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CAIE Further Paper 4 2021 June Q6

14 marks Standard +0.8

The continuous random variable $X$ has probability density function f given by $$f(x) = \begin{cases} \frac{1}{8} & 0 \leq x < 1, \\ \frac{1}{28}(8 - x) & 1 \leq x \leq 8, \\ 0 & \text{otherwise}. \end{cases}$$

Find the cumulative distribution function of $X$. [3]

Find the value of the constant $a$ such that P$(X \leq a) = \frac{5}{7}$. [3]

The random variable $Y$ is given by $Y = \sqrt[3]{X}$.

Find the probability density function of $Y$. [5]

CAIE M2 2014 June Q3

Standard +0.3

3 A light elastic string has natural length 0.8 m and modulus of elasticity 16 N . One end of the string is attached to a fixed point $O$, and a particle $P$ of mass 0.4 kg is attached to the other end of the string. The particle $P$ hangs in equilibrium vertically below $O$.

Show that the extension of the string is 0.2 m . $P$ is projected vertically downwards from the equilibrium position. $P$ first comes to instantaneous rest at the point where $O P = 1.4 \mathrm {~m}$.
Calculate the speed at which $P$ is projected.
Find the speed of $P$ at the first instant when the string subsequently becomes slack.

CAIE M2 2014 June Q4

Standard +0.8

4 A particle $P$ is projected with speed $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $40 ^ { \circ }$ above the horizontal from a point $O$ on horizontal ground.

Find the height of $P$ above the ground when $P$ has speed $18 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Calculate the length of time for which the speed of $P$ is less than $18 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, and find the horizontal distance travelled by $P$ during this time.

Find the speed and direction of motion of the ball when it reaches the ground.
Calculate the distance from the foot of the tower to the point where the ball reaches the ground.

CAIE M2 2013 June Q7

Challenging +1.2

7 A small ball $B$ of mass 0.2 kg moves in a narrow fixed smooth cylindrical tube $O A$ of length 1 m , closed at the end $A$. When the ball has displacement $x \mathrm {~m}$ from $O$, it has velocity $v \mathrm {~ms} ^ { - 1 }$ in the direction $O A$ and experiences a resisting force of magnitude $\frac { k } { 1 - x } \mathrm {~N}$.

\includegraphics[max width=\textwidth, alt={}, center]{10abedc3-c814-47c0-8ed4-849ef325feca-4_186_805_488_715} The tube is fixed in a horizontal position and $B$ is projected from $O$ towards $A$ with velocity $1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ (see diagram). Given that $B$ comes to instantaneous rest after travelling 0.55 m , show that $k = 0.1803$, correct to 4 significant figures.
The tube is now fixed in a vertical position with $O$ above $A$. The ball $B$ is released from rest at $O$. Calculate the speed of $B$ after it has descended 0.1 m . \end{document}

CAIE FP2 2013 November Q3

Standard +0.8

3 hours
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF10) \section*{READ THESE INSTRUCTIONS FIRST} If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use a soft pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.
Where a numerical value is necessary, take the acceleration due to gravity to be $10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
The use of a calculator is expected, where appropriate.
Results obtained solely from a graphic calculator, without supporting working or reasoning, will not receive credit.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
[0pt] The number of marks is given in brackets [ ] at the end of each question or part question.

CAIE FP2 2013 November Q1

Challenging +1.2

1 \includegraphics[max width=\textwidth, alt={}, center]{c1aae41e-530c-4db4-8959-8afe223c4dbc-2_547_423_260_861} Three identical uniform rods, $A B , B C$ and $C D$, each of mass $M$ and length $2 a$, are rigidly joined to form three sides of a square. A uniform circular disc, of mass $\frac { 2 } { 3 } M$ and radius $a$, has the opposite ends of one of its diameters attached to $A$ and $D$ respectively. The disc and the rods all lie in the same plane (see diagram). Find the moment of inertia of the system about the axis $A D$.

