Questions — CAIE FP2 (515 questions)

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CAIE FP2 2019 November Q4
9 marks Challenging +1.2
A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\) and \(P\) is held with the string taut and horizontal. The particle \(P\) is projected vertically downwards with speed \(\sqrt{(2ag)}\) so that it begins to move along a circular path. The string becomes slack when \(OP\) makes an angle \(\theta\) with the upward vertical through \(O\).
  1. Show that \(\cos \theta = \frac{2}{3}\). [5]
  2. Find the greatest height, above the horizontal through \(O\), reached by \(P\) in its subsequent motion. [4]
CAIE FP2 2019 November Q5
12 marks Challenging +1.8
\includegraphics{figure_5} A thin uniform rod \(AB\) has mass \(\lambda M\) and length \(2a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere (spherical shell) with centre \(O\), mass \(3M\) and radius \(a\). The end \(B\) of the rod is rigidly attached to the surface of a uniform solid sphere with centre \(C\), mass \(5M\) and radius \(a\). The rod lies along the line joining the centres of the spheres, so that \(CBAO\) is a straight line. The horizontal axis \(L\) is perpendicular to the rod and passes through the point of the rod that is a distance \(\frac{1}{2}a\) from \(B\) (see diagram). The object consisting of the rod and the two spheres can rotate freely about \(L\).
  1. Show that the moment of inertia of the object about \(L\) is \(\left(\frac{408 + 7\lambda}{12}\right)Ma^2\). [6]
The period of small oscillations of the object about \(L\) is \(5\pi\sqrt{\left(\frac{2a}{g}\right)}\).
  1. Find the value of \(\lambda\). [6]
CAIE FP2 2019 November Q6
7 marks Challenging +1.2
A random sample of 9 members is taken from the large number of members of a sports club, and their heights are measured. The heights of all the members of the club are assumed to be normally distributed. A 95% confidence interval for the population mean height, \(\mu\) metres, is calculated from the data as \(1.65 \leqslant \mu \leqslant 1.85\).
  1. Find an unbiased estimate for the population variance. [3]
  2. Denoting the height of a member of the club by \(x\) metres, find \(\Sigma x^2\) for this sample of 9 members. [4]
CAIE FP2 2019 November Q7
7 marks Standard +0.3
The time, \(T\) days, before an electrical component develops a fault has distribution function F given by $$\mathrm{F}(t) = \begin{cases} 1 - e^{-at} & t \geqslant 0, \\ 0 & \text{otherwise}, \end{cases}$$ where \(a\) is a positive constant. The mean value of \(T\) is 200.
  1. Write down the value of \(a\). [1]
  2. Find the probability that an electrical component of this type develops a fault in less than 150 days. [2]
A piece of equipment contains \(n\) of these components, which develop faults independently of each other. The probability that, after 150 days, at least one of the \(n\) components has not developed a fault is greater than 0.99.
  1. Find the smallest possible value of \(n\). [4]
CAIE FP2 2019 November Q8
9 marks Challenging +1.2
A random sample of 8 elephants from region \(A\) is taken and their weights, \(x\) tonnes, are recorded. (1 tonne = 1000 kg.) The results are summarised as follows. $$\Sigma x = 32.4 \quad \Sigma x^2 = 131.82$$ A random sample of 10 elephants from region \(B\) is taken. Their weights give a sample mean of 3.78 tonnes and an unbiased variance estimate of 0.1555 tonnes\(^2\). The distributions of the weights of elephants in regions \(A\) and \(B\) are both assumed to be normal with the same population variance. Test at the 10% significance level whether the mean weight of elephants in region \(A\) is the same as the mean weight of elephants in region \(B\). [9]
CAIE FP2 2019 November Q9
10 marks Standard +0.8
A random sample of five pairs of values of \(x\) and \(y\) is taken from a bivariate distribution. The values are shown in the following table, where \(p\) and \(q\) are constants.
\(x\)12345
\(y\)4\(p\)\(q\)21
The equation of the regression line of \(y\) on \(x\) is \(y = -0.5x + 3.5\).
  1. Find the values of \(p\) and \(q\). [7]
  2. Find the value of the product moment correlation coefficient. [3]
CAIE FP2 2019 November Q10
10 marks Standard +0.8
The random variable \(X\) has probability density function f given by $$\mathrm{f}(x) = \begin{cases} \frac{1}{30}\left(\frac{8}{x^2} + 3x^2 - 14\right) & 2 \leqslant x \leqslant 4, \\ 0 & \text{otherwise}. \end{cases}$$
  1. Find the distribution function of \(X\). [3]
The random variable \(Y\) is defined by \(Y = X^2\).
  1. Find the probability density function of \(Y\). [4]
  2. Find the value of \(y\) such that \(\mathrm{P}(Y < y) = 0.8\). [3]
CAIE FP2 2019 November Q11
28 marks Challenging +1.2
Answer only one of the following two alternatives. EITHER The points \(A\) and \(B\) are a distance 1.2 m apart on a smooth horizontal surface. A particle \(P\) of mass \(\frac{2}{3}\) kg is attached to one end of a light spring of natural length 0.6 m and modulus of elasticity 10 N. The other end of the spring is attached to the point \(A\). A second light spring, of natural length 0.4 m and modulus of elasticity 20 N, has one end attached to \(P\) and the other end attached to \(B\).
  1. Show that when \(P\) is in equilibrium \(AP = 0.75\) m. [3]
The particle \(P\) is displaced by 0.05 m from the equilibrium position towards \(A\) and then released from rest.
  1. Show that \(P\) performs simple harmonic motion and state the period of the motion. [6]
  2. Find the speed of \(P\) when it passes through the equilibrium position. [2]
  3. Find the speed of \(P\) when its acceleration is equal to half of its maximum value. [3]
OR The number of puncture repairs carried out each week by a small repair shop is recorded over a period of 40 weeks. The results are shown in the following table.
Number of repairs in a week012345\(\geqslant 6\)
Number of weeks61596310
  1. Calculate the mean and variance for the number of repairs in a week and comment on the possible suitability of a Poisson distribution to model the data. [3]
Records over a longer period of time indicate that the mean number of repairs in a week is 1.6. The following table shows some of the expected frequencies, correct to 3 decimal places, for a period of 40 weeks using a Poisson distribution with mean 1.6.
Number of repairs in a week012345\(\geqslant 6\)
Expected frequency8.07612.92110.3375.5132.205\(a\)\(b\)
  1. Show that \(a = 0.706\) and find the value of the constant \(b\). [3]
  2. Carry out a goodness of fit test of a Poisson distribution with mean 1.6, using a 10% significance level. [8]
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 \(B\),
  1. show that \(P = \frac { 15 } { 34 } \mathrm {~W}\),
  2. 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.
  1. Find the value of the product moment correlation coefficient for the sample.
  2. Test, at the \(10 \%\) significance level, whether there is evidence of non-zero correlation between the variables.
  3. Find the mean values of \(x\) and \(y\) for this sample.
  4. 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 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,
  1. test, at the \(10 \%\) significance level, whether the population mean length of fish of this species is greater than 15.8 cm ,
  2. 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 \(\frac { 11 } { 7 } l\).
  1. Show that the modulus of elasticity of the string is \(\frac { 7 } { 5 } \mathrm { mg }\).
  2. Show that \(P\) moves in simple harmonic motion about the equilibrium position and state the period of the motion.
  3. 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 .
  4. Test, at the \(5 \%\) significance level, whether there is evidence of positive correlation between the variables.
  5. Given that \(b - a = 0.5\), find the values of \(a\) and \(b\).
  6. 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 \(z = 5 y\).
  7. 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.