Questions — OCR (4619 questions)

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OCR Pure 1 2018 December Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{a16ab26f-21fb-4a73-8b94-c16bef611bcb-7_524_714_274_246} The diagram shows the graph of \(y = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x - \frac { 1 } { 3 } \pi \right)\), which crosses the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\).
  1. Determine the coordinates of the points \(A\) and \(B\).
  2. Give full details of a sequence of three geometrical transformations which transform the graph of \(y = \tan ^ { - 1 } x\) to the graph of \(y = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x - \frac { 1 } { 3 } \pi \right)\). The equation \(x = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x - \frac { 1 } { 3 } \pi \right)\) has only one root.
  3. Show by calculation that this root lies between \(x = 0\) and \(x = 1\).
  4. Use the iterative formula \(x _ { n + 1 } = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x _ { n } - \frac { 1 } { 3 } \pi \right)\), with a suitable starting value, to find the root correct to 3 significant figures. Show the result of each iteration.
  5. Using the diagram in the Printed Answer Booklet, show how the iterative process converges to the root.
OCR Pure 1 2018 December Q11
11 In this question you must show detailed reasoning. A function f is given by \(\mathrm { f } ( x ) = \frac { x - 4 } { ( x + 2 ) ( x - 1 ) } + \frac { 3 x + 1 } { ( x + 3 ) ( x - 1 ) }\).
  1. Show that \(\mathrm { f } ( x )\) can be written as \(\frac { 2 ( 2 x + 5 ) } { ( x + 2 ) ( x + 3 ) }\).
  2. Given that \(\int _ { a } ^ { a + 4 } \mathrm { f } ( x ) \mathrm { d } x = 2 \ln 3\), find the value of the positive constant \(a\).
OCR Pure 1 2018 December Q12
12
  1. By first writing \(\tan 3 \theta\) as \(\tan ( 2 \theta + \theta )\), show that \(\tan 3 \theta = \frac { 3 \tan \theta - \tan ^ { 3 } \theta } { 1 - 3 \tan ^ { 2 } \theta }\).
  2. Hence show that there are always exactly two different values of \(\theta\) between \(0 ^ { \circ }\) and \(180 ^ { \circ }\) which satisfy the equation
    \(3 \tan 3 \theta = \tan \theta + k\),
    where \(k\) is a non-zero constant. \section*{END OF QUESTION PAPER} \section*{OCR
    Oxford Cambridge and RSA}
OCR Stats 1 2018 December Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{166bcf11-c812-4077-91c8-916b093cbbd0-04_515_732_447_664} The diagram shows the curve \(y = \sqrt { x } - 3\). The shaded region is bounded by the curve and the two axes. Find the exact area of the shaded region.
\(2 \mathrm { f } ( x )\) is a cubic polynomial in which the coefficient of \(x ^ { 3 }\) is 1 . The equation \(\mathrm { f } ( x ) = 0\) has exactly two roots.
  1. Sketch a possible graph of \(y = \mathrm { f } ( x )\). It is now given that the two roots are \(x = 2\) and \(x = 3\).
  2. Find, in expanded form, the two possible polynomials \(\mathrm { f } ( x )\).
OCR Stats 1 2018 December Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{166bcf11-c812-4077-91c8-916b093cbbd0-05_796_1653_260_205} The diagram shows the graph of \(y = \mathrm { g } ( x )\).
In the printed answer booklet, using the same scale as in this diagram, sketch the curves
  1. \(\quad y = \frac { 3 } { 2 } \mathrm {~g} ( x )\),
  2. \(y = \mathrm { g } \left( \frac { 1 } { 2 } x \right)\).
OCR Stats 1 2018 December Q4
4 In this question you must show detailed reasoning.
  1. Show that \(\cos A + \sin A \tan A = \sec A\).
  2. Solve the equation \(\tan 2 \theta = 3 \tan \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
OCR Stats 1 2018 December Q5
5 Points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\). Point \(C\) lies on \(A B\) such that \(A C : C B = p : 1\).
  1. Show that the position vector of \(C\) is \(\frac { 1 } { p + 1 } ( \mathbf { a } + p \mathbf { b } )\). It is now given that \(\mathbf { a } = 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }\) and \(\mathbf { b } = - 6 \mathbf { i } + 4 \mathbf { j } + 12 \mathbf { k }\), and that \(C\) lies on the \(y\)-axis.
  2. Find the value of \(p\).
  3. Write down the position vector of \(C\).
OCR Stats 1 2018 December Q6
6 The table shows information about three geometric series. The three geometric series have different common ratios.
First
term
Common
ratio
Number
of terms
Last
term
Series 112\(n _ { 1 }\)1024
Series 21\(r _ { 2 }\)\(n _ { 2 }\)1024
Series 31\(r _ { 3 }\)\(n _ { 3 }\)1024
  1. Find \(n _ { 1 }\).
  2. Given that \(r _ { 2 }\) is an integer less than 10 , find the value of \(r _ { 2 }\) and the value of \(n _ { 2 }\).
  3. Given that \(r _ { 3 }\) is not an integer, find a possible value for the sum of all the terms in Series 3 .
OCR Stats 1 2018 December Q7
7
  1. Show that, if \(n\) is a positive integer, then \(\left( x ^ { n } - 1 \right)\) is divisible by \(( x - 1 )\).
  2. Hence show that, if \(k\) is a positive integer, then \(2 ^ { 8 k } - 1\) is divisible by 17.
OCR Stats 1 2018 December Q8
8 Use a suitable trigonometric substitution to find \(\int \frac { x ^ { 2 } } { \sqrt { 1 - x ^ { 2 } } } \mathrm {~d} x\).
