Questions — CAIE FP2 (515 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
CAIE FP2 2010 November Q8
7 marks Standard +0.3
The owner of three driving schools, \(A\), \(B\) and \(C\), wished to assess whether there was an association between passing the driving test and the school attended. He selected a random sample of learner drivers from each of his schools and recorded the numbers of passes and failures at each school. The results that he obtained are shown in the table below.
Driving school attended
\(A\)\(B\)\(C\)
Passes231517
Failures272543
Using a \(\chi^2\)-test and a 5% level of significance, test whether there is an association between passing or failing the driving test and the driving school attended. [7]
CAIE FP2 2010 November Q9
10 marks Challenging +1.2
A national athletics coach suspects that, on average, 200-metre runners' indoor times exceed their outdoor times by more than 0.1 seconds. In order to test this, the coach randomly selects eight 200-metre runners and records their indoor and outdoor times. The results, in seconds, are shown in the table.
Runner\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Indoor time21.521.820.921.221.421.421.221.0
Outdoor time21.121.720.720.921.321.021.120.8
Stating suitable hypotheses and any necessary assumption that you make, test the coach's suspicion at the 2.5% level of significance. [10]
CAIE FP2 2010 November Q10
13 marks Standard +0.3
For each month of a certain year, a weather station recorded the average rainfall per day, \(x\) mm, and the average amount of sunshine per day, \(y\) hours. The results are summarised below. \(n = 12\), \(\Sigma x = 24.29\), \(\Sigma x^2 = 50.146\), \(\Sigma y = 45.8\), \(\Sigma y^2 = 211.16\), \(\Sigma xy = 88.415\).
  1. Find the mean values, \(\bar{x}\) and \(\bar{y}\). [1]
  2. Calculate the gradient of the line of regression of \(y\) on \(x\). [2]
  3. Use the answers to parts (i) and (ii) to obtain the equation of the line of regression of \(y\) on \(x\). [2]
  4. Find the product moment correlation coefficient and comment, in context, on its value. [4]
  5. Stating your hypotheses, test at the 1% level of significance whether there is negative correlation between average rainfall per day and average amount of sunshine per day. [4]
CAIE FP2 2010 November Q11
28 marks Standard +0.3
Answer only one of the following two alternatives. **EITHER** A particle of mass 0.1 kg lies on a smooth horizontal table on the line between two points \(A\) and \(B\) on the table, which are 6 m apart. The particle is joined to \(A\) by a light elastic string of natural length 2 m and modulus of elasticity 60 N, and to \(B\) by a light elastic string of natural length 1 m and modulus of elasticity 20 N. The mid-point of \(AB\) is \(M\), and \(O\) is the point between \(M\) and \(B\) at which the particle can rest in equilibrium. Show that \(MO = 0.2\) m. [4] The particle is held at \(M\) and then released. Show that the equation of motion is $$\frac{\mathrm{d}^2y}{\mathrm{d}t^2} = -500y,$$ where \(y\) metres is the displacement from \(O\) in the direction \(OB\) at time \(t\) seconds, and state the period of the motion. [5] For the instant when the particle is 0.3 m from \(M\) for the first time, find
  1. the speed of the particle, [2]
  2. the time taken, after release, to reach this position. [3]
**OR** The continuous random variable \(T\) has a negative exponential distribution with probability density function given by $$\mathrm{f}(t) = \begin{cases} \lambda\mathrm{e}^{-\lambda t} & t \geqslant 0, \\ 0 & \text{otherwise.} \end{cases}$$ Show that for \(t \geqslant 0\) the distribution function is given by F\((t) = 1 - \mathrm{e}^{-\lambda t}\). [2] The table below shows some values of F\((t)\) for the case when the mean is 20. Find the missing value. [2]
\(t\)0510152025303540
F\((t)\)00.22120.39350.63210.71350.77690.82620.8647
It is thought that the lifetime of a species of insect under laboratory conditions has a negative exponential distribution with mean 20 hours. When observation starts there are 100 insects, which have been randomly selected. The lifetimes of the insects, in hours, are summarised in the table below.
