Questions - CAIE 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

CAIE FP2 2009 June Q1

1 A line $O P$ of fixed length $l$ rotates in a plane about the fixed point $O$. At time $t = 0$, the line is at the position $O A$. At time $t$, angle $A O P = \theta$ radians and $\frac { \mathrm { d } \theta } { \mathrm { d } t } = \sin \theta$. Show that, for all $t$, the magnitude of the acceleration of $P$ is equal to the magnitude of its velocity.

CAIE FP2 2009 June Q2

2 The tip of a sewing-machine needle oscillates vertically in simple harmonic motion through a distance of 2.10 cm . It takes 2.25 s to perform 100 complete oscillations. Find, in $\mathrm { m } \mathrm { s } ^ { - 1 }$, the maximum speed of the tip of the needle. Show that the speed of the tip when it is at a distance of 0.5 cm from a position of instantaneous rest is $2.50 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, correct to 3 significant figures.

CAIE FP2 2009 June Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{15ed1dfc-8188-4e20-9c0b-ce31af35f0b6-2_513_711_890_717} A uniform lamina of mass $m$ is bounded by concentric circles with centre $O$ and radii $a$ and $2 a$. The lamina is free to rotate about a fixed smooth horizontal axis $T$ which is tangential to the outer rim (see diagram). Show that the moment of inertia of the lamina about $T$ is $\frac { 21 } { 4 } m a ^ { 2 }$. When hanging at rest, with $O$ vertically below $T$, the lamina is given an angular speed $\omega$ about $T$. The lamina comes to instantaneous rest in the subsequent motion. Neglecting air resistance, find the set of possible values of $\omega$.

CAIE FP2 2009 June Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{15ed1dfc-8188-4e20-9c0b-ce31af35f0b6-3_512_983_267_580} A uniform sphere rests on a horizontal plane. The sphere has centre $O$, radius 0.6 m and weight 36 N . A uniform rod $A B$, of weight 14 N and length 1 m , rests with $A$ in contact with the plane and $B$ in contact with the sphere at the end of a horizontal diameter. The point of contact of the sphere with the plane is $C$, and $A , B , C$ and $O$ lie in the same vertical plane (see diagram). The contacts at $A , B$ and $C$ are rough and the system is in equilibrium. By taking moments about $C$ for the system, show that the magnitude of the normal contact force at $A$ is 10 N . Show that the magnitudes of the frictional forces at $A , B$ and $C$ are equal. The coefficients of friction at $A , B$ and $C$ are all equal to $\mu$. Find the smallest possible value of $\mu$.

CAIE FP2 2009 June Q5

5 Two spheres $A$ and $B$, of equal radius, have masses $m _ { 1 }$ and $m _ { 2 }$ respectively. They lie at rest on a smooth horizontal plane. Sphere $A$ is projected directly towards sphere $B$ with speed $u$ and, as a result of the collision, $A$ is brought to rest. Show that

the speed of $B$ immediately after the collision cannot exceed $u$,
$m _ { 1 } \leqslant m _ { 2 }$.
\includegraphics[max width=\textwidth, alt={}, center]{15ed1dfc-8188-4e20-9c0b-ce31af35f0b6-3_273_611_1745_767} After the collision, $B$ hits a smooth vertical wall which is at an angle of $60 ^ { \circ }$ to the direction of motion of $B$ (see diagram). In the impact with the wall $B$ loses $\frac { 2 } { 3 }$ of its kinetic energy. Find the coefficient of restitution between $B$ and the wall and show that the direction of motion of $B$ turns through $90 ^ { \circ }$.

CAIE FP2 2009 June Q6

6 marks

6 The times taken by employees in a factory to complete a certain task have a normal distribution with mean $\mu$ seconds and standard deviation $\sigma$ seconds, both of which are unknown. Based on a random sample of 20 employees, the symmetric $95 \%$ confidence interval for $\mu$ is $( 481,509 )$. Calculate a symmetric $90 \%$ confidence interval for $\mu$.
[0pt] [6]

CAIE FP2 2009 June Q7

7 An experiment was carried out to determine how much weedkiller to apply per $100 \mathrm {~m} ^ { 2 }$ in a large field. Ten $100 \mathrm {~m} ^ { 2 }$ areas of the field were randomly chosen and sprayed with predetermined volumes of the weedkiller. The volume of the weedkiller is denoted by $x$ litres and the number of weeds that survived is denoted by $y$. The results are given in the table.

$x$	0.10	0.15	0.20	0.25	0.30	0.35	0.40	0.45	0.50	0.55
$y$	48	40	44	35	39	24	10	13	9	6

$$\left[ \Sigma x = 3.25 , \Sigma x ^ { 2 } = 1.2625 , \Sigma y = 268 , \Sigma y ^ { 2 } = 9548 , \Sigma x y = 66.10 . \right]$$ It is given that the product moment correlation coefficient for the data is - 0.951 , correct to 3 decimal places.

Calculate the equation of a suitable regression line, giving a reason for your choice of line.
Estimate the best volume of weedkiller to apply, and comment on the reliability of your estimate.

CAIE FP2 2009 June Q8

8 Part of a research study of identical twins who had been separated at birth involved a random sample of 9 pairs, in which one twin had been raised by the natural parents and the other by adoptive parents. The IQ scores of these twins were measured, with the following results.

