Questions - CAIE FP2 - A-Level Maths

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CAIE FP2 2017 Specimen Q10 EITHER

Challenging +1.8

\includegraphics[max width=\textwidth, alt={}]{3b311657-f609-4e8d-81e6-b0cbc7a8cbae-18_598_601_440_772}

An object is formed by attaching a thin uniform rod $P Q$ to a uniform rectangular lamina $A B C D$. The lamina has mass $m$, and $A B = D C = 6 a , B C = A D = 3 a$. The rod has mass $M$ and length $3 a$. The end $P$ of the rod is attached to the mid-point of $A B$. The rod is perpendicular to $A B$ and in the plane of the lamina (see diagram).

Show that the moment of inertia of the object about a smooth horizontal axis $l _ { 1 }$, through $Q$ and perpendicular to the plane of the lamina, is $3 ( 8 m + M ) a ^ { 2 }$.
Show that the moment of inertia of the object about a smooth horizontal axis $l _ { 2 }$, through the mid-point of $P Q$ and perpendicular to the plane of the lamina, is $\frac { 3 } { 4 } ( 17 m + M ) a ^ { 2 }$.
Find expressions for the periods of small oscillations of the object about the axes $l _ { 1 }$ and $l _ { 2 }$, and verify that these periods are equal when $m = M$.

CAIE FP2 2017 Specimen Q10 OR

Standard +0.8

A farmer $A$ grows two types of potato plants, Royal and Majestic. A random sample of 10 Royal plants is taken and the potatoes from each plant are weighed. The total mass of potatoes on a plant is $x \mathrm {~kg}$. The data are summarised as follows. $$\Sigma x = 42.0 \quad \Sigma x ^ { 2 } = 180.0$$ A random sample of 12 Majestic plants is taken. The total mass of potatoes on a plant is $y \mathrm {~kg}$. The data are summarised as follows. $$\Sigma y = 57.6 \quad \Sigma y ^ { 2 } = 281.5$$

Test, at the $5 \%$ significance level, whether the population mean mass of potatoes from Royal plants is the same as the population mean mass of potatoes from Majestic plants. You may assume that both distributions are normal and you should state any additional assumption that you make.
A neighbouring farmer $B$ grows Crown potato plants. His plants produce 3.8 kg of potatoes per plant, on average. Farmer $A$ claims that her Royal plants produce a higher mean mass of potatoes than Farmer $B$ 's Crown plants.
Test, at the $5 \%$ significance level, whether Farmer $A$ 's claim is justified.

CAIE FP2 2018 June Q5

12 marks Challenging +1.2

Show that the moment of inertia of the object about the axis $l$ is $180 M a ^ { 2 }$.
Show that small oscillations of the object about the axis $l$ are approximately simple harmonic, and state the period.

CAIE FP2 2019 June Q4

11 marks Challenging +1.3

Find the moment of inertia of the object, consisting of the rod and two spheres, about $L$.
The object is pivoted at $A$ so that it can rotate freely about $L$. The object is released from rest with the rod making an angle of $60 ^ { \circ }$ to the downward vertical. The greatest angular speed attained by the object in the subsequent motion is $\frac { 9 } { 20 } \sqrt { } \left( \frac { g } { a } \right)$.
Find the value of $k$.

CAIE FP2 2017 November Q5

12 marks Challenging +1.3

Show that the moment of inertia of the system, consisting of frame and small object, about an axis through $O$ perpendicular to the plane of the frame, is $\frac { 169 } { 3 } m a ^ { 2 }$.
Show that small oscillations of the system about this axis are approximately simple harmonic and state their period.

CAIE FP2 2017 Specimen Q3

11 marks Standard +0.8

Find the value of $k$.
The particle $P$ is released from rest at a point between $A$ and $B$ where both strings are taut. Show that $P$ performs simple harmonic motion and state the period of the motion.
In the case where $P$ is released from rest at a distance $0.2 a \mathrm {~m}$ from $M$, the speed of $P$ is $0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when $P$ is $0.05 a \mathrm {~m}$ from $M$. Find the value of $a$.

the angular speed of the smaller disc, [2]
the magnitude of the acceleration of a point $P$ on the circumference of the larger disc, and the angle between the direction of this acceleration and $PA$. [7]

CAIE FP2 2010 June Q5

13 marks Challenging +1.8

\includegraphics{figure_5} A light elastic band, of total natural length $a$ and modulus of elasticity $\frac{1}{2}mg$, is stretched over two small smooth pins fixed at the same horizontal level and at a distance $a$ apart. A particle of mass $m$ is attached to the lower part of the band and when the particle is in equilibrium the sloping parts of the band each make an angle $\beta$ with the vertical (see diagram). Express the tension in the band in terms of $m$, $g$ and $\beta$, and hence show that $\beta = \frac{1}{4}\pi$. [4] The particle is given a velocity of magnitude $\sqrt{(ag)}$ vertically downwards. At time $t$ the displacement of the particle from its equilibrium position is $x$. Show that, neglecting air resistance, $$\ddot{x} = -\frac{2g}{a}x.$$ [3] Show that the particle passes through the level of the pins in the subsequent motion, and find the time taken to reach this level for the first time. [6]

CAIE FP2 2010 June Q6

5 marks Standard +0.3

The lifetime, $X$ days, of a particular insect is such that $\log_{10} X$ has a normal distribution with mean $1.5$ and standard deviation $0.2$. Find the median lifetime. [3] Find also P$(X \geqslant 50)$. [2]

