclat t ual t
$t$ magtud $t$ ultat $c$ actg

CAIE FP2 2014 June Q3

3 A atcl attacd $t d$ a lgt xtbl tg lgt $\quad T t d$ t tg attacd t a xd t t atcl agg ulbum t $t$ tg tcal $t$ ga tal mul magtud t tg mak a agl t t dad tcal at $\quad$ t $t$

tat - $\pi \quad c \quad \theta$
It g tat $\quad { } _ { \pi } ^ { \alpha } \quad \theta \quad$ a t ctat Jut $\quad$ yg yu a dcb tmt ac tllg ca
(a)
(b)

CAIE FP2 2014 June Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{e2ff0097-b2a1-4901-b880-4ef4505c9cbe-3_529_606_260_767}

A um d	ad lgt	T d t ulbum a mt g		$t$
$t d \quad$ tg $a u g t$	al la $T$ dtac		ad tagl bt		ad
t tal	- A atcl ma	- attacd t t d at		dagam
Fd t mal act at	ad dduc tat	-
$T$ cct ct bt td ad t la		tat

CAIE FP2 2014 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{e2ff0097-b2a1-4901-b880-4ef4505c9cbe-3_543_704_1338_708}

A um ctagula lama	$c$	ad		a ma		A um
ccula lama adu - a ma	-	T t lama a xd tgt t am la
t t ct ccdg at t t	dagam A atcl		ma -		- attacd at
T ytm t tat abut a xd mt tal ax tug					ad dcula $t$ t
la	$t$ ytm abut $t$ ax				$\_\_\_\_$
						$\mu \geqslant \leqslant$
T ytm lad m $t t$	tal ad	$b l$	Fd $t m$			$- t$
gatt agula d tub	ut mt gg t alu		cct $t$ dcmal lac

CAIE FP2 2014 June Q6

6 Emly at a atcula cmay a $b$ ty $t$ duc abc $t$ cmay dcd $t t$
$t u$ ac day at ay tm bt am ad m F a adm aml mly
$t u m b u a b c t y a b$
a $g$ t llg tabl
$k g u a c$ daym am $t m T$
duc xtm ad all mly $t k$
ad $t$ ya at $t$ tduct xtm

Emly
$B$
$A t$

a ad aml tt t tt at t gcac ll t t ulat ma umb
$u$ abc a dcad ll g t tduct xtm

CAIE FP2 2014 June Q7

7 Jam t a dcu atdly
a attmt $t$ ac a uccul $t A t$ cutd a
uccul $t$ dtac acd mt $F$ ac $t$ t bablty tat Jam
uccul - ddtly all $t$ t Fd t bablty tat Jam tak

xactly $t$ tact $t$ uccul $t$
$m$ ta $t$ tac $t$ tuccul $t$ I $d t$ ualy a cmtt a dcut mut $t$ mt $t$ at mt $x$
attmt a uccul $t$ acd ut $t$ a tak Fd t bablty tat
Jam ual t cmtt Cl at dcut Fact $\quad t$ bablty tat ll ac a $t$
mt - ddtly all t t Fd t bablty tat xactly Jam ad
Cl ual t cmtt

CAIE FP2 2014 June Q8

8 A adm aml tak $m$ t adult ul d ty ca Tult a
at a tad clad by aggu ad
$g$ t llg tabl

	Hatcback	Etat	Ctbl
d ya
Bt ad ya
O ya

Tt at t gcac ll $t$
d ty ca d
dt aggu

CAIE FP2 2014 June Q9

9 T ctuu adm aabl
a dtbut uct $F g$ by
Z
$F \pi \theta$ Fd talu
c $P$
$\pi$
$\theta$ T adm aabl dd by $l \quad$ Fd t dtbut uct
Fd t bablty dty uct
ad ktc t ga Aumg tat lgt a mally dtbutd

tt at $t$ gcac ll $t$ t ulat ma lgt $t c$ gat ta cm

calculat a cdc tal $t$ ulat ma lgt $t c$ EITHE

A atcl			gt latc tg atual lgt	\multirow{4}{*}{}

T atcl		$t$ ad ac $t$ gatt $g t$	\multirow{4}{*}{}
t lgt t tg
(i) tat $t$ mdulu latcty $t$ tg	-
(ii) tat $m m l$ amc mt abut $t$ ulbum $t$ ad tat $t$ d t mt
(iii)		$t$ ual t al t maxmum alu
O
$F$ a adm aml $b$ ad t uat $t g l$	at a alu	θt uat $t g l$
ad
ctly aml	a ctat $T$ duct mmt $c$		lat cct $t$
(i) Tt at $t$ gcac ll $t$ t dc $t$ clat bt $t$ aabl
(ii)	d talu
(iii)	alu taml data d t alu			ad ktc $t$
$F$ ac $t$ a alu	$\pi$		cdd
(iv)	t uat $\operatorname { tg } l$	utyg yu a	ad d talu $t$

