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CAIE Further Paper 3 2024 November Q1

5 marks Challenging +1.2

1 A particle $P$ is projected with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle $\tan ^ { - 1 } 2$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. When $P$ has travelled a distance 56 m horizontally from $O$, it is at a vertical height $H \mathrm {~m}$ above the plane. When $P$ has travelled a distance 84 m horizontally from $O$, it is at a vertical height $\frac { 1 } { 2 } H \mathrm {~m}$ above the plane. Find, in either order, the value of $u$ and the value of $H$. \includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-02_2718_38_106_2009}

CAIE Further Paper 3 2024 November Q3

8 marks Challenging +1.2

3 A particle $P$ of mass $m \mathrm {~kg}$ is attached to one end of a light elastic string of natural length 2 m and modulus of elasticity $2 m g \mathrm {~N}$. The other end of the string is attached to a fixed point $O$. The particle $P$ hangs in equilibrium vertically below $O$. The particle $P$ is pulled down vertically a distance $d$ m below its equilibrium position and released from rest.

Given that the particle just reaches $O$ in the subsequent motion, find the value of $d$. \includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-04_2717_33_109_2014} \includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-05_2725_35_99_20}
Hence find the speed of $P$ when it is 2 m below $O$. \includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-06_785_729_255_708} An object is formed by removing a cylinder of radius $\frac { 2 } { 3 } a$ and height $k h ( k < 1 )$ from a uniform solid cylinder of radius $a$ and height $h$. The vertical axes of symmetry of the two cylinders coincide. The upper faces of the two cylinders are in the same plane as each other. The points $A$ and $B$ are the opposite ends of a diameter of the upper face of the object (see diagram).
Find, in terms of $h$ and $k$, the distance of the centre of mass of the object from $A B$.

When the object is suspended from $A$, the angle between $A B$ and the vertical is $\theta$, where $\tan \theta = \frac { 3 } { 2 }$.
Given that $h = \frac { 8 } { 3 } a$, find the possible values of $k$.

CAIE Further Paper 3 2024 November Q5

7 marks Challenging +1.2

5 A particle $P$ of mass 2 kg moving on a horizontal straight line has displacement $x \mathrm {~m}$ from a fixed point $O$ on the line and velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at time $t$ s. The only horizontal force acting on $P$ is a variable force $F \mathrm {~N}$ which can be expressed as a function of $t$. It is given that $$\frac { v } { x } = \frac { 3 - t } { 1 + t }$$ and when $t = 0 , x = 5$.

Find an expression for $x$ in terms of $t$. \includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-09_2725_35_99_20}
Find the magnitude of $F$ when $t = 3$. \includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-10_559_1257_255_445} A particle $P$ of mass 0.05 kg is attached to one end of a light inextensible string of length 1 m . The other end of the string is attached to a fixed point $O$. A particle $Q$ of mass 0.04 kg is attached to one end of a second light inextensible string. The other end of this string is attached to $P$. The particle $P$ moves in a horizontal circle of radius 0.8 m with angular speed $\omega \operatorname { rad~s } ^ { - 1 }$. The particle $Q$ moves in a horizontal circle of radius 1.4 m also with angular speed $\omega \mathrm { rad } \mathrm { s } ^ { - 1 }$. The centres of the circles are vertically below $O$, and $O , P$ and $Q$ are always in the same vertical plane. The strings $O P$ and $P Q$ remain at constant angles $\alpha$ and $\beta$ respectively to the vertical (see diagram).
Find the tension in the string $O P$. \includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-10_2718_42_107_2007} \includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-11_2725_35_99_20}
Find the value of $\omega$.
Find the value of $\beta$.

CAIE Further Paper 3 2024 November Q7

10 marks Challenging +1.8

7 A particle $P$ is projected with speed $u$ at an angle $\tan ^ { - 1 } \left( \frac { 4 } { 3 } \right)$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. When $P$ is moving horizontally, it strikes a smooth inclined plane at the point $A$. This plane is inclined to the horizontal at an angle $\alpha$, and the line of greatest slope through $A$ lies in the vertical plane through $O$ and $A$. As a result of the impact, $P$ moves vertically upwards. The coefficient of restitution between $P$ and the inclined plane is $e$.

