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OCR S2 2016 June Q8
13 marks Standard +0.3
It is known that the lifetime of a certain species of animal in the wild has mean 13.3 years. A zoologist reads a study of 50 randomly chosen animals of this species that have been kept in zoos. According to the study, for these 50 animals the sample mean lifetime is 12.48 years and the population variance is 12.25 years\(^2\).
  1. Test at the 5% significance level whether these results provide evidence that animals of this species that have been kept in zoos have a shorter expected lifetime than those in the wild. [7]
  2. Subsequently the zoologist discovered that there had been a mistake in the study. The quoted variance of 12.25 years\(^2\) was in fact the sample variance. Determine whether this makes a difference to the conclusion of the test. [5]
  3. Explain whether the Central Limit Theorem is needed in these tests. [1]
OCR S2 2016 June Q9
6 marks Challenging +1.3
The random variable \(R\) has the distribution Po\((\lambda)\). A significance test is carried out at the 1% level of the null hypothesis H\(_0\): \(\lambda = 11\) against H\(_1\): \(\lambda > 11\), based on a single observation of \(R\). Given that in fact the value of \(\lambda\) is 14, find the probability that the result of the test is incorrect, and give the technical name for such an incorrect outcome. You should show the values of any relevant probabilities. [6]
OCR S3 2012 January Q1
6 marks Moderate -0.8
In a test of association of two factors, \(A\) and \(B\), a \(2 \times 2\) contingency table yielded \(5.63\) for the value of \(\chi^2\) with Yates' correction.
  1. State the null hypothesis and alternative hypothesis for the test. [1]
  2. State how Yates' correction is applied, and whether it increases or decreases the value of \(\chi^2\). [2]
  3. Carry out the test at the \(2\frac{1}{2}\%\) significance level. [3]
OCR S3 2012 January Q2
7 marks Standard +0.3
An investigation in 2007 into the incidence of tuberculosis (TB) in badgers in a certain area found that 42 out of a random sample of 190 badgers tested positive for TB. In 2010, 48 out of a random sample of 150 badgers tested positive for TB.
  1. Assuming that the population proportions of badgers with TB are the same in 2007 and 2010, obtain the best estimate of this proportion. [1]
  2. Carry out a test at the \(2\frac{1}{2}\%\) significance level of whether the population proportion of badgers with TB increased from 2007 to 2010. [6]
OCR S3 2012 January Q3
8 marks Standard +0.3
The continuous random variable \(U\) has a normal distribution with unknown mean \(\mu\) and known variance 1. A random sample of four observations of \(U\) gave the values \(3.9, 2.1, 4.6\) and \(1.4\).
  1. Calculate a \(90\%\) confidence interval for \(\mu\). [3]
  2. The probability that the sum of four random observations of \(U\) is less than 11 is denoted by \(p\). For each of the end points of the confidence interval in part (i) calculate the corresponding value of \(p\). [5]
OCR S3 2012 January Q4
10 marks Standard +0.3
\(X\) is a continuous random variable with the distribution N\((48.5, 12.5^2)\). The values of \(X\) are transformed to standardised values of \(Y\), using the equation \(Y = aX + b\), where \(a\) and \(b\) are constants with \(a > 0\).
  1. Find values of \(a\) and \(b\) for which the mean and standard deviation of \(Y\) are 40 and 10 respectively. [4]
  2. State the distribution of \(Y\). [1]
Two randomly chosen standardised values are denoted by \(Y_1\) and \(Y_2\).
  1. Calculate the probability that \(Y_2\) is at least 10 greater than \(Y_1\). [5]
OCR S3 2012 January Q5
10 marks Standard +0.3
A statistician suggested that the weekly sales \(X\) thousand litres at a petrol station could be modelled by the following probability density function. $$\text{f}(x) = \begin{cases} \frac{1}{40}(2x + 3) & 0 \leqslant x < 5, \\ 0 & \text{otherwise.} \end{cases}$$
  1. Show that, using this model, P\((a < X < a + 1) = \frac{a + 2}{20}\) for \(0 \leqslant a < 4\). [3]
Sales in 100 randomly chosen weeks gave the following grouped frequency table.
