Questions — OCR (4907 questions)

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OCR H240/03 2017 Specimen Q10
7 marks Standard +0.3
A body of mass 20 kg is on a rough plane inclined at angle \(\alpha\) to the horizontal. The body is held at rest on the plane by the action of a force of magnitude \(P\) N. The force is acting up the plane in a direction parallel to a line of greatest slope of the plane. The coefficient of friction between the body and the plane is \(\mu\).
  1. When \(P = 100\), the body is on the point of sliding down the plane. Show that \(g \sin \alpha = g\mu \cos \alpha + 5\). [4]
  2. When \(P\) is increased to 150, the body is on the point of sliding up the plane. Use this, and your answer to part (a), to find an expression for \(\alpha\) in terms of \(g\). [3]
OCR H240/03 2017 Specimen Q11
9 marks Standard +0.3
In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the directions east and north respectively. A particle of mass 0.12 kg is moving so that its position vector \(\mathbf{r}\) metres at time \(t\) seconds is given by \(\mathbf{r} = 2t^2\mathbf{i} + (5t^2 - 4t)\mathbf{j}\).
  1. Show that when \(t = 0.7\) the bearing on which the particle is moving is approximately \(044°\). [3]
  2. Find the magnitude of the resultant force acting on the particle at the instant when \(t = 0.7\). [4]
  3. Determine the times at which the particle is moving on a bearing of \(045°\). [2]
OCR H240/03 2017 Specimen Q12
14 marks Standard +0.3
A girl is practising netball. She throws the ball from a height of 1.5 m above horizontal ground and aims to get the ball through a hoop. The hoop is 2.5 m vertically above the ground and is 6 m horizontally from the point of projection. The situation is modelled as follows.
  • The initial velocity of the ball has magnitude \(U\) m s\(^{-1}\).
  • The angle of projection is \(40°\).
  • The ball is modelled as a particle.
  • The hoop is modelled as a point.
This is shown on the diagram below. \includegraphics{figure_12}
  1. For \(U = 10\), find
    1. the greatest height above the ground reached by the ball [5]
    2. the distance between the ball and the hoop when the ball is vertically above the hoop. [4]
  2. Calculate the value of \(U\) which allows her to hit the hoop. [3]
  3. How appropriate is this model for predicting the path of the ball when it is thrown by the girl? [1]
  4. Suggest one improvement that might be made to this model. [1]
OCR H240/03 2017 Specimen Q13
8 marks Standard +0.3
Particle \(A\), of mass \(m\) kg, lies on the plane \(\Pi_1\) inclined at an angle of \(\tan^{-1} \frac{4}{3}\) to the horizontal. Particle \(B\), of \(4m\) kg, lies on the plane \(\Pi_2\) inclined at an angle of \(\tan^{-1} \frac{4}{3}\) to the horizontal. The particles are attached to the ends of a light inextensible string which passes over a smooth pulley at \(P\). The coefficient of friction between particle \(A\) and \(\Pi_1\) is \(\frac{1}{4}\) and plane \(\Pi_2\) is smooth. Particle \(A\) is initially held at rest such that the string is taut and lies in a line of greatest slope of each plane. This is shown on the diagram below. \includegraphics{figure_13}
  1. Show that when \(A\) is released it accelerates towards the pulley at \(\frac{7g}{15}\) m s\(^{-2}\). [6]
  2. Assuming that \(A\) does not reach the pulley, show that it has moved a distance of \(\frac{1}{4}\) m when its speed is \(\sqrt{\frac{7g}{30}}\) m s\(^{-1}\). [2]
OCR H240/03 2017 Specimen Q14
8 marks Standard +0.8
A uniform ladder \(AB\) of mass 35 kg and length 7 m rests with its end \(A\) on rough horizontal ground and its end \(B\) against a rough vertical wall. The ladder is inclined at an angle of \(45°\) to the horizontal. A man of mass 70 kg is standing on the ladder at a point \(C\), which is \(x\) metres from \(A\). The coefficient of friction between the ladder and the wall is \(\frac{1}{4}\) and the coefficient of friction between the ladder and the ground is \(\frac{1}{2}\). The system is in limiting equilibrium. Find \(x\). [8]