Lifetime (hours)\(0-5\)\(5-10\)\(10-15\)\(15-20\)\(20-25\)\(25-30\)\(30-35\)\(35-40\)\(\geqslant 40\)
Frequency2020119985117
Calculate the expected values for each interval, assuming a negative exponential model with a mean of 20 hours, giving your values correct to 2 decimal places. [3] Perform a \(\chi^2\)-test of goodness of fit, at the 5% level of significance, in order to test whether a negative exponential distribution, with a mean of 20 hours, is a suitable model for the lifetime of this species of insect under laboratory conditions. [7]
CAIE FP2 2014 November Q1
5 marks Standard +0.3
Two smooth spheres \(A\) and \(B\), of equal radii and masses \(2m\) and \(m\) respectively, lie at rest on a smooth horizontal table. The spheres \(A\) and \(B\) are projected directly towards each other with speeds \(4u\) and \(3u\) respectively. The coefficient of restitution between the spheres is \(e\). Find the set of values of \(e\) for which the direction of motion of \(A\) is reversed in the collision. [5]
CAIE FP2 2014 November Q2
5 marks Standard +0.8
\includegraphics{figure_2} A small smooth ball \(P\) is moving on a smooth horizontal plane with speed \(4\text{ m s}^{-1}\). It strikes a smooth vertical barrier at an angle \(\alpha\) (see diagram). The coefficient of restitution between \(P\) and the barrier is \(0.4\). Given that the speed of \(P\) is halved as a result of the collision, find the value of \(\alpha\). [5]
CAIE FP2 2014 November Q3
10 marks Challenging +1.2
\includegraphics{figure_3} A smooth cylinder of radius \(a\) is fixed with its axis horizontal. The point \(O\) is the centre of a circular cross-section of the cylinder. The line \(AOB\) is a diameter of this circular cross-section and the radius \(OA\) makes an angle \(\alpha\) with the upward vertical (see diagram). It is given that \(\cos \alpha = \frac{3}{5}\). A particle \(P\) of mass \(m\) moves on the inner surface of the cylinder in the plane of the cross-section. The particle passes through \(A\) with speed \(u\) along the surface in the downwards direction. The magnitude of the reaction between \(P\) and the inner surface of the sphere is \(R_A\) when \(P\) is at \(A\), and is \(R_B\) when \(P\) is at \(B\). It is given that \(R_B = 10R_A\). Show that \(u^2 = ag\). [6] The particle loses contact with the surface of the cylinder when \(OP\) makes an angle \(\theta\) with the upward vertical. Find the value of \(\cos \theta\). [4]
CAIE FP2 2014 November Q4
11 marks Challenging +1.2
\includegraphics{figure_4} A uniform rod \(AB\), of length \(l\) and mass \(m\), rests in equilibrium with its lower end \(A\) on a rough horizontal floor and the end \(B\) against a smooth vertical wall. The rod is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{4}{3}\), and is in a vertical plane perpendicular to the wall. The rod is supported by a light spring \(CD\) which is in compression in a vertical line with its lower end \(D\) fixed on the floor. The upper end \(C\) is attached to the rod at a distance \(\frac{4l}{5}\) from \(B\) (see diagram). The coefficient of friction at \(A\) between the rod and the floor is \(\frac{1}{2}\) and the system is in limiting equilibrium.
  1. Show that the normal reaction of the floor at \(A\) has magnitude \(\frac{1}{2}mg\) and find the force in the spring. [7]
  2. Given that the modulus of elasticity of the spring is \(2mg\), find the natural length of the spring. [4]
CAIE FP2 2014 November Q5
12 marks Challenging +1.2
The points \(A\) and \(B\) are on a smooth horizontal table at a distance \(8a\) apart. A particle \(P\) of mass \(m\) lies on the table on the line \(AB\), between \(A\) and \(B\). The particle is attached to \(A\) by a light elastic string of natural length \(3a\) and modulus of elasticity \(6mg\), and to \(B\) by a light elastic string of natural length \(2a\) and modulus of elasticity \(mg\). In equilibrium, \(P\) is at the point \(O\) on \(AB\).
  1. Show that \(AO = 3.6a\). [4]
The particle is released from rest at the point \(C\) on \(AB\), between \(A\) and \(B\), where \(AC = 3.4a\).