Twin pair	1	2	3	4	5	6	7	8	9
IQ of twin raised by natural parents	82	92	115	132	88	95	112	83	123
IQ of twin raised by adoptive parents	92	88	115	134	97	94	107	88	130

It may be assumed that the difference in IQ scores has a normal distribution. The mean IQ scores of separated twins raised by natural parents and by adoptive parents are denoted by $\mu _ { N }$ and $\mu _ { A }$ respectively. Obtain a $90 \%$ confidence interval for $\mu _ { N } - \mu _ { A }$. One of the researchers claimed that there was no evidence of a difference between the two population means. State, giving a reason, whether the confidence interval supports this claim.

CAIE FP2 2009 June Q9

9 The proportions of blood types $\mathrm { A } , \mathrm { B } , \mathrm { AB }$ and O in the Australian population are $38 \% , 10 \% , 3 \%$ and $49 \%$ respectively. In order to test whether the population in Sydney conforms to these figures, a random sample of 200 residents is selected. The table shows the observed frequencies of these types in the sample.

Blood Type	A	B	AB	O
Frequency	57	24	9	110

Carry out a suitable test at the 5\% significance level. Find the smallest sample size that could be used for the test.

CAIE FP2 2009 June Q10

10 The number of hits per minute on a particular website has a Poisson distribution with mean 0.8. The time between successive hits is denoted by $T$ minutes. Show that $\mathrm { P } ( T > t ) = \mathrm { e } ^ { - 0.8 t }$ and hence show that $T$ has a negative exponential distribution. Using a suitable approximation, which should be justified, find the probability that the time interval between the 1st hit and the 51st hit exceeds one hour.

CAIE FP2 2009 June Q11 EITHER

\includegraphics[max width=\textwidth, alt={}]{15ed1dfc-8188-4e20-9c0b-ce31af35f0b6-5_250_878_808_632}

Two particles $A$ and $B$, of equal mass $m$, are connected by a light elastic string of natural length $a$ and modulus of elasticity $4 m g$. Particle $A$ rests on a rough horizontal table at a distance $a$ from the edge of the table. The string passes over a small smooth pulley $P$ fixed at the edge of the table. At time $t = 0 , B$ is released from rest at $P$ and falls vertically. At time $t , B$ has fallen a distance $x$, without $A$ slipping (see diagram). Show that $$\ddot { x } = - \frac { g } { a } ( 4 x - a ) .$$ Deduce that, while $A$ does not slip, $B$ moves in simple harmonic motion and identify the centre of the motion. Given that the coefficient of friction between $A$ and the table is $\frac { 1 } { 3 }$, find the value of $x$ when $A$ starts to slip, and the corresponding value of $t$, expressing this answer in the form $k \sqrt { } \left( \frac { a } { g } \right)$. Give the value of $k$ correct to 3 decimal places.

CAIE FP2 2009 June Q11 OR

A study was made of the acidity levels in farmland on opposite sides of an island. The levels were measured at six randomly chosen points on the eastern side and at five randomly chosen points on the western side. The values obtained, in suitable units, are denoted by $x _ { E }$ and $x _ { W }$ respectively. The sample means $\bar { x } _ { E }$ and $\bar { x } _ { W }$, and unbiased estimates of the two population variances, $s _ { E } ^ { 2 }$ and $s _ { W } ^ { 2 }$, are as follows. $$\bar { x } _ { E } = 5.035 , s _ { E } ^ { 2 } = 0.0231 , \bar { x } _ { W } = 4.782 , s _ { W } ^ { 2 } = 0.0195 .$$ The population means on the eastern and western sides are denoted by $\mu _ { E }$ and $\mu _ { W }$ respectively. State suitable hypotheses for a test for a difference between the mean acidity levels on the two sides of the island. Stating any required assumptions, obtain the rejection region for a test at the $5 \%$ significance level of whether the mean acidity levels differ on the two sides of the island. Give the conclusion of the test. Find the largest value of $a$ for which the samples above provide evidence at the $5 \%$ significance level that $\mu _ { E } - \mu _ { W } > a$.

CAIE FP2 2010 June Q1

1 A particle $P$, of mass 0.2 kg , moves in simple harmonic motion along a straight line under the action of a resultant force of magnitude $F \mathrm {~N}$. The distance between the end-points of the motion is 0.6 m , and the period of the motion is 0.5 s . Find the greatest value of $F$ during the motion.

CAIE FP2 2010 June Q2

2 A uniform $\operatorname { rod } A B$ of weight $W$ rests in equilibrium with $A$ in contact with a rough vertical wall. The rod is in a vertical plane perpendicular to the wall, and is supported by a force of magnitude $P$ acting at $B$ in this vertical plane. The rod makes an angle of $60 ^ { \circ }$ with the wall, and the force makes an angle of $30 ^ { \circ }$ with the rod (see diagram). Find the value of $P$. Find also the set of possible values of the coefficient of friction between the rod and the wall.

		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

\(x\)	0.10	0.15	0.20	0.25	0.30	0.35	0.40	0.45	0.50	0.55
\(y\)	48	40	44	35	39	24	10	13	9	6

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

Questions — CAIE FP2 (474 questions)

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CAIE FP2 2009 June Q11 EITHER

CAIE FP2 2009 June Q11 OR

CAIE FP2 2010 June Q1

CAIE FP2 2010 June Q2