Questions — OCR PURE (137 questions)

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OCR PURE Q1
1 In this question you must show detailed reasoning.
  1. Express \(3 ^ { \frac { 7 } { 2 } }\) in the form \(a \sqrt { b }\), where \(a\) is an integer and \(b\) is a prime number.
  2. Express \(\frac { \sqrt { 2 } } { 1 - \sqrt { 2 } }\) in the form \(c + d \sqrt { e }\), where \(c\) and \(d\) are integers and \(e\) is a prime number.
OCR PURE Q2
2
  1. The equation \(x ^ { 2 } + 3 x + k = 0\) has repeated roots. Find the value of the constant \(k\).
  2. Solve the inequality \(6 + x - x ^ { 2 } > 0\).
OCR PURE Q3
3
  1. Solve the equation \(\sin ^ { 2 } \theta = 0.25\) for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\).
  2. In this question you must show detailed reasoning. Solve the equation \(\tan 3 \phi = \sqrt { 3 }\) for \(0 ^ { \circ } \leqslant \phi < 90 ^ { \circ }\).
OCR PURE Q4
4
  1. It is given that \(y = x ^ { 2 } + 3 x\).
    (a) Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    (b) Find the values of \(x\) for which \(y\) is increasing.
  2. Find \(\int ( 3 - 4 \sqrt { x } ) \mathrm { d } x\).
    \(5 N\) is an integer that is not divisible by 3 . Prove that \(N ^ { 2 }\) is of the form \(3 p + 1\), where \(p\) is an integer.
OCR PURE Q6
6 Sketch the following curves.
  1. \(y = \frac { 2 } { x }\)
  2. \(y = x ^ { 3 } - 6 x ^ { 2 } + 9 x\)
    \(7 \quad O A B C\) is a parallelogram with \(\overrightarrow { O A } = \mathbf { a }\) and \(\overrightarrow { O C } = \mathbf { c } . P\) is the midpoint of \(A C\).
    \includegraphics[max width=\textwidth, alt={}, center]{3fdf2b2d-20a9-4d68-b760-57ec529b5893-4_298_735_383_657}
  3. Find the following in terms of \(\mathbf { a }\) and \(\mathbf { c }\), simplifying your answers.
OCR PURE Q9
9 Jo is investigating the popularity of a certain band amongst students at her school. She decides to survey a sample of 100 students.
  1. State an advantage of using a stratified sample rather than a simple random sample.
  2. Explain whether it would be reasonable for Jo to use her results to draw conclusions about all students in the UK.
OCR PURE Q10
10 The probability distribution of a random variable \(X\) is given in the table.
\(x\)0246
\(\mathrm { P } ( X = x )\)\(\frac { 3 } { 8 }\)\(\frac { 5 } { 16 }\)\(4 p\)\(p\)
  1. Find the value of \(p\).
  2. Two values of \(X\) are chosen at random. Find the probability that the product of these values is 0 .
OCR PURE Q11
11 The probability that Janice sees a kingfisher on any particular day is 0.3 . She notes the number, \(X\), of days in a week on which she sees a kingfisher.
  1. State one necessary condition for \(X\) to have a binomial distribution. Assume now that \(X\) has a binomial distribution.
  2. Find the probability that, in a week, Janice sees a kingfisher on exactly 2 days. Each week Janice notes the number of days on which she sees a kingfisher.
  3. Find the probability that Janice sees a kingfisher on exactly 2 days in a week during at least 4 of 6 randomly chosen weeks.
OCR PURE Q12
12 It is known that \(20 \%\) of plants of a certain type suffer from a fungal disease, when grown under normal conditions. Some plants of this type are grown using a new method. A random sample of 250 of these plants is chosen, and it is found that 36 suffer from the disease. Test, at the \(2 \%\) significance level, whether there is evidence that the new method reduces the proportion of plants which suffer from the disease.