Show that $e \tan ^ { 2 } \alpha = 1$. \includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-12_2713_33_111_2017} In its subsequent motion, the greatest height reached by $P$ above $A$ is $\frac { 3 } { 16 }$ of the vertical height of $A$ above the horizontal plane.
Find the value of $e$.
If you use the following page to complete the answer to any question, the question number must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-14_2715_33_109_2012}

CAIE Further Paper 3 2020 Specimen Q1

4 marks Challenging +1.2

1 A ch ld's ty co ists

CAIE Further Paper 3 2020 Specimen Q2

8 marks Standard +0.3

2 A lig elastic strig hsa tn al leg h $a$ ad md B 6 elasticity 2 mg . Or ed 6 th strig is attach d œ fiæ $\Phi \quad n A$. Tb ob rend te strig s attach d œ $\mathbf { P }$ rticle 6 mass $2 m$.

Fid in terms $6 a$, th ex en in 6 th strig wh $n$th $\boldsymbol { \rho }$ rticle $\mathbf { h }$ g freely in eq lib im b low $A$.
Th $\mathbf { p }$ rticle is released rm rest at $A$. Fid in terms $\mathbf { 6 } \quad a$, th $\dot { \mathbf { d } }$ stance

CAIE Further Paper 3 2020 Specimen Q3

10 marks Challenging +1.2

3 A $\boldsymbol { \rho }$ rticle $P 6$ mass $m$ g falls frm rest $\mathbf { d } \quad \mathrm { rg }$ avty. The re is a resistiv fo ce 6 mag td $m k v ^ { 2 } \mathrm {~N}$, wh re $v \mathrm {~ms} ^ { - 1 }$ is th se e $\boldsymbol { C }$ after it $\mathbf { h }$ s fallera id stan e $x \mathrm {~m}$ ad $k$ is a $\mathbf { p }$ itie co tan.

Bys b in ra ra p p iate d fferen ial eq tin ,s how th t $$v ^ { 2 } = \frac { g } { k } \left( 1 - \mathrm { e } ^ { - 2 k x } \right)$$ [] It is $\mathbf { N }$ it $\mathbf { h } \mathrm { t } k = 0.01$. The speed of $P \mathrm { w } \mathbf { h } \mathrm { n } x \mathbf { b } \mathrm {~cm}$ es larg ap $\mathbf { o } \mathrm { ch } \mathrm { s } V \mathrm {~ms} ^ { - 1 }$.
1. Fid $V$ correct to 2 decimal places.
2. Hen e fidw far $P \mathbf { b }$ s fallenw $\mathbf { b }$ n ts sp ed $\mathrm { s } \frac { 1 } { 2 } V \mathrm {~ms} ^ { - 1 }$.
  \includegraphics[max width=\textwidth, alt={}]{1e5941a9-14eb-441c-8d39-fd62695446ac-08_319_908_255_580}
  Two in fo m smo h sp res $A$ ad $B \mathbf { 6 }$ eq $l$ radili masses $m$ ad $2 m$ resp ctie ly. Sp re $B$ is at rest $\mathbf { n }$ a smo hb izn al sn face. Sp ere $A$ is mi $\mathbf { g } \mathbf { n }$ th sn face with sp ed $u$ at an ag e $\mathbf { b } \boldsymbol { B }$ to th lin 6 cen res $6 A$ and $B$ wh n it cb lid s with $B$ (see $\dot { \mathbf { d } }$ ag am). Th ce fficien

CAIE Further Paper 3 2020 Specimen Q5

10 marks Standard +0.8

5 A $\mathbf { p }$ rticle $P \mathbf { 6 }$ mass $m$ is attach d to e. a lig in k en ib e strig leg h $a$. Tb o b re. th strig s attach d or fiæ $\Phi \quad n O$.

\includegraphics[max width=\textwidth, alt={}, center]{1e5941a9-14eb-441c-8d39-fd62695446ac-10_314_768_397_651} Th $\boldsymbol { p }$ rticle $P$ mo sinab izo al circle with a co tan ag arsp ed $\omega$ with th strig in lie d at $\theta$ to $\mathrm { b } \quad \mathrm { m }$ ard rtical th $\mathrm { g } \quad O$ (see $\dot { \mathrm { d } } \mathrm { ag } \mathrm { am }$ ). Sw that $\omega ^ { 2 } = \frac { 2 g } { a }$.
Th $\mathbf { p }$ rticle w has at restad stan e $a$ rtically b low $O$. It isth $\mathrm { np } \dot { \mathrm { p } }$ ected b izb ally so th t it $\mathbf { b } \mathbf { g }$ s to m in a rtical circle with cen re $O$. Wh n th strig maks an ag e $\boldsymbol { \theta }$, with th d $\mathbb { w }$ ard rtical th $\mathbf { g } O$, th ang arse $\operatorname { d } P$ is $\sqrt { \frac { 2 g } { a } }$. Tb strig first $\mathbf { g }$ s slack when $O P$ mak s ara g e $\theta$ with b ard rtical th ob $O$. Fid b le 6 co $\theta$.