OCR AS Pure 2017 Specimen Q1
5 marks Easy -1.8
The diagram below shows the graph of \(y = f(x)\). \includegraphics{figure_1}
  1. On the diagram in the Printed Answer Booklet draw the graph of \(y = f(x + 3)\). [2]
  2. Describe fully the transformation which transforms the graph of \(y = f(x)\) to the graph of \(y = -f(x)\). [1]
The point \((2, 3)\) lies on the graph of \(y = g(x)\). State the coordinates of its image when \(y = g(x)\) is transformed to
  1. \(y = 4g(x)\) [1]
  2. \(y = g(4x)\). [1]
OCR AS Pure 2017 Specimen Q2
5 marks Standard +0.3
In this question you must show detailed reasoning. Solve the equation \(2\cos^2 x = 2 - \sin x\) for \(0° \leq x \leq 180°\). [5]
OCR AS Pure 2017 Specimen Q3
7 marks Moderate -0.8
The number of members of a social networking site is modelled by \(m = 150e^{2t}\), where \(m\) is the number of members and \(t\) is time in weeks after the launch of the site.
  1. State what this model implies about the relationship between \(m\) and the rate of change of \(m\). [2]
  2. What is the significance of the integer 150 in the model? [1]
  3. Find the week in which the model predicts that the number of members first exceeds 60 000. [3]
  4. The social networking site only expects to attract 60 000 members. Suggest how the model could be refined to take account of this. [1]
OCR AS Pure 2017 Specimen Q4
6 marks Moderate -0.8
The points \(A\), \(B\) and \(C\) have position vectors \(\begin{pmatrix} -2 \\ 1 \end{pmatrix}\), \(\begin{pmatrix} 2 \\ 5 \end{pmatrix}\) and \(\begin{pmatrix} 6 \\ 3 \end{pmatrix}\) respectively. \(M\) is the midpoint of \(BC\).
  1. Find the position vector of the point \(D\) such that \(\overrightarrow{BC} = \overrightarrow{AD}\). [3]
  2. Find the magnitude of \(\overrightarrow{AM}\). [3]
OCR AS Pure 2017 Specimen Q5
7 marks Moderate -0.8
A doctors' surgery starts a campaign to reduce missed appointments. The number of missed appointments for each of the first five weeks after the start of the campaign is shown below.
Number of weeks after the start (\(x\))12345
Number of missed appointments (\(y\))235149995938
This data could be modelled by an equation of the form \(y = pq^x\) where \(p\) and \(q\) are constants.
  1. Show that this relationship may be expressed in the form \(\log_{10} y = mx + c\), expressing \(m\) and \(c\) in terms of \(p\) and/or \(q\). [2]
The diagram below shows \(\log_{10} y\) plotted against \(x\), for the given data. \includegraphics{figure_5}
  1. Estimate the values of \(p\) and \(q\). [3]
  2. Use the model to predict when the number of missed appointments will fall below 20. Explain why this answer may not be reliable. [2]
OCR AS Pure 2017 Specimen Q6
5 marks Standard +0.3
  1. A student suggests that, for any prime number between 20 and 40, when its digits are squared and then added, the sum is an odd number. For example, 23 has digits 2 and 3 which gives \(2^2 + 3^2 = 13\), which is odd. Show by counter example that this suggestion is false. [2]
  2. Prove that the sum of the squares of any three consecutive positive integers cannot be divided by 3. [3]
OCR AS Pure 2017 Specimen Q7
5 marks Standard +0.8
Differentiate \(f(x) = x^4\) from first principles. [5]
OCR AS Pure 2017 Specimen Q8
10 marks Standard +0.3
A curve has equation \(y = kx^{\frac{1}{2}}\) where \(k\) is a constant. The point \(P\) on the curve has \(x\)-coordinate 4. The normal to the curve at \(P\) is parallel to the line \(2x + 3y = 0\) and meets the \(x\)-axis at the point \(Q\). The line \(PQ\) is the radius of a circle centre \(P\). Show that \(k = \frac{1}{2}\). Find the equation of the circle. [10]