OCR Stats 1 2018 December Q9
9 Research has shown that drug A is effective in \(32 \%\) of patients with a certain disease. In a trial, drug B is given to a random sample of 1000 patients with the disease, and it is found that the drug is effective in 290 of these patients. Test at the \(2.5 \%\) significance level whether there is evidence that drug B is effective in a lower proportion of patients than drug A .
OCR Stats 1 2018 December Q10
10 Using the 2001 UK census results and some software, Javid intended to calculate the mean number of people who travelled to work by underground, metro, light rail or tram (UMLT) for all 348 Local Authorities. However, Javid noticed that for one LA the entry in the UMLT column is a dash, rather than a 0 . See the extract below.
Data extract for one LA in 2001
Work
mainly at or
from home
UMLTTrain
Bus,
minibus or
coach
295-44
Javid felt that it was not clear how this LA was to be treated so he decided to omit it from his calculation.
  1. Explain how the omission of this LA affects Javid's calculation of the mean. The value of the mean that Javid obtained was 2046.3.
  2. Calculate the value of the mean when this LA is not removed. Javid finds that the corresponding mean for all Local Authorities for 2011 is 2860.8. In order to compare the means for the two years, Javid also finds the total number of employees in each of these years. His results are given below.
    Year20012011
    Total number of
    employees
    		2362775326526336
  3. Show that a higher proportion of employees used the metro to travel to work in 2011 than in 2001.
  4. Suggest a reason for this increase.
OCR Stats 1 2018 December Q11
11 Laxmi wishes to test whether there is linear correlation between the mass and the height of adult males.
  1. State, with a reason, whether Laxmi should use a 1-tail or a 2-tail test. Laxmi chooses a random sample of 40 adult males and calculates Pearson's product-moment correlation coefficient, \(r\). She finds that \(r = 0.2705\).
  2. Use the table below to carry out the test at the \(5 \%\) significance level. \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Critical values of Pearson's product-moment correlation coefficient.}
    \multirow{2}{*}{}1-tail test5\%2.5\%1\%0.5\%
    2-tail test10\%5\%2.5\%1\%
    \multirow{4}{*}{\(n\)}380.27090.32020.37600.4128
    390.26730.31600.37120.4076
    400.26380.31200.36650.4026
    410.26050.30810.36210.3978
    \end{table}
OCR Stats 1 2018 December Q12
12 Paul drew a cumulative frequency graph showing information about the numbers of people in various age-groups in a certain region X. He forgot to include the scale on the cumulative frequency axis, as shown below.
\includegraphics[max width=\textwidth, alt={}, center]{166bcf11-c812-4077-91c8-916b093cbbd0-09_758_936_408_561}
  1. Find an estimate of the median age of the population of region X .
  2. Find an estimate of the proportion of people aged over 60 in region X . Sonika drew similar cumulative graphs for another two regions, Y and Z , but she included the scales on the cumulative frequency axes, as shown below.
    \includegraphics[max width=\textwidth, alt={}, center]{166bcf11-c812-4077-91c8-916b093cbbd0-10_748_935_358_116}
    \includegraphics[max width=\textwidth, alt={}, center]{166bcf11-c812-4077-91c8-916b093cbbd0-10_746_940_358_1011}
  3. Find an age group, of width 20 years, in which region Z has approximately 3 times as many people as region Y.
  4. State one advantage and one disadvantage of using Sonika's two diagrams to compare the populations in Regions Y and Z.
  5. Without calculation state, with a reason, which of regions Y or Z has the greater proportion of people aged under 40.
OCR Stats 1 2018 December Q13
13 The marks of 24 students in a test had mean \(m\) and standard deviation \(\sqrt { 6 }\). Two new students took the same test. Their marks were \(m - 4\) and \(m + 4\). Show that the standard deviation of the marks of all 26 students is 2.60 , correct to 3 significant figures.
OCR Stats 1 2018 December Q14
14 Mr Jones has 3 tins of beans and 2 tins of pears. His daughter has removed the labels for a school project, and the tins are identical in appearance. Mr Jones opens tins in turn until he has opened at least 1 tin of beans and at least 1 tin of pears. He does not open any remaining tins.
  1. Draw a tree diagram to illustrate this situation, labelling each branch with its associated probability.
  2. Find the probability that Mr Jones opens exactly 3 tins.
  3. It is given that the last tin Mr Jones opens is a tin of pears. Find the probability that he opens exactly 3 tins.
OCR Stats 1 2018 December Q15
15 A fair dice is thrown 1000 times and the number, \(X\), of throws on which the score is 6 is noted.
    1. State the distribution of \(X\).
    2. Explain why a normal distribution would be an appropriate approximation to the distribution of \(X\).
  2. Use a normal distribution to find two positive integer values, \(a\) and \(b\), such that \(\mathrm { P } ( a < X < b ) \approx 0.4\).