\(x\)\(0 \leqslant x < 1\)\(1 \leqslant x < 2\)\(2 \leqslant x < 3\)\(3 \leqslant x < 4\)\(4 \leqslant x < 5\)
Frequency1612183024
  1. Carry out a goodness of fit test at the \(10\%\) significance level of whether f\((x)\) fits the data. [7]
OCR S3 2012 January Q6
13 marks Standard +0.3
The continuous random variable \(Y\) has probability density function given by $$\text{f}(y) = \begin{cases} -\frac{1}{4}y & -2 < y < 0, \\ \frac{1}{4}y & 0 \leqslant y \leqslant 2, \\ 0 & \text{otherwise.} \end{cases}$$ Find
  1. the interquartile range of \(Y\), [4]
  2. Var\((Y)\), [5]
  3. E\((|Y|)\). [4]
OCR S3 2012 January Q7
18 marks Standard +0.3
The manufacturer's specification for batteries used in a certain electronic game is that the mean lifetime should be 32 hours. The manufacturer tests a random sample of 10 batteries made in Factory A, and the lifetimes (\(x\) hours) are summarised by \(n = 10\), \(\sum x = 289.0\) and \(\sum x^2 = 8586.19\). It may be assumed that the population of lifetimes has a normal distribution.
  1. Carry out a one-tail test at the \(5\%\) significance level of whether the specification is being met. [7]
  2. Justify the use of a one-tail test in this context. [1]
Batteries made with the same specification are also made in Factory B. The lifetimes of these batteries are also normally distributed. A random sample of 12 batteries from this factory was tested. The lifetimes are summarised by \(n = 12\), \(\sum x = 363.0\) and \(\sum x^2 = 11290.95\).
    1. State what further assumption must be made in order to test whether there is any difference in the mean lifetimes of batteries made at the two factories. Use the data to comment on whether this assumption is reasonable. [3]
    2. Carry out the test at the \(10\%\) significance level. [7]
OCR M1 Q1
7 marks Moderate -0.3
\includegraphics{figure_1} A light inextensible string has its ends attached to two fixed points \(A\) and \(B\). The point \(A\) is vertically above \(B\). A smooth ring \(R\) of mass \(m\) kg is threaded on the string and is pulled by a force of magnitude \(1.6\) N acting upwards at \(45°\) to the horizontal. The section \(AR\) of the string makes an angle of \(30°\) with the downward vertical and the section \(BR\) is horizontal (see diagram). The ring is in equilibrium with the string taut.
  1. Give a reason why the tension in the part \(AR\) of the string is the same as that in the part \(BR\). [1]
  2. Show that the tension in the string is \(0.754\) N, correct to 3 significant figures. [3]
  3. Find the value of \(m\). [3]
OCR M1 Q2
7 marks Standard +0.3
\includegraphics{figure_2} Particles \(A\) and \(B\), of masses \(0.2\) kg and \(0.3\) kg respectively, are attached to the ends of a light inextensible string. Particle \(A\) is held at rest at a fixed point and \(B\) hangs vertically below \(A\). Particle \(A\) is now released. As the particles fall the air resistance acting on \(A\) is \(0.4\) N and the air resistance acting on \(B\) is \(0.25\) N (see diagram). The downward acceleration of each of the particles is \(a\) m s\(^{-2}\) and the tension in the string is \(T\) N.
  1. Write down two equations in \(a\) and \(T\) obtained by applying Newton's second law to \(A\) and to \(B\). [4]
  2. Find the values of \(a\) and \(T\). [3]
OCR M1 Q3
8 marks Standard +0.3
Two small spheres \(P\) and \(Q\) have masses \(0.1\) kg and \(0.2\) kg respectively. The spheres are moving directly towards each other on a horizontal plane and collide. Immediately before the collision \(P\) has speed \(4\) m s\(^{-1}\) and \(Q\) has speed \(3\) m s\(^{-1}\). Immediately after the collision the spheres move away from each other, \(P\) with speed \(u\) m s\(^{-1}\) and \(Q\) with speed \((3.5 - u)\) m s\(^{-1}\).
  1. Find the value of \(u\). [4]
After the collision the spheres both move with deceleration of magnitude \(5\) m s\(^{-2}\) until they come to rest on the plane.
  1. Find the distance \(PQ\) when both \(P\) and \(Q\) are at rest. [4]
OCR M1 Q4
9 marks Standard +0.3
A particle moves downwards on a smooth plane inclined at an angle \(\alpha\) to the horizontal. The particle passes through the point \(P\) with speed \(u\) m s\(^{-1}\). The particle travels \(2\) m during the first \(0.8\) s after passing through \(P\), then a further \(6\) m in the next \(1.2\) s. Find
  1. the value of \(u\) and the acceleration of the particle, [7]
  2. the value of \(\alpha\) in degrees. [2]
OCR M1 Q5
12 marks Standard +0.8
\includegraphics{figure_5} Two small rings \(A\) and \(B\) are attached to opposite ends of a light inextensible string. The rings are threaded on a rough wire which is fixed vertically. \(A\) is above \(B\). A horizontal force is applied to a point \(P\) of the string. Both parts \(AP\) and \(BP\) of the string are taut. The system is in equilibrium with angle \(BAP = \alpha\) and angle \(ABP = \beta\) (see diagram). The weight of \(A\) is \(2\) N and the tensions in the parts \(AP\) and \(BP\) of the string are \(7\) N and \(T\) N respectively. It is given that \(\cos \alpha = 0.28\) and \(\sin \alpha = 0.96\), and that \(A\) is in limiting equilibrium.