OCR AS Pure 2017 Specimen Q9
5 marks Moderate -0.8
The diagram below shows the velocity-time graph of a car moving along a straight road, where \(v\) m s\(^{-1}\) is the velocity of the car at time \(t\) s after it passes through the point \(A\). \includegraphics{figure_9}
  1. Calculate the acceleration of the car at \(t = 6\). [2]
  2. Jasmit says "The distance travelled by the car during the first 20 seconds of the car's motion is more than five times its displacement from \(A\) after the first 20 seconds of the car's motion". Give evidence to support Jasmit's statement. [3]
OCR AS Pure 2017 Specimen Q10
10 marks Moderate -0.3
A student is attempting to model the flight of a boomerang. She throws the boomerang from a fixed point \(O\) and catches it when it returns to \(O\). She suggests the model for the displacement, \(s\) metres, after \(t\) seconds is given by \(s = 9t^2 - \frac{3}{2}t^3\), \(0 \leq t \leq 6\). For this model,
  1. determine what happens at \(t = 6\), [2]
  2. find the greatest displacement of the boomerang from \(O\), [4]
  3. find the velocity of the boomerang 1 second before the student catches it, [2]
  4. find the acceleration of the boomerang 1 second before the student catches it. [2]
OCR AS Pure 2017 Specimen Q11
10 marks Moderate -0.8
In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the directions east and north respectively. Distance is measured in metres and time in seconds. A ship of mass 100 000 kg is being towed by two tug boats. • The cables attaching each tug to the ship are horizontal. • One tug produces a force of \((350\mathbf{i} + 400\mathbf{j})\) N. • The other tug produces a force of \((250\mathbf{i} - 400\mathbf{j})\) N. • The total resistance to motion is 200 N. • At the instant when the tugs begin to tow the ship, it is moving east at a speed of 1.5 m s\(^{-1}\).
  1. Explain why the ship continues to move directly east. [2]
  2. Find the acceleration of the ship. [2]
  3. Find the time which the ship takes to move 400 m while it is being towed. Find its speed after moving that distance. [6]
OCR Further Statistics 2017 Specimen Q1
6 marks Moderate -0.8
The table below shows the typical stopping distances \(d\) metres for a particular car travelling at \(v\) miles per hour.
\(v\)203040506070
\(d\)132436527294
  1. State each of the following words that describe the variable \(v\). Independent \quad Dependent \quad Controlled \quad Response [1]
  2. Calculate the equation of the regression line of \(d\) on \(v\). [2]
  3. Use the equation found in part (ii) to estimate the typical stopping distance when this car is travelling at 45 miles per hour. [1]
It is given that the product moment correlation coefficient for the data is 0.990 correct to three significant figures.
  1. Explain whether your estimate found in part (iii) is reliable. [2]
OCR Further Statistics 2017 Specimen Q2
6 marks Standard +0.3
The mass \(J\) kg of a bag of randomly chosen Jersey potatoes is a normally distributed random variable with mean 1.00 and standard deviation 0.06. The mass \(K\) kg of a bag of randomly chosen King Edward potatoes is an independent normally distributed random variable with mean 0.80 and standard deviation 0.04.
  1. Find the probability that the total mass of 6 bags of Jersey potatoes and 8 bags of King Edward potatoes is greater than 12.70 kg. [3]
  2. Find the probability that the mass of one bag of King Edward potatoes is more than 75\% of the mass of one bag of Jersey potatoes. [3]
OCR Further Statistics 2017 Specimen Q3
8 marks Standard +0.3
A game is played as follows. A fair six-sided dice is thrown once. If the score obtained is even, the amount of money, in £, that the contestant wins is half the score on the dice, otherwise it is twice the score on the dice.