CAIE FP2 2013 November Q4

Challenging +1.8

4 \includegraphics[max width=\textwidth, alt={}, center]{c1aae41e-530c-4db4-8959-8afe223c4dbc-3_563_572_258_785} A uniform circular disc, with centre $O$ and weight $W$, rests in equilibrium on a horizontal floor and against a vertical wall. The plane of the disc is vertical and perpendicular to the wall. The disc is in contact with the floor at $A$ and with the wall at $B$. A force of magnitude $P$ acts tangentially on the disc at the point $C$ on the edge of the disc, where the radius $O C$ makes an angle $\theta$ with the upward vertical, and $\tan \theta = \frac { 4 } { 3 }$ (see diagram). The coefficient of friction between the disc and the floor and between the disc and the wall is $\frac { 1 } { 2 }$. Show that the sum of the magnitudes of the frictional forces at $A$ and $B$ is equal to $P$. Given that the equilibrium is limiting at both $A$ and $B$,

show that $P = \frac { 15 } { 34 } \mathrm {~W}$,
find the ratio of the magnitude of the normal reaction at $A$ to the magnitude of the normal reaction at $B$.

CAIE FP2 2013 November Q9

Standard +0.3

9 For a random sample of 10 observations of pairs of values $( x , y )$, the equations of the regression lines of $y$ on $x$ and of $x$ on $y$ are $$y = 4.21 x - 0.862 \quad \text { and } \quad x = 0.043 y + 6.36 ,$$ respectively.

Find the value of the product moment correlation coefficient for the sample.
Test, at the $10 \%$ significance level, whether there is evidence of non-zero correlation between the variables.
Find the mean values of $x$ and $y$ for this sample.
Estimate the value of $x$ when $y = 2.3$ and comment on the reliability of your answer.

CAIE FP2 2013 November Q11

Challenging +1.8

11 Answer only one of the following two alternatives.
EITHER
A smooth sphere, with centre $O$ and radius $a$, is fixed on a smooth horizontal plane $\Pi$. A particle $P$ of mass $m$ is projected horizontally from the highest point of the sphere with speed $\sqrt { } \left( \frac { 2 } { 5 } g a \right)$. While $P$ remains in contact with the sphere, the angle between $O P$ and the upward vertical is denoted by $\theta$. Show that $P$ loses contact with the sphere when $\cos \theta = \frac { 4 } { 5 }$. Subsequently the particle collides with the plane $\Pi$. The coefficient of restitution between $P$ and $\Pi$ is $\frac { 5 } { 9 }$. Find the vertical height of $P$ above $\Pi$ when the vertical component of the velocity of $P$ first becomes zero.
OR
A factory produces bottles of spring water. The manager decides to assess the performance of the two machines that are used to fill the bottles with water. He selects a random sample of 60 bottles filled by the first machine $X$ and a random sample of 80 bottles filled by the second machine $Y$. The volumes of water, $x$ and $y$, measured in appropriate units, are summarised as follows. $$\Sigma x = 58.2 \quad \Sigma x ^ { 2 } = 85.8 \quad \Sigma y = 97.6 \quad \Sigma y ^ { 2 } = 188.6$$ A test at the $\alpha \%$ significance level shows that the mean volume of water in bottles filled by machine $X$ is less than the mean volume of water in bottles filled by machine $Y$. Find the set of possible values of $\alpha$.

CAIE Further Paper 2 2020 Specimen Q0

Standard +0.3

0 & 2 & 2
- 1 & 1 & 3 \end{array} \right) .$$

Find the eigenvalues of $\mathbf { A }$.
Use the characteristic equation of $\mathbf { A }$ to find $\mathbf { A } ^ { - 1 }$.