  1. Show that \(P\) moves in simple harmonic motion and state the period. [6]
  2. Find the greatest speed of \(P\). [2]
CAIE FP2 2014 November Q6
5 marks Challenging +1.2
A random sample of 50 observations of a random variable \(X\) and a random sample of 60 observations of a random variable \(Y\) are taken. The results for the sample means, \(\bar{x}\) and \(\bar{y}\), and the unbiased estimates for the population variances, \(s_x^2\) and \(s_y^2\), respectively, are as follows. $$\bar{x} = 25.4 \quad \bar{y} = 23.6 \quad s_x^2 = 23.2 \quad s_y^2 = 27.8$$ A test, at the \(\alpha\%\) significance level, of the null hypothesis that the population means of \(X\) and \(Y\) are equal against the alternative hypothesis that they are not equal is carried out. Given that the null hypothesis is not rejected, find the set of possible values of \(\alpha\). [5]
CAIE FP2 2014 November Q7
6 marks Standard +0.3
The time, \(T\) seconds, between successive cars passing a particular checkpoint on a wide road has probability density function f given by $$f(t) = \begin{cases} \frac{1}{100}e^{-0.01t} & t \geq 0, \\ 0 & \text{otherwise.} \end{cases}$$
  1. State the expected value of \(T\). [1]
  2. Find the median value of \(T\). [3]
Sally wishes to cross the road at this checkpoint and she needs 20 seconds to complete the crossing. She decides to start out immediately after a car passes. Find the probability that she will complete the crossing before the next car passes. [2]
CAIE FP2 2014 November Q8
9 marks Challenging +1.2
The numbers of a particular type of laptop computer sold by a store on each of 100 consecutive Saturdays are summarised in the following table.
Number sold01234567\(\geq 8\)
Number of Saturdays7203916142110
Fit a Poisson distribution to the data and carry out a goodness of fit test at the 2.5% significance level. [9]
CAIE FP2 2014 November Q9
11 marks Standard +0.3
A random sample of 10 pairs of values of \(x\) and \(y\) is given in the following table.
\(x\)466827121495
\(y\)24686109865
  1. Find the equation of the regression line of \(y\) on \(x\). [4]
  2. Find the product moment correlation coefficient for the sample. [2]
  3. Find the estimated value of \(y\) when \(x = 10\), and comment on the reliability of this estimate. [2]
  4. Another sample of \(N\) pairs of data from the same population has the same product moment correlation coefficient as the first sample given. A test, at the 1% significance level, on this second sample indicates that there is sufficient evidence to conclude that there is positive correlation. Find the set of possible values of \(N\). [3]
CAIE FP2 2014 November Q10
12 marks Standard +0.8
The continuous random variable \(X\) has probability density function f given by $$f(x) = \begin{cases} \frac{1}{2} & 1 \leq x \leq 3, \\ 0 & \text{otherwise.} \end{cases}$$ The random variable \(Y\) is defined by \(Y = X^3\). Find the distribution function of \(Y\). [5] Sketch the graph of the probability density function of \(Y\). [3] Find the probability that \(Y\) lies between its median value and its mean value. [4]
CAIE FP2 2014 November Q11
28 marks Challenging +1.8
Answer only one of the following two alternatives. **EITHER** \includegraphics{figure_11a} A uniform plane object consists of three identical circular rings, \(X\), \(Y\) and \(Z\), enclosed in a larger circular ring \(W\). Each of the inner rings has mass \(m\) and radius \(r\). The outer ring has mass \(3m\) and radius \(R\). The centres of the inner rings lie at the vertices of an equilateral triangle of side \(2r\). The outer ring touches each of the inner rings and the rings are rigidly joined together. The fixed axis \(AB\) is the diameter of \(W\) that passes through the centre of \(X\) and the point of contact of \(Y\) and \(Z\) (see diagram). It is given that \(R = \left(1 + \frac{2}{3}\sqrt{3}\right)r\).
  1. Show that the moment of inertia of the object about \(AB\) is \(\left(7 + 2\sqrt{3}\right)mr^2\). [8]
The line \(CD\) is the diameter of \(W\) that is perpendicular to \(AB\). A particle of mass \(9m\) is attached to \(D\). The object is now held with its plane horizontal. It is released from rest and rotates freely about the fixed horizontal axis \(AB\).