  1. Find the coefficient of friction between the wire and the ring \(A\). [7]
  2. By considering the forces acting at \(P\), show that \(T \cos \beta = 1.96\). [2]
  3. Given that there is no frictional force acting on \(B\), find the mass of \(B\). [3]
OCR M1 Q6
12 marks Standard +0.3
A particle of mass \(0.04\) kg is acted on by a force of magnitude \(P\) N in a direction at an angle \(\alpha\) to the upward vertical.
  1. The resultant of the weight of the particle and the force applied to the particle acts horizontally. Given that \(\alpha = 20°\) find
    1. the value of \(P\), [3]
    2. the magnitude of the resultant, [2]
    3. the magnitude of the acceleration of the particle. [2]
  2. It is given instead that \(P = 0.08\) and \(\alpha = 90°\). Find the magnitude and direction of the resultant force on the particle. [5]
OCR M1 Q7
17 marks Standard +0.3
\includegraphics{figure_7} A car \(P\) starts from rest and travels along a straight road for \(600\) s. The \((t, v)\) graph for the journey is shown in the diagram. This graph consists of three straight line segments. Find
  1. the distance travelled by \(P\), [3]
  2. the deceleration of \(P\) during the interval \(500 < t < 600\). [2]
Another car \(Q\) starts from rest at the same instant as \(P\) and travels in the same direction along the same road for \(600\) s. At time \(t\) s after starting the velocity of \(Q\) is \((600t^2 - t^3) \times 10^{-6}\) m s\(^{-1}\).
  1. Find an expression in terms of \(t\) for the acceleration of \(Q\). [2]
  2. Find how much less \(Q\)'s deceleration is than \(P\)'s when \(t = 550\). [2]
  3. Show that \(Q\) has its maximum velocity when \(t = 400\). [2]
  4. Find how much further \(Q\) has travelled than \(P\) when \(t = 400\). [6]
OCR M1 Q1
7 marks Standard +0.2
\includegraphics{figure_1} Particles \(P\) and \(Q\), of masses \(0.3\) kg and \(0.4\) kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The system is in motion with the string taut and with each of the particles moving vertically. The downward acceleration of \(P\) is \(a\) m s\(^{-2}\) (see diagram).
  1. Show that \(a = -1.4\). [4]
Initially \(P\) and \(Q\) are at the same horizontal level. \(P\)'s initial velocity is vertically downwards and has magnitude \(2.8\) m s\(^{-1}\).
  1. Assuming that \(P\) does not reach the floor and that \(Q\) does not reach the pulley, find the time taken for \(P\) to return to its initial position. [3]
OCR M1 Q2
7 marks Moderate -0.3
\includegraphics{figure_2} An object of mass \(0.08\) kg is attached to one end of a light inextensible string. The other end of the string is attached to the underside of the roof inside a furniture van. The van is moving horizontally with constant acceleration \(1.25\) m s\(^{-2}\). The string makes a constant angle \(\alpha\) with the downward vertical and the tension in the string is \(T\) N (see diagram).
  1. By applying Newton's second law horizontally to the object, find the value of \(T \sin \alpha\). [2]
  2. Find the value of \(T\). [5]
OCR M1 Q3
11 marks Moderate -0.3
A motorcyclist starts from rest at a point \(O\) and travels in a straight line. His velocity after \(t\) seconds is \(v\) m s\(^{-1}\), for \(0 \leq t \leq T\), where \(v = 7.2t - 0.45t^2\). The motorcyclist's acceleration is zero when \(t = T\).
  1. Find the value of \(T\). [4]
  2. Show that \(v = 28.8\) when \(t = T\). [1]
For \(t \geq T\) the motorcyclist travels in the same direction as before, but with constant speed \(28.8\) m s\(^{-1}\).