CAIE M1 2014 June Q4

Standard +0.3

4 A particle $P$ moves on a straight line, starting from rest at a point $O$ of the line. The time after $P$ starts to move is $t \mathrm {~s}$, and the particle moves along the line with constant acceleration $\frac { 1 } { 4 } \mathrm {~m} \mathrm {~s} ^ { - 2 }$ until it passes through a point $A$ at time $t = 8$. After passing through $A$ the velocity of $P$ is $\frac { 1 } { 2 } t ^ { \frac { 2 } { 3 } } \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Find the acceleration of $P$ immediately after it passes through $A$. Hence show that the acceleration of $P$ decreases by $\frac { 1 } { 12 } \mathrm {~m} \mathrm {~s} ^ { - 2 }$ as it passes through $A$.
Find the distance moved by $P$ from $t = 0$ to $t = 27$.

CAIE M1 2014 June Q5

Standard +0.3

5 \includegraphics[max width=\textwidth, alt={}, center]{77976dad-c055-45fd-93fe-e37fa8e9ae22-3_343_691_254_725} A light inextensible rope has a block $A$ of mass 5 kg attached at one end, and a block $B$ of mass 16 kg attached at the other end. The rope passes over a smooth pulley which is fixed at the top of a rough plane inclined at an angle of $30 ^ { \circ }$ to the horizontal. Block $A$ is held at rest at the bottom of the plane and block $B$ hangs below the pulley (see diagram). The coefficient of friction between $A$ and the plane is $\frac { 1 } { \sqrt { 3 } }$. Block $A$ is released from rest and the system starts to move. When each of the blocks has moved a distance of $x \mathrm {~m}$ each has speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Write down the gain in kinetic energy of the system in terms of $v$.
Find, in terms of $x$,
(a) the loss of gravitational potential energy of the system,
(b) the work done against the frictional force.
Show that $21 v ^ { 2 } = 220 x$.

CAIE FP2 2014 June Q10

Standard +0.3

10 The lengths of a random sample of eight fish of a certain species are measured, in cm, as follows. $$\begin{array} { l l l l l l l l } 17.3 & 15.8 & 18.2 & 15.6 & 16.0 & 18.8 & 15.3 & 15.0 \end{array}$$ Assuming that lengths are normally distributed,

test, at the $10 \%$ significance level, whether the population mean length of fish of this species is greater than 15.8 cm ,
calculate a $95 \%$ confidence interval for the population mean length of fish of this species.

CAIE FP2 2014 June Q11

Challenging +1.2

11 Answer only one of the following two alternatives.
EITHER
A particle $P$ of mass $m$ is suspended from a fixed point by a light elastic string of natural length $l$, and hangs in equilibrium. The particle is pulled vertically down to a position where the length of the string is $\frac { 13 } { 7 } l$. The particle is released from rest in this position and reaches its greatest height when the length of the string is $\frac { 11 } { 7 } l$.

Show that the modulus of elasticity of the string is $\frac { 7 } { 5 } \mathrm { mg }$.
Show that $P$ moves in simple harmonic motion about the equilibrium position and state the period of the motion.
Find the time after release when the speed of $P$ is first equal to half of its maximum value.
OR
For a random sample of 12 observations of pairs of values $( x , y )$, the equation of the regression line of $y$ on $x$ and the equation of the regression line of $x$ on $y$ are $$y = b x + 4.5 \quad \text { and } \quad x = a y + c$$ respectively, where $a , b$ and $c$ are constants. The product moment correlation coefficient for the sample is 0.6 .
Test, at the $5 \%$ significance level, whether there is evidence of positive correlation between the variables.
Given that $b - a = 0.5$, find the values of $a$ and $b$.
Given that the sum of the $x$-values in the sample data is 66, find the value of $c$ and sketch the two regression lines on the same diagram. For each of the 12 pairs of values of $( x , y )$ in the sample, another variable $z$ is considered, where $z = 5 y$.
State the coefficient of $x$ in the equation of the regression line of $z$ on $x$ and find the value of the product moment correlation coefficient between $x$ and $z$, justifying your answer.