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OCR Pure 1 2018 September Q1
5 marks Easy -1.8
1 Solve the following inequalities.
  1. \(- 5 < 3 x + 1 < 14\)
  2. \(4 x ^ { 2 } + 3 > 28\)
OCR Pure 1 2018 September Q2
6 marks Easy -1.2
2 Vector \(\mathbf { v } = a \mathbf { i } + 0.6 \mathbf { j }\), where \(a\) is a constant.
  1. Given that the direction of \(\mathbf { v }\) is \(45 ^ { \circ }\), state the value of \(a\).
  2. Given instead that \(\mathbf { v }\) is parallel to \(8 \mathbf { i } + 3 \mathbf { j }\), find the value of \(a\).
  3. Given instead that \(\mathbf { v }\) is a unit vector, find the possible values of \(a\).
OCR Pure 1 2018 September Q3
6 marks Moderate -0.3
3
  1. The diagram below shows the graphs of \(y = | 3 x - 2 |\) and \(y = | 2 x + 1 |\). \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-4_423_682_1110_694} On the diagram in your Printed Answer Booklet, give the coordinates of the points of intersection of the graphs with the coordinate axes.
  2. Solve the equation \(| 2 x + 1 | = | 3 x - 2 |\).
OCR Pure 1 2018 September Q4
7 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-5_487_789_251_639} The diagram shows the triangle \(A O B\), in which angle \(A O B = 0.8\) radians, \(O A = 7 \mathrm {~cm}\) and \(O B = 10 \mathrm {~cm}\). \(C D\) is the arc of a circle with centre \(O\) and radius \(O C\). The area of the triangle \(A O B\) is twice the area of the sector COD
  1. Find the length \(O C\).
  2. Find the perimeter of the region \(A B C D\).
OCR Pure 1 2018 September Q5
6 marks Moderate -0.3
5 A student was asked to solve the equation \(2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0\). The student's attempt is written out below. $$\begin{aligned} & 2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0 \\ & 4 \log _ { 3 } x - 3 \log _ { 3 } x - 2 = 0 \\ & \log _ { 3 } x - 2 = 0 \\ & \log _ { 3 } x = 2 \\ & x = 8 \end{aligned}$$
  1. Identify the two mistakes that the student has made.
  2. Solve the equation \(2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0\), giving your answers in an exact form.
OCR Pure 1 2018 September Q6
7 marks Standard +0.3
6 In this question you must show detailed reasoning. A curve has equation \(y = \frac { 2 x } { 3 x - 1 } + \sqrt { 5 x + 1 }\). Show that the equation of the tangent to the curve at the point where \(x = 3\) is \(19 x - 32 y + 95 = 0\).
OCR Pure 1 2018 September Q7
11 marks Moderate -0.3
7 A line has equation \(y = 2 x\) and a circle has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 16 y + 56 = 0\).
  1. Show that the line does not meet the circle.
  2. (a) Find the equation of the line through the centre of the circle that is perpendicular to the line \(y = 2 x\).
    (b) Hence find the shortest distance between the line \(y = 2 x\) and the circle, giving your answer in an exact form.
OCR Pure 1 2018 September Q8
9 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-6_533_524_246_772} The diagram shows a container which consists of a cylinder with a solid base and a hemispherical top. The radius of the cylinder is \(r \mathrm {~cm}\) and the height is \(h \mathrm {~cm}\). The container is to be made of thin plastic. The volume of the container is \(45 \pi \mathrm {~cm} ^ { 3 }\).
  1. Show that the surface area of the container, \(A \mathrm {~cm} ^ { 2 }\), is given by $$A = \frac { 5 } { 3 } \pi r ^ { 2 } + \frac { 90 \pi } { r } .$$ [The volume of a sphere is \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\) and the surface area of a sphere is \(S = 4 \pi r ^ { 2 }\).]
  2. Use calculus to find the minimum surface area of the container, justifying that it is a minimum.
  3. Suggest a reason why the manufacturer would wish to minimise the surface area.
OCR Pure 1 2018 September Q9
8 marks Standard +0.3
9 An analyst believes that the sales of a particular electronic device are growing exponentially. In 2015 the sales were 3.1 million devices and the rate of increase in the annual sales is 0.8 million devices per year.
  1. Find a model to represent the annual sales, defining any variables used.
  2. In 2017 the sales were 5.2 million devices. Determine whether this is consistent with the model in part (i).
  3. The analyst uses the model in part (i) to predict the sales for 2025. Comment on the reliability of this prediction.
OCR Pure 1 2018 September Q10
13 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-7_579_764_255_651} The diagram shows the graph of \(\mathrm { f } ( x ) = \ln ( 3 x + 1 ) - x\), which has a stationary point at \(x = \alpha\). A student wishes to find the non-zero root \(\beta\) of the equation \(\ln ( 3 x + 1 ) - x = 0\) using the Newton-Raphson method.
  1. (a) Determine the value of \(\alpha\).
    (b) Explain why the Newton-Raphson method will fail if \(\alpha\) is used as the initial value.
