Questions — OCR PURE (139 questions)

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OCR PURE Q1
8 marks Easy -1.2
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 Q2
4 marks Moderate -0.5
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 Q3
10 marks Standard +0.3
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 Q4
6 marks Standard +0.3
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 Q5
5 marks Moderate -0.3
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 Q6
6 marks Standard +0.8
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 Q7
11 marks Moderate -0.8
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 Q8
3 marks Easy -1.8
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 Q9
4 marks Moderate -0.3
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 Q10
6 marks Easy -1.8
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 Q11
8 marks Standard +0.3
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 Q12
4 marks Moderate -0.5
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
6 marks Easy -1.2
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
5 marks Moderate -0.3
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
8 marks Standard +0.3
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 \(\frac { \sin ^ { 2 } \theta - 1 + \cos \theta } { 1 - \cos \theta } \equiv \cos \theta \quad ( \cos \theta \neq 1 )\).
OCR PURE Q4
5 marks Easy -1.2
4
  1. Expand \(( 1 + x ) ^ { 4 }\).
  2. Use your expansion to determine the exact value of \(1002 ^ { 4 }\).
OCR PURE Q5
8 marks Standard +0.3
5 The function f is defined by \(\mathrm { f } ( x ) = ( x + a ) ( x + 3 a ) ( x - b )\) where \(a\) and \(b\) are positive integers.
  1. On the axes in the Printed Answer Booklet, sketch the curve \(y = \mathrm { f } ( x )\).
  2. On your sketch show, in terms of \(a\) and \(b\), the coordinates of the points where the curve meets the axes. It is now given that \(a = 1\) and \(b = 4\).
  3. Find the total area enclosed between the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis.
OCR PURE Q6
13 marks Moderate -0.3
6 In this question you must show detailed reasoning.
  1. Solve the inequality \(x ^ { 2 } + x - 6 > 0\), giving your answer in set notation.
  2. Solve the equation \(x ^ { 3 } - 7 x ^ { \frac { 3 } { 2 } } - 8 = 0\).
  3. Find the exact solution of the equation \(\left( 3 ^ { x } \right) ^ { 2 } = 3 \times 2 ^ { x }\).
OCR PURE Q7
5 marks Moderate -0.3
7 Determine the points of intersection of the curve \(3 x y + x ^ { 2 } + 14 = 0\) and the line \(x + 2 y = 4\).
OCR PURE Q8
5 marks Easy -1.2
8 The histogram shows information about the lengths, \(l\) centimetres, of a sample of worms of a certain species. \includegraphics[max width=\textwidth, alt={}, center]{4c6b7c92-2fc9-4d4f-a199-8e70f34e5eed-5_904_1284_488_242} The number of worms in the sample with lengths in the class \(3 \leqslant l < 4\) is 30 .
  1. Find the number of worms in the sample with lengths in the class \(0 \leqslant l < 2\).
  2. Find an estimate of the number of worms in the sample with lengths in the range \(4.5 \leqslant l < 5.5\).
OCR PURE Q9
8 marks Easy -1.8
9 A researcher is studying changes in behaviour in travelling to work by people who live outside London, between 2001 and 2011. He chooses the 15 Local Authorities (LAs) outside London with the largest decreases in the percentage of people driving to work, and arranges these in descending order. The table shows the changes in percentages from 2001 to 2011 in various travel categories, for these Local Authorities.
Local AuthorityWork mainly at or from homeUnderground, metro, light rail, tramTrainBus, minibus or coachDriving a car or vanPassenger in a car or vanBicycleOn foot
Brighton and Hove3.20.11.50.8-8.2-1.52.12.3
Cambridge2.20.01.61.2-7.4-1.03.10.6
Elmbridge2.90.44.10.2-6.6-0.70.3-0.3
Oxford2.00.00.6-0.4-5.2-1.12.22.1
Epsom and Ewell1.60.43.91.1-5.2-0.90.0-0.6
Watford0.72.03.10.4-4.5-1.20.0-0.1
Tandridge3.30.24.0-0.1-4.5-1.10.0-1.3
Mole Valley3.30.11.90.3-4.4-0.70.2-0.3
St Albans2.30.33.4-0.3-4.3-1.20.3-0.2
Chiltern2.91.41.40.1-4.2-0.6-0.2-0.8
Exeter0.70.01.0-0.6-4.2-1.51.73.4
Woking2.10.13.70.0-4.2-1.3-0.10.0
Reigate and Banstead1.80.13.20.6-4.1-1.00.1-0.2
Waverley4.30.12.5-0.5-3.9-0.9-0.3-0.9
Guildford2.70.12.40.2-3.6-1.20.0-0.3
  1. Explain why these LAs are not necessarily the 15 LAs with the largest decreases in the percentage of people driving to work.
  2. The researcher wants to talk to those LAs outside London which have been most successful in encouraging people to change to cycling or walking to work.
    Suggest four LAs that he should talk to and why.
  3. The researcher claims that Waverley is the LA outside London which has had the largest increase in the number of people working mainly at or from home.
    Does the data support his claim? Explain your answer.
  4. Which two categories have replaced driving to work for the highest percentages of workers in these LAs? Support your answer with evidence from the table.
  5. The researcher suggested that there would be strong correlation between the decrease in the percentage driving to work and the increase in percentage working mainly at or from home. Without calculation, use data from the table to comment briefly on this suggestion.
OCR PURE Q10
8 marks Standard +0.3
10 Some packets of a certain kind of biscuit contain a free gift. The manufacturer claims that the proportion of packets containing a free gift is 1 in 4 . Marisa suspects that this claim is not true, and that the true proportion is less than 1 in 4 . She chooses 20 packets at random and finds that exactly 1 contains the free gift.
  1. Use a binomial model to test the manufacturer's claim, at the \(2.5 \%\) significance level. The packets are packed in boxes, with each box containing 40 packets. Marisa chooses three boxes at random and finds that one box contains 19 packets with the free gift and the other two boxes contain no packets with the free gift.
  2. Give a reason why this suggests that the binomial model used in part (a) may not be appropriate.
OCR PURE Q1
7 marks Easy -1.2
1 In the triangle \(A B C , A B = 3 , B C = 4\) and angle \(A B C = 30 ^ { \circ }\). Find the following.
  1. The area of the triangle.
  2. The length \(A C\).
  3. The angle \(A C B\).
OCR PURE Q2
4 marks Easy -1.2
2 The number of people, \(n\), living in a small town is changing over time. In an attempt to predict the future growth of the town, a researcher uses the following model for \(n\) in terms of \(t\), where \(t\) is the time in years from the start of the research. \(n = 12500 + \frac { 5000 } { t }\), for \(t \geqslant 1\) Find the rate of change of \(n\) when \(t = 5\).
OCR PURE Q3
6 marks Standard +0.3
3 The diagram shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a cubic polynomial in \(x\). This diagram is repeated in the Printed Answer Booklet. \includegraphics[max width=\textwidth, alt={}, center]{d44919ed-806d-48c0-9726-c5fd67764504-03_896_1467_1382_244}
  1. State the values of \(x\) for which \(\mathrm { f } ( x ) < \frac { 1 } { 2 }\), giving your answer in set notation.
  2. On the diagram in the Printed Answer Booklet, draw the graph of \(y = \mathrm { f } ( - x )\).
  3. Explain how you can tell that \(\mathrm { f } ( x )\) cannot be expressed as the product of three real linear factors.