OCR PURE Q13
13 The radar diagrams illustrate some population figures from the 2011 census results.
\includegraphics[max width=\textwidth, alt={}, center]{3fdf2b2d-20a9-4d68-b760-57ec529b5893-6_712_764_303_248}
\includegraphics[max width=\textwidth, alt={}, center]{3fdf2b2d-20a9-4d68-b760-57ec529b5893-6_709_757_303_1137} Each radius represents an age group, as follows:
Radius123456
Age
group
\(0 - 17\)\(18 - 29\)\(30 - 44\)\(45 - 59\)\(60 - 74\)\(75 +\)
The distance of each dot from the centre represents the number of people in the relevant age group.
  1. The scales on the two diagrams are different. State an advantage and a disadvantage of using different scales in order to make comparisons between the ages of people in these two Local Authorities.
  2. Approximately how many people aged 45 to 59 were there in Liverpool?
  3. State the main two differences between the age profiles of the two Local Authorities.
  4. James makes the following claim.
    "Assuming that there are no significant movements of population either into or out of the two regions, the 2021 census results are likely to show an increase in the number of children in Liverpool and a decrease in the number of children in Rutland." Use the radar diagrams to give a justification for this claim.
OCR PURE 2066 Q1
1 It is given that \(\mathrm { f } ( x ) = 3 x - \frac { 5 } { x ^ { 3 } }\).
Find
  1. \(\mathrm { f } ^ { \prime } ( x )\),
  2. \(\mathrm { f } ^ { \prime \prime } ( x )\),
  3. \(\int \mathrm { f } ( x ) \mathrm { d } x\).
OCR PURE 2066 Q2
2 The circle \(x ^ { 2 } + y ^ { 2 } - 4 x + k y + 12 = 0\) has radius 1.
Find the two possible values of the constant \(k\).
OCR PURE 2066 Q3
3 In this question you must show detailed reasoning.
  1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 8 x + 3\).
    1. Show that \(f ( 1 ) = 0\).
    2. Solve the equation \(\mathrm { f } ( x ) = 0\).
  2. Hence solve the equation \(2 \sin ^ { 3 } \theta + 3 \sin ^ { 2 } \theta - 8 \sin \theta + 3 = 0\) for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\).
OCR PURE 2066 Q4
4
  1. Find the coordinates of the stationary points on the curve \(y = x ^ { 3 } - 6 x ^ { 2 } + 9 x\).
  2. The equation \(x ^ { 3 } - 6 x ^ { 2 } + 9 x + k = 0\) has exactly one real root. Using your answers from part (a) or otherwise, find the range of possible values of \(k\).
OCR PURE 2066 Q5
5
  1. Prove that the following statement is not true.
    \(m\) is an odd number greater than \(1 \Rightarrow m ^ { 2 } + 4\) is prime.
  2. By considering separately the case when \(n\) is odd and the case when \(n\) is even, prove that the following statement is true.
    \(n\) is a positive integer \(\Rightarrow n ^ { 2 } + 1\) is not a multiple of 4 .
OCR PURE 2066 Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{68f1107f-f188-4698-934e-8fd593b25418-4_442_661_840_260} The diagram shows triangle \(A B C\), with \(A B = x \mathrm {~cm} , A C = y \mathrm {~cm}\) and angle \(B A C = 60 ^ { \circ }\). It is given that the area of the triangle is \(( x + y ) \sqrt { 3 } \mathrm {~cm} ^ { 2 }\).
  1. Show that \(4 x + 4 y = x y\). When the vertices of the triangle are placed on the circumference of a circle, \(A C\) is a diameter of the circle.
  2. Determine the value of \(x\) and the value of \(y\).
OCR PURE 2066 Q7
7
  1. Write down an expression for the gradient of the curve \(y = \mathrm { e } ^ { k x }\).
  2. The line L is a tangent to the curve \(y = \mathrm { e } ^ { \frac { 1 } { 2 } x }\) at the point where \(x = 2\). Show that L passes through the point \(( 0,0 )\).
  3. Find the coordinates of the point of intersection of the curves \(y = 3 \mathrm { e } ^ { x }\) and \(y = 1 - 2 \mathrm { e } ^ { \frac { 1 } { 2 } x }\).
OCR PURE 2066 Q8
8
  1. Joseph drew a histogram to show information about one Local Authority. He used data from the "Age structure by LA 2011" tab in the large data set. The table shows an extract from the data that he used.