  1. Find the displacement of the motorcyclist from \(O\) when \(t = 31\). [6]
OCR M1 Q4
11 marks Moderate -0.3
\includegraphics{figure_4} A block of mass \(2\) kg is at rest on a rough horizontal plane, acted on by a force of magnitude \(12\) N at an angle of \(15°\) upwards from the horizontal (see diagram).
  1. Find the frictional component of the contact force exerted on the block by the plane. [2]
  2. Show that the normal component of the contact force exerted on the block by the plane has magnitude \(16.5\) N, correct to 3 significant figures. [2]
It is given that the block is on the point of sliding.
  1. Find the coefficient of friction between the block and the plane. [2]
The force of magnitude \(12\) N is now replaced by a horizontal force of magnitude \(20\) N. The block starts to move.
  1. Find the acceleration of the block. [5]
OCR M1 Q5
11 marks Standard +0.3
A man drives a car on a horizontal straight road. At \(t = 0\), where the time \(t\) is in seconds, the car runs out of petrol. At this instant the car is moving at \(12\) m s\(^{-1}\). The car decelerates uniformly, coming to rest when \(t = 8\). The man then walks back along the road at \(0.7\) m s\(^{-1}\) until he reaches a petrol station a distance of \(420\) m from his car. After his arrival at the petrol station it takes him \(250\) s to obtain a can of petrol. He is then given a lift back to his car on a motorcycle. The motorcycle starts from rest and accelerates uniformly until its speed is \(20\) m s\(^{-1}\); it then decelerates uniformly, coming to rest at the stationary car at time \(t = T\).
  1. Sketch the shape of the \((t, v)\) graph for the man for \(0 \leq t \leq T\). [Your sketch need not be drawn to scale; numerical values need not be shown.] [5]
  2. Find the deceleration of the car for \(0 < t < 8\). [2]
  3. Find the value of \(T\). [4]
OCR M1 Q6
11 marks Standard +0.3
\includegraphics{figure_6} A smooth ring \(R\) of weight \(W\) N is threaded on a light inextensible string. The ends of the string are attached to fixed points \(A\) and \(B\), where \(A\) is vertically above \(B\). A horizontal force of magnitude \(P\) N acts on \(R\). The system is in equilibrium with the string taut; \(AR\) makes an angle \(\alpha\) with the downward vertical and \(BR\) makes an angle \(\beta\) with the upward vertical (see Fig. 1).
  1. By considering the vertical components of the forces acting on \(R\), show that \(\alpha < \beta\). [3]
  1. \includegraphics{figure_6ii} It is given that when \(P = 14\), \(AR = 0.4\) m, \(BR = 0.3\) m and the distance of \(R\) from the vertical line \(AB\) is \(0.24\) m (see Fig. 2). Find
    1. the tension in the string, [3]
    2. the value of \(W\). [3]
  2. For the case when \(P = 0\),
    1. describe the position of \(R\), [1]
    2. state the tension in the string. [1]
OCR M1 Q7
14 marks Standard +0.8
\includegraphics{figure_7} \(PQ\) is a line of greatest slope, of length \(4\) m, on a smooth plane inclined at \(30°\) to the horizontal. Particles \(A\) and \(B\), of masses \(0.15\) kg and \(0.5\) kg respectively, move along \(PQ\) with \(A\) below \(B\). The particles are both moving upwards, \(A\) with speed \(8\) m s\(^{-1}\) and \(B\) with speed \(2\) m s\(^{-1}\), when they collide at the mid-point of \(PQ\) (see diagram). Particle \(A\) is instantaneously at rest immediately after the collision.
  1. Show that \(B\) does not reach \(Q\) in the subsequent motion. [8]
  2. Find the time interval between the instant of \(A\)'s arrival at \(P\) and the instant of \(B\)'s arrival at \(P\). [6]
OCR M1 Q1
5 marks Moderate -0.3
Each of two wagons has an unloaded mass of \(1200\) kg. One of the wagons carries a load of mass \(m\) kg and the other wagon is unloaded. The wagons are moving towards each other on the same rails, each with speed \(3\) m s\(^{-1}\), when they collide. Immediately after the collision the loaded wagon is at rest and the speed of the unloaded wagon is \(5\) m s\(^{-1}\). Find the value of \(m\). [5]
OCR M1 Q2
6 marks Moderate -0.3
\includegraphics{figure_2} Forces of magnitudes \(6.5\) N and \(2.5\) N act at a point in the directions shown. The resultant of the two forces has magnitude \(R\) N and acts at right angles to the force of magnitude \(2.5\) N (see diagram).
  1. Show that \(\theta = 22.6°\), correct to 3 significant figures. [3]
  2. Find the value of \(R\). [3]