CAIE Further Paper 3 2020 Specimen Q6

9 marks Standard +0.8

6 A $\mathbf { p }$ rticle $P$ is $\mathrm { p } \dot { \mathbf { j } }$ ected with sp ed $u$ at an ag $\mathrm { e } \alpha$ ab tb $\mathbf { b }$ izn al frm $\mathrm { a } \dot { \mathbf { p } }$ n $O \mathbf { n }$ ab izb al p ae ad mo s freely d rg av ty. Th $\mathbf { b }$ izo al ad rtical d sp acemen s $\mathbf { 6 } P$ frm $O$ at a sb eq $n$ time $t$ are $\mathbf { d } \mathbf { n }$ edy $\quad x$ ad $y$ resp ctie ly.

Derie the eq tirib the trajecto y $P$ irt b fo m $$y = x \tan \alpha - \frac { g x ^ { 2 } } { 2 u ^ { 2 } } \sec ^ { 2 } \alpha$$
Tb g eatestb in $6 P$ ab th p au is o edy $H$. Wh $\mathrm { n } P$ is ata $\mathbf { b }$ ig $6 \frac { 3 } { 4 } H$, it $\mathbf { a }$ strac lled ab izn ald stan e $d$. Gie it $\mathbf { h } \tan \alpha = \Im$ id it erms $6 H$,tb twœ sibed le $\mathrm { s } \varnothing d$. If B e th follw ig lin dpg to cm p ete th an wer(s) to ay q stin (s), th q stin $\mathrm { m } \quad \mathbf { b } \quad \mathrm { r } ( \mathrm { s } )$ ms tb clearlys n n

CAIE S1 2020 June Q1

5 marks Moderate -0.8

1 The score when two fair six-sided dice are thrown is the sum of the two numbers on the upper faces.

Show that the probability that the score is 4 is $\frac { 1 } { 12 }$.
The two dice are thrown repeatedly until a score of 4 is obtained. The number of throws taken is denoted by the random variable $X$.
Find the mean of $X$.
Find the probability that a score of 4 is first obtained on the 6th throw.
Find $\mathrm { P } ( X < 8 )$.

CAIE S1 2020 June Q2

6 marks Standard +0.3

Find the number of different arrangements that can be made from the 9 letters of the word JEWELLERY in which the three Es are together and the two Ls are together.
Find the number of different arrangements that can be made from the 9 letters of the word JEWELLERY in which the two Ls are not next to each other.

CAIE S1 2020 June Q3

7 marks Moderate -0.8

3 A company produces small boxes of sweets that contain 5 jellies and 3 chocolates. Jemeel chooses 3 sweets at random from a box.

Draw up the probability distribution table for the number of jellies that Jemeel chooses.
The company also produces large boxes of sweets. For any large box, the probability that it contains more jellies than chocolates is 0.64 . 10 large boxes are chosen at random.
Find the probability that no more than 7 of these boxes contain more jellies than chocolates.

CAIE S1 2020 June Q4

4 marks Standard +0.8

4 In a music competition, there are 8 pianists, 4 guitarists and 6 violinists. 7 of these musicians will be selected to go through to the final. How many different selections of 7 finalists can be made if there must be at least 2 pianists, at least 1 guitarist and more violinists than guitarists?

CAIE S1 2020 June Q5

8 marks Easy -1.2

5 On Mondays, Rani cooks her evening meal. She has a pizza, a burger or a curry with probabilities $0.35,0.44,0.21$ respectively. When she cooks a pizza, Rani has some fruit with probability 0.3 . When she cooks a burger, she has some fruit with probability 0.8 . When she cooks a curry, she never has any fruit.

Draw a fully labelled tree diagram to represent this information.
Find the probability that Rani has some fruit.
Find the probability that Rani does not have a burger given that she does not have any fruit.

CAIE S1 2020 June Q6

9 marks Standard +0.8

6 The lengths of female snakes of a particular species are normally distributed with mean 54 cm and standard deviation 6.1 cm .