  1. Find, in terms of \(g\) and \(r\), the angular speed of the object when it has rotated through \(60°\). [6]
**OR** Fish of a certain species live in two separate lakes, \(A\) and \(B\). A zoologist claims that the mean length of fish in \(A\) is greater than the mean length of fish in \(B\). To test his claim, he catches a random sample of 8 fish from \(A\) and a random sample of 6 fish from \(B\). The lengths of the 8 fish from \(A\), in appropriate units, are as follows. $$15.3 \quad 12.0 \quad 15.1 \quad 11.2 \quad 14.4 \quad 13.8 \quad 12.4 \quad 11.8$$ Assuming a normal distribution, find a 95% confidence interval for the mean length of fish in \(A\). [5] The lengths of the 6 fish from \(B\), in the same units, are as follows. $$15.0 \quad 10.7 \quad 13.6 \quad 12.4 \quad 11.6 \quad 12.6$$ Stating any assumptions that you make, test at the 5% significance level whether the mean length of fish in \(A\) is greater than the mean length of fish in \(B\). [7] Calculate a 95% confidence interval for the difference in the mean lengths of fish from \(A\) and from \(B\). [2]
CAIE FP2 2015 November Q1
9 marks Challenging +1.2
\includegraphics{figure_1} A uniform ladder \(AB\), of length \(3a\) and weight \(W\), rests with the end \(A\) in contact with smooth horizontal ground and the end \(B\) against a smooth vertical wall. One end of a light inextensible rope is attached to the ladder at the point \(C\), where \(AC = a\). The other end of the rope is fixed to the point \(D\) at the base of the wall and the rope \(DC\) is in the same vertical plane as the ladder \(AB\). The ladder rests in equilibrium in a vertical plane perpendicular to the wall, with the ladder making an angle \(\theta\) with the horizontal and the rope making an angle \(\alpha\) with the horizontal (see diagram). It is given that \(\tan \alpha = 2\tan \theta\). Find, in terms of \(W\) and \(\alpha\), the tension in the rope and the magnitudes of the forces acting on the ladder at \(A\) and at \(B\). [9]
CAIE FP2 2015 November Q2
10 marks Standard +0.8
A small uniform sphere \(A\), of mass \(2m\), is moving with speed \(u\) on a smooth horizontal surface when it collides directly with a small uniform sphere \(B\), of mass \(m\), which is at rest. The spheres have equal radii and the coefficient of restitution between them is \(e\). Find expressions for the speeds of \(A\) and \(B\) immediately after the collision. [4] Subsequently \(B\) collides with a vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(0.4\). After \(B\) has collided with the wall, the speeds of \(A\) and \(B\) are equal. Find \(e\). [2] Initially \(B\) is at a distance \(d\) from the wall. Find the distance of \(B\) from the wall when it next collides with \(A\). [4]
CAIE FP2 2015 November Q3
11 marks Challenging +1.3
\(A\) and \(B\) are two fixed points on a smooth horizontal surface, with \(AB = 3a\) m. One end of a light elastic string, of natural length \(a\) m and modulus of elasticity \(mg\) N, is attached to the point \(A\). The other end of this string is attached to a particle \(P\) of mass \(m\) kg. One end of a second light elastic string, of natural length \(ka\) m and modulus of elasticity \(2mg\) N, is attached to \(B\). The other end of this string is attached to \(P\). Given that the system is in equilibrium when \(P\) is at \(M\), the mid-point of \(AB\), find the value of \(k\). [3] 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. [5] In the case where \(P\) is released from rest at a distance \(0.2a\) m from \(M\), the speed of \(P\) is \(0.7\) m s\(^{-1}\) when \(P\) is \(0.05a\) m from \(M\). Find the value of \(a\). [3]
CAIE FP2 2015 November Q4
13 marks Challenging +1.8