  2. Show that the Newton-Raphson iterative formula for finding \(\beta\) can be written as $$x _ { n + 1 } = \frac { 3 x _ { n } - \left( 3 x _ { n } + 1 \right) \ln \left( 3 x _ { n } + 1 \right) } { 2 - 3 x _ { n } } .$$
  3. Apply the iterative formula in part (ii) with initial value \(x _ { 1 } = 1\) to find the value of \(\beta\) correct to 5 significant figures. You should show the result of each iteration.
  4. Use a change of sign method to verify that the value of \(\beta\) found in part (iii) is correct to 5 significant figures.
OCR Pure 1 2018 September Q11
12 marks Challenging +1.2
11 In this question you must show detailed reasoning. The \(n\)th term of a geometric progression is denoted by \(g _ { n }\) and the \(n\)th term of an arithmetic progression is denoted by \(a _ { n }\). It is given that \(g _ { 1 } = a _ { 1 } = 1 + \sqrt { 5 } , g _ { 3 } = a _ { 2 }\) and \(g _ { 4 } + a _ { 3 } = 0\). Given also that the geometric progression is convergent, show that its sum to infinity is \(4 + 2 \sqrt { 5 }\).
OCR Pure 1 2018 September Q12
10 marks Challenging +1.2
12 The gradient function of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } \sin 2 x } { 2 \cos ^ { 2 } 4 y - 1 }\).
  1. Find an equation for the curve in the form \(\mathrm { f } ( y ) = g ( x )\). The curve passes through the point \(\left( \frac { 1 } { 4 } \pi , \frac { 1 } { 12 } \pi \right)\).
  2. Find the smallest positive value of \(y\) for which \(x = 0\). \section*{END OF QUESTION PAPER}
OCR Stats 1 2018 September Q1
7 marks Moderate -0.8
1
  1. Differentiate the following with respect to \(x\).
    (a) \(\frac { 1 } { ( 3 x - 4 ) ^ { 2 } }\) (b) \(\frac { \ln ( x + 2 ) } { x }\)
  2. Find \(\int \mathrm { e } ^ { ( 2 x + 3 ) } \mathrm { d } x\).
OCR Stats 1 2018 September Q2
9 marks Easy -1.3
2
  1. Ben saves his pocket money as follows.
    Each week he puts money into his piggy bank (which pays no interest). In the first week he puts in 10p. In the second week he puts in 12p. In the third week he puts in 14p, and so on. How much money does Ben have in his piggy bank after 25 weeks?
  2. On January 1st Shirley invests \(\pounds 500\) in a savings account that pays compound interest at \(3 \%\) per annum. She makes no further payments into this account. The interest is added on 31st December each year.
    (a) Find the number of years after which her investment will first be worth more than \(\pounds 600\).
    (b) State an assumption that you have made in answering part (ii)(a).
OCR Stats 1 2018 September Q3
4 marks Standard +0.3
3 Use small angle approximations to estimate the solution of the equation \(\frac { \cos \frac { 1 } { 2 } \theta } { 1 + \sin \theta } = 0.825\), if \(\theta\) is small enough to neglect terms in \(\theta ^ { 3 }\) or above.
OCR Stats 1 2018 September Q4
4 marks Moderate -0.5
4 Prove that the sum of the squares of any two consecutive integers is of the form \(4 k + 1\), where \(k\) is an integer.
OCR Stats 1 2018 September Q5
3 marks Easy -1.2
5 The diagram shows the graph of \(y = \sin x ^ { \circ }\) for \(0 \leqslant x \leqslant 360\). \includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-4_597_965_1909_539} Find an equation for the transformed curve when the curve \(y = \sin x ^ { \circ }\) is reflected in
  1. the \(x\)-axis,
  2. the line \(y = 0.5\).
OCR Stats 1 2018 September Q6
9 marks Standard +0.3
6
  1. Find the coefficient of \(x ^ { 4 }\) in the expansion of \(( 3 x - 2 ) ^ { 10 }\).
  2. In the expansion of \(( 1 + 2 x ) ^ { n }\), where \(n\) is a positive integer, the coefficients of \(x ^ { 7 }\) and \(x ^ { 8 }\) are equal. Find the value of \(n\).
  3. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(\frac { 1 } { \sqrt { 4 + x } }\).
OCR Stats 1 2018 September Q7
7 marks Standard +0.3
7 The diagram shows part of the curve \(y = x ^ { 2 }\) for \(0 \leqslant x \leqslant p\), where \(p\) is a constant. \includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-5_736_543_669_762} The area \(A\) of the region enclosed by the curve, the \(x\)-axis and the line \(x = p\) is given approximately by the sum \(S\) of the areas of \(n\) rectangles, each of width \(h\), where \(h\) is small and \(n h = p\). The first three such rectangles are shown in the diagram.
  1. Find an expression for \(S\) in terms of \(n\) and \(h\).
  2. Use the identity \(\sum _ { r = 1 } ^ { n } r ^ { 2 } \equiv \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\) to show that \(S = \frac { 1 } { 6 } p ( p + h ) ( 2 p + h )\).