Find the probability that a randomly chosen female snake of this species has length between 50 cm and 60 cm .
The lengths of male snakes of this species also have a normal distribution. A scientist measures the lengths of a random sample of 200 male snakes of this species. He finds that 32 have lengths less than 45 cm and 17 have lengths more than 56 cm .
Find estimates for the mean and standard deviation of the lengths of male snakes of this species.

CAIE S1 2020 June Q7

11 marks Moderate -0.8

7 The numbers of chocolate bars sold per day in a cinema over a period of 100 days are summarised in the following table.

Number of chocolate bars sold	$1 - 10$	$11 - 15$	$16 - 30$	$31 - 50$	$51 - 60$
Number of days	18	24	30	20	8

Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{3ada18de-c4f7-4049-9032-46b796be83c3-12_1203_1399_833_415}
What is the greatest possible value of the interquartile range for the data?
Calculate estimates of the mean and standard deviation of the number of chocolate bars sold.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE S1 2004 June Q1

4 marks Easy -1.2

1 Two cricket teams kept records of the number of runs scored by their teams in 8 matches. The scores are shown in the following table.

Team $A$	150	220	77	30	298	118	160	57
Team $B$	166	142	170	93	111	130	148	86

Find the mean and standard deviation of the scores for team $A$. The mean and standard deviation for team $B$ are 130.75 and 29.63 respectively.
State with a reason which team has the more consistent scores.

CAIE S1 2004 June Q2

5 marks Easy -1.8

2 In a recent survey, 640 people were asked about the length of time each week that they spent watching television. The median time was found to be 20 hours, and the lower and upper quartiles were 15 hours and 35 hours respectively. The least amount of time that anyone spent was 3 hours, and the greatest amount was 60 hours.

On graph paper, show these results using a fully labelled cumulative frequency graph.
Use your graph to estimate how many people watched more than 50 hours of television each week.

CAIE S1 2004 June Q3

5 marks Moderate -0.8

3 Two fair dice are thrown. Let the random variable $X$ be the smaller of the two scores if the scores are different, or the score on one of the dice if the scores are the same.

Copy and complete the following table to show the probability distribution of $X$.
$x$ 1 2 3 4 5 6
$\mathrm { P } ( X = x )$
Find $\mathrm { E } ( X )$.

CAIE S1 2004 June Q4

8 marks Moderate -0.3

4 Melons are sold in three sizes: small, medium and large. The weights follow a normal distribution with mean 450 grams and standard deviation 120 grams. Melons weighing less than 350 grams are classified as small.

Find the proportion of melons which are classified as small.
The rest of the melons are divided in equal proportions between medium and large. Find the weight above which melons are classified as large.

CAIE S1 2004 June Q5

8 marks Moderate -0.8

The menu for a meal in a restaurant is as follows. \begin{displayquote} Starter Course
Melon
or
Soup
or
Smoked Salmon \end{displayquote} \begin{displayquote} Main Course
Chicken
or
Steak
or
Lamb Cutlets
or
Vegetable Curry
or
Fish \end{displayquote} \begin{displayquote} Dessert Course
Cheesecake
or
Ice Cream
or
Apple Pie
All the main courses are served with salad and either
new potatoes or french fries.
1. How many different three-course meals are there?
2. How many different choices are there if customers may choose only two of the three courses?
In how many ways can a group of 14 people eating at the restaurant be divided between three tables seating 5, 5 and 4? \end{displayquote}

CAIE S1 2004 June Q6

9 marks Moderate -0.3

6 When Don plays tennis, $65 \%$ of his first serves go into the correct area of the court. If the first serve goes into the correct area, his chance of winning the point is $90 \%$. If his first serve does not go into the correct area, Don is allowed a second serve, and of these, $80 \%$ go into the correct area. If the second serve goes into the correct area, his chance of winning the point is $60 \%$. If neither serve goes into the correct area, Don loses the point.

Draw a tree diagram to represent this information.
Using your tree diagram, find the probability that Don loses the point.
Find the conditional probability that Don's first serve went into the correct area, given that he loses the point.

CAIE S1 2004 June Q7

11 marks Standard +0.3

7 A shop sells old video tapes, of which 1 in 5 on average are known to be damaged.

A random sample of 15 tapes is taken. Find the probability that at most 2 are damaged.
Find the smallest value of $n$ if there is a probability of at least 0.85 that a random sample of $n$ tapes contains at least one damaged tape.
A random sample of 1600 tapes is taken. Use a suitable approximation to find the probability that there are at least 290 damaged tapes.