B A PAGE B A PAGE B A PAGE

CAIE FP2 2013 November Q8

8 Te ananye neet mnent tenm e unt $n$ en $y$ $$\left\{ { } ^ { t } \right. \text { tyen ty }$$ te e
ee $n \quad e$ tentnt
i $t t$
$t$ nnttu nmy en mnent e tnte tye $\{$ $\underset { n m m } { n m m e } \quad e \quad t n \quad$ e $\quad \lambda \quad$ teeut $n$ teee n ne \section*{n} $e$ et ey
i nte ve te utmant $e \quad t n e$ ent te me
ii Tet tie n neee tete $e$ ene n ne et n eten te e
iii $n$ tenan $u$ te me
iv tmeteu en mantee ty yune

CAIE FP2 2013 November Q10

10 Cutna ee $e$ ee $n$ ee $n$ tey ee $n m m e$ me ut na $n$ ene ut na tenume ee $n e \quad n \quad e \quad n \quad n t e n t e$


$e$
ene

Tet tite neee atee eateden eeene me $n$ eme ut na
$A \quad e n m m e \quad n t e n t n t$ ee tentee $t$ unttte t enteme me nteet w $n$ neee y te tet

		me ut na			eme ut na
	$t$	$n$	$n$	$e$	$n$	e ent
$t$	$t$	$u$	$e$	$t$	eent n u n te

$11 y \quad$ one $g a a$ \begin{table}[h]

\captionsetup{labelformat=empty} \caption{TH}

$m$

a a

x атаа

$\sqrt { - } { } ^ { a }$

\multirow[b]{3}{*}{(}

\end{table} \begin{verbatim} ay $g$ a maag a ma \end{verbatim}

\includegraphics[max width=\textwidth, alt={}]{bd7c631e-eb5a-42c7-9d17-d9ed16f616d8-5_40_17_1009_239}

\begin{verbatim} , та та аа а тта а ) а тата ута a \end{verbatim} LNE LAN PAGE

Edexcel FP2 Q4

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

Find the set of values of $x$ for which $\frac { x ^ { 2 } } { x - 2 } > 2 x$.
(Total 6 marks)

Edexcel FP2 2006 January Q3

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

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
  $\theta = \frac { \pi } { 2 }$.
Show that the area of $R$ is $\frac { a ^ { 2 } } { 16 } ( 3 \sqrt { 3 } - 4 )$.

Edexcel FP2 2006 January Q5

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

7. $$( 1 + 2 x ) \frac { \mathrm { d } y } { \mathrm {~d} x } = x + 4 y ^ { 2 }$$

Show that $$( 1 + 2 x ) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 1 + 2 ( 4 y - 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x }$$
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 $x = 0$,
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

8. In the Argand diagram the point $P$ represents the complex number $z$. Given that arg $\left( \frac { z - 2 \mathrm { i } } { z + 2 } \right) = \frac { \pi } { 2 }$,

sketch the locus of $P$,
deduce the value of $| \mathrm { z } + 1 - \mathrm { i } |$. The transformation $T$ from the $z$-plane to the $w$-plane is defined by $$w = \frac { 2 ( 1 + \mathrm { i } ) } { z + 2 } , \quad z \neq - 2$$
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

Find the set of values for which

$$| x - 1 | > 6 x - 1$$

Edexcel FP2 2002 June Q2

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.
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

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

5. Using algebra, find the set of values of $x$ for which $2 x - 5 > \frac { 3 } { x }$.

Edexcel FP2 2002 June Q6

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

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

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}

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.
Calculate the value of $a$.
Find the area of the surface of the pool. (6)

		me ut na			eme ut na
	\(t\)	\(n\)	\(n\)	\(e\)	\(n\)	e ent
\(t\)	\(t\)	\(u\)	\(e\)	\(t\)	eent n u n te

A um ctagula lama	\(c\)	ad		a ma		A um
ccula lama adu - a ma	-	T t lama a xd tgt t am la
t t ct ccdg at t t	dagam A atcl		ma -		- attacd at
T ytm t tat abut a xd mt tal ax tug					ad dcula \(t\) t
la	\(t\) ytm abut \(t\) ax				\(\_\_\_\_\)
						\(\mu \geqslant \leqslant\)
T ytm lad m \(t t\)	tal ad	\(b l\)	Fd \(t m\)			\(- t\)
gatt agula d tub	ut mt gg t alu		cct \(t\) dcmal lac

Emly
\(B\)
\(A t\)

Questions FP2 (1157 questions)

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CAIE FP2 2014 June Q1

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Edexcel FP2 2006 January Q1

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Edexcel FP2 2002 June Q8