CAIE FP2 2010 June Q7

7 marks Standard +0.8

The continuous random variable $X$ has distribution function given by $$\text{F}(x) = \begin{cases} 0 & x < 0, \\ 1 - e^{-\frac{x}{4}} & x \geqslant 0. \end{cases}$$ For a random value of $X$, find the probability that $2$ lies between $X$ and $4X$. [3] Find also the expected value of the width of the interval $(X, 4X)$. [4]

CAIE FP2 2010 June Q8

9 marks Challenging +1.2

An examination involved writing an essay. In order to compare the time taken to write the essay by students from two large colleges, a sample of $12$ students from college A and a sample of $8$ students from college B were randomly selected. The times, $t_A$ and $t_B$, taken for these students to write the essay were measured, correct to the nearest minute, and are summarised by $n_A = 12$, $\Sigma t_A = 257$, $\Sigma t_A^2 = 5629$, $n_B = 8$, $\Sigma t_B = 206$, $\Sigma t_B^2 = 5359$. Stating any required assumptions, calculate a $95\%$ confidence interval for the difference in the population means. [8] State, giving a reason, whether your confidence interval supports the statement that the population means, for the two colleges, are equal. [1]

CAIE FP2 2010 June Q9

9 marks Moderate -0.3

A set of $20$ pairs of bivariate data $(x, y)$ is summarised by $$\Sigma x = 200, \quad \Sigma x^2 = 2125, \quad \Sigma y = 240, \quad \Sigma y^2 = 8245.$$ The product moment correlation coefficient is $-0.992$.

What does the value of the product moment correlation coefficient indicate about a scatter diagram of the data points? [1]
Find the equation of the regression line of $y$ on $x$. [6]
The equation of the regression line of $x$ on $y$ is $x = a' + b'y$. Find the value of $b'$. [2]

CAIE FP2 2010 June Q10

13 marks Standard +0.3

Three new flu vaccines, $A$, $B$ and $C$, were tested on $500$ volunteers. The vaccines were assigned randomly to the volunteers and $178$ received $A$, $149$ received $B$ and $173$ received $C$. During the following winter, of the volunteers given $A$ caught flu, $29$ of the volunteers given $B$ caught flu, and $16$ of the volunteers given $C$ caught flu. Carry out a suitable test for independence at the $5\%$ significance level. [10] Without using a statistical test, decide which of the vaccines appears to be most effective. [3]

CAIE FP2 2010 June Q11

28 marks Challenging +1.8

Answer only one of the following two alternatives. **EITHER** \includegraphics{figure_11a} A uniform disc, of mass $4m$ and radius $a$, and a uniform ring, of mass $m$ and radius $2a$, each have centre $O$. A wheel is made by fixing three uniform rods, $OA$, $OB$ and $OC$, each of mass $m$ and length $2a$, to the disc and the ring, as shown in the diagram. Show that the moment of inertia of the wheel about an axis through $A$, perpendicular to the plane of the wheel, is $42ma^2$. [5] The axis through $A$ is horizontal, and the wheel can rotate freely about this axis. The wheel is released from rest with $O$ above the level of $A$ and $AO$ making an angle of $30°$ with the horizontal. Find the angular speed of the wheel when $AO$ is horizontal. [3] When $AO$ is horizontal the disc becomes detached from the wheel. Find the angle that $AO$ makes with the horizontal when the wheel first comes to instantaneous rest. [6] **OR** The continuous random variable $T$ has probability density function given by $$f(t) = \begin{cases} 0 & t < 2, \\ \frac{2}{(t-1)^3} & t \geqslant 2. \end{cases}$$

Find the distribution function of $T$, and find also P$(T > 5)$. [3]
Consecutive independent observations of $T$ are made until the first observation that exceeds $5$ is obtained. The random variable $N$ is the total number of observations that have been made up to and including the observation exceeding $5$. Find P$(N > E(N))$. [3]
Find the probability density function of $Y$, where $Y = \frac{1}{T-1}$. [8]

CAIE FP2 2012 June Q1

4 marks Moderate -0.3

A circular flywheel of radius 0.3 m, with moment of inertia about its axis 18 kg m$^2$, is rotating freely with angular speed 6 rad s$^{-1}$. A tangential force of constant magnitude 48 N is applied to the rim of the flywheel, in order to slow the flywheel down. Find the time taken for the angular speed of the flywheel to be reduced to 2 rad s$^{-1}$. [4]

CAIE FP2 2012 June Q2

6 marks Standard +0.8

Two particles, of masses $3m$ and $m$, are moving in the same straight line towards each other with speeds $2u$ and $u$ respectively. When they collide, the impulse acting on each particle has magnitude $4mu$. Show that the total loss in kinetic energy is $\frac{4}{5}mu^2$. [6]

CAIE FP2 2012 June Q3

10 marks Challenging +1.8

A particle $P$ of mass $m$ is projected horizontally with speed $\sqrt{\left(\frac{1}{2}ga\right)}$ from the lowest point of the inside of a fixed hollow smooth sphere of internal radius $a$ and centre $O$. The angle between $OP$ and the downward vertical at $O$ is denoted by $\theta$. Show that, as long as $P$ remains in contact with the inner surface of the sphere, the magnitude of the reaction between the sphere and the particle is $\frac{5}{2}mg(1 + 2\cos \theta)$. [4] Find the speed of $P$

when it loses contact with the sphere, [3]
when, in the subsequent motion, it passes through the horizontal plane containing $O$. (You may assume that this happens before $P$ comes into contact with the sphere again.) [3]