    Age group0 to 4
    Frequency2143
    Joseph used a scale of \(1 \mathrm {~cm} = 1000\) units on the frequency density axis. Calculate the height of the histogram block for the 0 to 4 class.
  2. Magdalene wishes to draw a statistical diagram to illustrate some of the data from the "Method of travel by LA 2011" tab in the large data set. State why she cannot draw a histogram.
OCR PURE 2066 Q9
9 The table shows information about the number of days absent last year by students in class 2A at a certain school.
Number of days absent012 to 45 to 1011 to 2021 to 30More than 30
Number of students71291010
  1. Calculate an estimate of the mean for these data.
  2. Find the median of these data. The headteacher is writing a report on the numbers of absences at her school. She wishes to include a figure for the average number of absences in class 2A. A governor suggests that she should quote the mean. The class teacher suggests that she should quote the median, because it is lower than the mean.
  3. Give another reason for using the median rather than the mean for the average number of absences in class 2A.
OCR PURE 2066 Q10
10 The table shows extracts from the "Method of travel by LA" tabs for 2001 and 2011 in the large data set.
Local authority (LA)All people in employmentUnderground, metro, light rail, tramTrainBus, minibus or coachMotorcycle, scooter or mopedDriving a car or van
LA1 20017922614369523520575122716052
LA1 201111855622486833630541122012445
LA2 20012036141901062153271256121690
LA2 20112278943231865137321038146644
LA3 20014299335482436327424105
LA3 20114901433828338019128981
LA4 2001101697656932175884645407
LA4 2011123218249513152427576354020
  1. In one of these four LAs a new tram system was opened in 2004. Suggest, with a reason taken from the data, which LA this could have been.
  2. Julian suggests that the figures for "Bus, minibus or coach" for LA1 show that some new bus routes were probably introduced in this LA between 2001 and 2011. Use data from the table to comment on this suggestion.
  3. In one of these four LAs a congestion charge on vehicles was introduced in 2003. Suggest, with a reason taken from the data, which LA this could have been.
OCR PURE 2066 Q11
11 It is known that, under the standard treatment for a certain disease, \(9.7 \%\) of patients with the disease experience side effects within one year. In a trial of a new treatment, a random sample of 450 patients with this disease was selected and the number \(X\) who experienced side effects within one year was noted.
  1. State one assumption needed in order to use a binomial model for \(X\). It was found that 51 of the 450 patients experienced side effects within one year.
  2. Test, at the \(10 \%\) significance level, whether the proportion of patients experiencing side effects within one year is greater under the new treatment than under the standard treatment.
OCR PURE 2066 Q12
12 The Venn diagram shows the numbers of students studying various subjects, in a year group of 100 students.
\includegraphics[max width=\textwidth, alt={}, center]{68f1107f-f188-4698-934e-8fd593b25418-7_554_910_347_244} A student is chosen at random from the 100 students. Then another student is chosen from the remaining students. Find the probability that the first student studies History and the second student studies Geography but not Psychology. \section*{END OF QUESTION PAPER} \section*{OCR} Oxford Cambridge and RSA
OCR PURE Q1
1
  1. Find \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x ^ { 3 } - 3 x + \frac { 5 } { x ^ { 2 } } \right)\).
  2. Find \(\int \left( 6 x ^ { 2 } - \frac { 2 } { x ^ { 3 } } \right) \mathrm { d } x\).
OCR PURE Q2
2 Points \(A\) and \(B\) have position vectors \(\binom { - 3 } { 4 }\) and \(\binom { 1 } { 2 }\) respectively.
Point \(C\) has position vector \(\binom { p } { 1 }\) and \(A B C\) is a straight line.
  1. Find \(p\). Point \(D\) has position vector \(\binom { q } { 1 }\) and angle \(A B D = 90 ^ { \circ }\).
  2. Determine the value of \(q\).
OCR PURE Q3
3 In this question you must show detailed reasoning.
  1. Solve the equation \(4 \sin ^ { 2 } \theta = \tan ^ { 2 } \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
  2. Prove that \(\frac { \sin ^ { 2 } \theta - 1 + \cos \theta } { 1 - \cos \theta } \equiv \cos \theta \quad ( \cos \theta \neq 1 )\).