  3. Show how to use this result to find \(A\) in terms of \(p\).
OCR Stats 1 2018 September Q8
9 marks Moderate -0.5
8 The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\), relative to an origin \(O\), in three dimensions. The figure \(O A P B S C T U\) is a cuboid, with vertices labelled as in the following diagram. \(M\) is the midpoint of \(A U\). \includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-5_557_1221_2087_420}
OCR Stats 1 2018 September Q9
12 marks Moderate -0.3
9 The finance department of a retail firm recorded the daily income each day for 300 days. The results are summarised in the histogram. \includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-6_689_1575_488_246}
  1. Find the number of days on which the daily income was between \(\pounds 4000\) and \(\pounds 6000\).
  2. Calculate an estimate of the number of days on which the daily income was between \(\pounds 2700\) and \(\pounds 3600\).
  3. Use the midpoints of the classes to show that an estimate of the mean daily income is \(\pounds 3275\). An estimate of the standard deviation of the daily income is \(\pounds 1060\). The finance department uses the distribution \(\mathrm { N } \left( 3275,1060 ^ { 2 } \right)\) to model the daily income, in pounds.
  4. Calculate the number of days on which, according to this model, the daily income would be between \(\pounds 4000\) and \(\pounds 6000\).
  5. It is given that approximately \(95 \%\) of values of the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\) lie within the range \(\mu \pm 2 \sigma\). Without further calculation, use this fact to comment briefly on whether the proposed model is a good fit to the data illustrated in the histogram.
OCR Stats 1 2018 September Q10
6 marks Easy -1.8
10 The table shows information, derived from the 2011 UK census, about the percentage of employees who used various methods of travel to work in four Local Authorities.
Local AuthorityUnderground, metro, light rail or tramTrainBusDriveWalk or cycle
A0.3\%4.5\%17\%52.8\%11\%
B0.2\%1.7\%1.7\%63.4\%11\%
C35.2\%3.0\%12\%11.7\%16\%
D8.9\%1.4\%9\%54.7\%10\%
One of the Local Authorities is a London borough and two are metropolitan boroughs, not in London.
  1. Which one of the Local Authorities is a London borough? Give a reason for your answer.
  2. Which two of the Local Authorities are metropolitan boroughs outside London? In each case give a reason for your answer.
  3. Describe one difference between the public transport available in the two metropolitan boroughs, as suggested by the table.
  4. Comment on the availability of public transport in Local Authority B as suggested by the table.
OCR Stats 1 2018 September Q11
8 marks Moderate -0.3
11 In an experiment involving a bivariate distribution ( \(X , Y\) ) a random sample of 7 pairs of values was obtained and Pearson's product-moment correlation coefficient \(r\) was calculated for these values.
  1. The value of \(r\) was found to be 0.894 . Use the table below to test, at the \(5 \%\) significance level, whether there is positive linear correlation in the population, stating your hypotheses and conclusion clearly.
    1-tail test 2-tail test5\%2.5\%1\%0.5\%
    10\%5\%2\%1\%
    \(n\)
    1----
    2----
    30.98770.99690.99950.9999
    40.90000.95000.98000.9900
    50.80540.87830.93430.9587
    60.72930.81140.88220.9587
    70.66940.75450.83290.9745
    80.62150.70670.78870.8343
    90.58820.66640.74980.7977
    100.54940.63190.71550.7646
    Scatter diagrams for four sets of bivariate data, are shown. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-8_380_371_301_191} \captionsetup{labelformat=empty} \caption{Diagram A}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-8_373_373_301_628} \captionsetup{labelformat=empty} \caption{Diagram B}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-8_378_373_301_1064} \captionsetup{labelformat=empty} \caption{Diagram C}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-8_378_373_301_1503} \captionsetup{labelformat=empty} \caption{Diagram D}
    \end{figure} It is given that \(r = 0.894\) for one of these diagrams.
  2. For each of the other diagrams, state how you can tell that \(r \neq 0.894\).
OCR Stats 1 2018 September Q12
8 marks Moderate -0.3
12 In the past, the time spent by customers in a certain shop had mean 10.5 minutes and standard deviation 4.2 minutes. Following a change of layout in the shop, the mean time spent in the shop by a random sample of 50 customers is found to be 12.0 minutes.
  1. Assuming that the standard deviation is unchanged, test at the \(1 \%\) significance level whether the mean time spent by customers in the shop has changed.
  2. Another random sample of 50 customers is chosen and a similar test at the \(1 \%\) significance level is carried out. Given that the population mean time has not changed, state the probability that the conclusion of the test will be that the population mean time has changed.
OCR Stats 1 2018 September Q13
7 marks Challenging +1.2
13 Bag A contains 3 black discs and 2 white discs only. Initially Bag B is empty. Discs are removed at random from bag A, and are placed in bag B, one at a time, until all 5 discs are in bag B.
  1. Write down the probability that the last disc that is placed in bag B is black.
  2. Find the probability that the first disc and the last disc that are placed in bag B are both black.
  3. Find the probability that, starting from when the first disc is placed in bag B , the number of black discs in bag B is always greater than the number of white discs in bag B.