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 \(O\). When \(P\) is hanging at rest vertically below \(O\), it is projected horizontally. In the subsequent motion \(P\) completes a vertical circle. The speed of \(P\) when it is at its highest point is \(u\). Show that the least possible value of \(u\) is \(\sqrt{(ag)}\). [2] It is now given that \(u = \sqrt{(ag)}\). When \(P\) passes through the lowest point of its path, it collides with, and coalesces with, a stationary particle of mass \(\frac{1}{4}m\). Find the speed of the combined particle immediately after the collision. [4] In the subsequent motion, when \(OP\) makes an angle \(\theta\) with the upward vertical the tension in the string is \(T\). Find an expression for \(T\) in terms of \(m\), \(g\) and \(\theta\). [5] Find the value of \(\cos \theta\) when the string becomes slack. [2]
CAIE FP2 2015 November Q5
5 marks Standard +0.3
A random sample of 10 observations of a normal variable \(X\) gave the following summarised data, where \(\bar{x}\) is the sample mean. $$\Sigma x = 222.8 \qquad \Sigma(x - \bar{x})^2 = 4.12$$ Find a 95% confidence interval for the population mean. [5]
CAIE FP2 2015 November Q6
8 marks Standard +0.3
A biased coin is tossed repeatedly until a head is obtained. The random variable \(X\) denotes the number of tosses required for a head to be obtained. The mean of \(X\) is equal to twice the variance of \(X\). Show that the probability that a head is obtained when the coin is tossed once is \(\frac{2}{3}\). [2] Find
  1. P(\(X = 4\)), [1]
  2. P(\(X > 4\)), [2]
  3. the least integer \(N\) such that P(\(X \leq N\)) \(> 0.999\). [3]
CAIE FP2 2015 November Q7
9 marks Standard +0.8
The continuous random variable \(X\) has probability density function given by $$f(x) = \begin{cases} \frac{1}{2t}x^2 & 1 \leq x \leq 4, \\ 0 & \text{otherwise}. \end{cases}$$ The random variable \(Y\) is defined by \(Y = X^2\). Show that \(Y\) has probability density function given by $$g(y) = \begin{cases} \left(\frac{1}{2t}\right)y^{\frac{1}{2}} & 1 \leq y \leq 16, \\ 0 & \text{otherwise}. \end{cases}$$ [5] Find
  1. the median value of \(Y\), [2]
  2. the expected value of \(Y\). [2]
CAIE FP2 2015 November Q8
10 marks Standard +0.8
The number of goals scored by a certain football team was recorded for each of 100 matches, and the results are summarised in the following table.
Number of goals0123456 or more
Frequency121631251330
Fit a Poisson distribution to the data, and test its goodness of fit at the 5% significance level. [10]
CAIE FP2 2015 November Q9
11 marks Standard +0.3
A random sample of 8 students is chosen from those sitting examinations in both Mathematics and French. Their marks in Mathematics, \(x\), and in French, \(y\), are summarised as follows. $$\Sigma x = 472 \qquad \Sigma x^2 = 29950 \qquad \Sigma y = 400 \qquad \Sigma y^2 = 21226 \qquad \Sigma xy = 24879$$ Another student scored 72 marks in the Mathematics examination but was unable to sit the French examination. Estimate the mark that this student would have obtained in the French examination. [5] Test, at the 5% significance level, whether there is non-zero correlation between marks in Mathematics and marks in French. [6]
CAIE FP2 2015 November Q10
28 marks Moderate -0.5
Answer only one of the following two alternatives. EITHER \includegraphics{figure_10a} An object is formed by attaching a thin uniform rod \(PQ\) to a uniform rectangular lamina \(ABCD\). The lamina has mass \(m\), and \(AB = DC = 6a\), \(BC = AD = 3a\). The rod has mass \(M\) and length \(3a\). The end \(P\) of the rod is attached to the mid-point of \(AB\). The rod is perpendicular to \(AB\) 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 \(3(8m + M)a^2\). [4] Show that the moment of inertia of the object about a smooth horizontal axis \(l_2\), through the mid-point of \(PQ\) and perpendicular to the plane of the lamina, is \(\frac{3}{4}(17m + M)a^2\). [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 \(m = M\). [8] OR 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 \(x\) kg. The data are summarised as follows. $$\Sigma x = 42.0 \qquad \Sigma x^2 = 180.0$$ A random sample of 12 Majestic plants is taken. The total mass of potatoes on a plant is \(y\) kg. The data are summarised as follows. $$\Sigma y = 57.6 \qquad \Sigma y^2 = 281.5$$ 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. [9] 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. [5]