  1. Find the probability distribution of the amount of money won by the contestant. [3]
  2. The contestant pays £5 for every time the dice is thrown. Find the standard deviation of the loss made by the contestant in 120 throws of the dice. [5]
OCR Further Statistics 2017 Specimen Q4
7 marks Challenging +1.2
A psychologist investigated the scores of pairs of twins on an aptitude test. Seven pairs of twins were chosen randomly, and the scores are given in the following table.
Elder twin65376079394088
Younger twin58396162502684
  1. Carry out an appropriate Wilcoxon test at the 10\% significance level to investigate whether there is evidence of a difference in test scores between the elder and the younger of a pair of twins. [6]
  2. Explain the advantage in this case of a Wilcoxon test over a sign test. [1]
OCR Further Statistics 2017 Specimen Q5
8 marks Standard +0.3
The number of goals scored by the home team in a randomly chosen hockey match is denoted by \(X\).
  1. In order for \(X\) to be modelled by a Poisson distribution it is assumed that goals scored are random events. State two other conditions needed for \(X\) to be modelled by a Poisson distribution in this context. [2]
Assume now that \(X\) can be modelled by the distribution Po\((1.9)\).
    1. Write down an expression for P\((X = r)\). [1]
    2. Hence find P\((X = 3)\). [1]
  2. Assume also that the number of goals scored by the away team in a randomly chosen hockey match has an independent Poisson distribution with mean \(\lambda\) between 1.31 and 1.32. Find an estimate for the probability that more than 3 goals are scored altogether in a randomly chosen match. [4]
OCR Further Statistics 2017 Specimen Q6
7 marks Standard +0.3
A bag contains 3 green counters, 3 blue counters and \(w\) white counters. Counters are selected at random, one at a time, with replacement, until a white counter is drawn. The total number of counters selected, including the white counter, is denoted by \(X\).
  1. In the case when \(w = 2\),
    1. write down the distribution of \(X\), [1]
    2. find \(P(3 < X \leq 7)\). [2]
  2. In the case when E\((X) = 2\), determine the value of \(w\). [2]
  3. In the case when \(w = 2\) and \(X = 6\), find the probability that the first five counters drawn alternate in colour. [2]
OCR Further Statistics 2017 Specimen Q7
9 marks Standard +0.3
Sweet pea plants grown using a standard plant food have a mean height of 1.6 m. A new plant food is used for a random sample of 49 randomly chosen plants and the heights, \(x\) metres, of this sample can be summarised by the following. $$n = 49$$ $$\sum x = 74.48$$ $$\sum x^2 = 120.8896$$ Test, at the 5\% significance level, whether, when the new plant food is used, the mean height of sweet pea plants is less than 1.6 m. [9]
OCR Further Statistics 2017 Specimen Q8
15 marks Standard +0.8
A continuous random variable \(X\) has probability density function given by $$f(x) = \begin{cases} 0.8e^{-0.8x} & x \geq 0, \\ 0 & x < 0. \end{cases}$$
  1. Find the mean and variance of \(X\). [4]
The lifetime of a certain organism is thought to have the same distribution as \(X\). The lifetimes in days of a random sample of 60 specimens of the organism were found. The observed frequencies, together with the expected frequencies correct to 3 decimal places, are given in the table.
Range\(0 \leq x < 1\)\(1 \leq x < 2\)\(2 \leq x < 3\)\(3 \leq x < 4\)\(x \geq 4\)
Observed24221031
Expected33.04014.8466.6712.9972.446
  1. Show how the expected frequency for \(1 \leq x < 2\) is obtained. [4]
  2. Carry out a goodness of fit test at the 5\% significance level. [7]
OCR Further Statistics 2017 Specimen Q9
9 marks Challenging +1.2
The continuous random variable \(X\) has cumulative distribution function given by $$F(x) = \begin{cases} 0 & x < 0, \\ \frac{1}{16}x^2 & 0 \leq x \leq 4, \\ 1 & x > 4. \end{cases}$$
  1. The random variable \(Y\) is defined by \(Y = \frac{1}{X^2}\). Find the cumulative distribution function of \(Y\). [5]
  2. Show that E\((Y)\) is not defined. [4]