  3. For your two values of \(a\) and \(b\), use the distribution of part (a)(i) to find the value of \(\mathrm { P } ( a < X < b )\), correct to 3 significant figures. \section*{OCR} Oxford Cambridge and RSA
OCR Mechanics 1 2018 December Q1
1 Use logarithms to solve the equation \(2 ^ { 3 x - 1 } = 3 ^ { x + 4 }\), giving your answer correct to 3 significant figures.
OCR Mechanics 1 2018 December Q2
2 In this question you must show detailed reasoning. Find the values of \(x\) for which the gradient of the curve \(y = \frac { 2 } { 3 } x ^ { 3 } + \frac { 5 } { 2 } x ^ { 2 } - 3 x + 7\) is positive. Give your answer in set notation.
OCR Mechanics 1 2018 December Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{918c616a-a0c7-4779-8d3c-84ddf1fa36d6-04_695_714_1087_248} The diagram shows a circle with centre \(( a , - a )\) that passes through the origin.
  1. Write down an equation for the circle in terms of \(a\).
  2. Given that the point \(( 1 , - 5 )\) lies on the circle, find the exact area of the circle.
OCR Mechanics 1 2018 December Q4
4 The first three terms of an arithmetic series are \(9 p , 8 p - 3,5 p\) respectively, where \(p\) is a constant.
Given that the sum of the first \(n\) terms of this series is - 1512 , find the value of \(n\).
OCR Mechanics 1 2018 December Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{918c616a-a0c7-4779-8d3c-84ddf1fa36d6-05_444_757_548_251} The functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined for \(x \geqslant 0\) by \(\mathrm { f } ( x ) = \frac { x } { x ^ { 2 } + 3 }\) and \(\mathrm { g } ( x ) = \mathrm { e } ^ { - 2 x }\). The diagram shows the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\). The equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) has exactly one real root \(\alpha\).
  1. Show that \(\alpha\) satisfies the equation \(\mathrm { h } ( x ) = 0\), where \(\mathrm { h } ( x ) = x ^ { 2 } + 3 - x \mathrm { e } ^ { 2 x }\).
  2. Hence show that a Newton-Raphson iterative formula for finding \(\alpha\) can be written in the form $$x _ { n + 1 } = \frac { x _ { n } ^ { 2 } \left( 1 - 2 \mathrm { e } ^ { 2 x _ { n } } \right) - 3 } { 2 x _ { n } - \left( 1 + 2 x _ { n } \right) \mathrm { e } ^ { 2 x _ { n } } } .$$
  3. Use this iterative formula, with a suitable initial value, to find \(\alpha\) correct to 3 decimal places. Show the result of each iteration. \section*{(d) In this question you must show detailed reasoning.} Find the exact value of \(x\) for which \(\operatorname { fg } ( x ) = \frac { 2 } { 13 }\).
OCR Mechanics 1 2018 December Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{918c616a-a0c7-4779-8d3c-84ddf1fa36d6-06_544_1232_251_260} The diagram shows the curve with parametric equations \(x = \ln \left( t ^ { 2 } - 4 \right) , \quad y = \frac { 4 } { t ^ { 2 } } , \quad\) where \(t > 2\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = \ln 5\) and \(x = \ln 12\).
  1. Show that the area of \(R\) is given by
    \(\int _ { a } ^ { b } \frac { 8 } { t \left( t ^ { 2 } - 4 \right) } \mathrm { d } t\),
    where \(a\) and \(b\) are constants to be determined.
  2. In this question you must show detailed reasoning. Hence find the exact area of \(R\), giving your answer in the form \(\ln k\), where \(k\) is a constant to be determined.
  3. Find a cartesian equation of the curve in the form \(y = \mathrm { f } ( x )\).
OCR Mechanics 1 2018 December Q7
7 A particle \(P\) moves with constant acceleration \(( 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\). At time \(t = 0\) seconds \(P\) is at the origin. At time \(t = 4\) seconds \(P\) has velocity \(( 2 \mathbf { i } + 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
  1. Find the displacement vector of \(P\) at time \(t = 4\) seconds.
  2. Find the speed of \(P\) at time \(t = 0\) seconds.
OCR Mechanics 1 2018 December Q8
8 A uniform ladder \(A B\), of weight 150 N and length 4 m , rests in equilibrium with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth vertical wall. The ladder is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = 3\). A man of weight 750 N is standing on the ladder at a distance \(x \mathrm {~m}\) from \(A\).
  1. Show that the magnitude of the frictional force exerted by the ground on the ladder is \(\frac { 25 } { 2 } ( 2 + 5 x ) \mathrm { N }\). The coefficient of friction between the ladder and the ground is \(\frac { 1 } { 4 }\).
  2. Find the greatest value of \(x\) for which equilibrium is possible.