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OCR H240/03 2018 June Q5
13 marks Standard +0.3
5
  1. Use the trapezium rule, with two strips of equal width, to show that $$\int _ { 0 } ^ { 4 } \frac { 1 } { 2 + \sqrt { x } } \mathrm {~d} x \approx \frac { 11 } { 4 } - \sqrt { 2 }$$
  2. Use the substitution \(x = u ^ { 2 }\) to find the exact value of $$\int _ { 0 } ^ { 4 } \frac { 1 } { 2 + \sqrt { x } } \mathrm {~d} x$$
  3. Using your answers to parts (i) and (ii), show that $$\ln 2 \approx k + \frac { \sqrt { 2 } } { 4 }$$ where \(k\) is a rational number to be determined.
OCR H240/03 2018 June Q6
8 marks Standard +0.3
6 It is given that the angle \(\theta\) satisfies the equation \(\sin \left( 2 \theta + \frac { 1 } { 4 } \pi \right) = 3 \cos \left( 2 \theta + \frac { 1 } { 4 } \pi \right)\).
  1. Show that \(\tan 2 \theta = \frac { 1 } { 2 }\).
  2. Hence find, in surd form, the exact value of \(\tan \theta\), given that \(\theta\) is an obtuse angle.
OCR H240/03 2018 June Q7
9 marks Standard +0.3
7 The gradient of the curve \(y = \mathrm { f } ( x )\) is given by the differential equation $$( 2 x - 1 ) ^ { 3 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y ^ { 2 } = 0$$ and the curve passes through the point \(( 1,1 )\). By solving this differential equation show that $$f ( x ) = \frac { a x ^ { 2 } - a x + 1 } { b x ^ { 2 } - b x + 1 }$$ where \(a\) and \(b\) are integers to be determined.
OCR H240/03 2018 June Q8
6 marks Moderate -0.8
8 In this question \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) denote unit vectors which are horizontal and vertically upwards respectively.
A particle of mass 5 kg , initially at rest at the point with position vector \(\binom { 2 } { 45 } \mathrm {~m}\), is acted on by gravity and also by two forces \(\binom { 15 } { - 8 } \mathrm {~N}\) and \(\binom { - 7 } { - 2 } \mathrm {~N}\).
  1. Find the acceleration vector of the particle.
  2. Find the position vector of the particle after 10 seconds.
OCR H240/03 2018 June Q9
9 marks Standard +0.3
9 A uniform plank \(A B\) has weight 100 N and length 4 m . The plank rests horizontally in equilibrium on two smooth supports \(C\) and \(D\), where \(A C = x \mathrm {~m}\) and \(C D = 0.5 \mathrm {~m}\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{d5ab20c8-afd5-473e-8238-96762bd3786d-6_181_1271_1101_395} The magnitude of the reaction of the support on the plank at \(C\) is 75 N . Modelling the plank as a rigid rod, find
  1. the magnitude of the reaction of the support on the plank at \(D\),
  2. the value of \(x\). A stone block, which is modelled as a particle, is now placed at the end of the plank at \(B\) and the plank is on the point of tilting about \(D\).
  3. Find the weight of the stone block.
  4. Explain the limitation of modelling
    1. the stone block as a particle,
    2. the plank as a rigid rod.
OCR H240/03 2018 June Q10
11 marks Standard +0.3
10 Three forces, of magnitudes \(4 \mathrm {~N} , 6 \mathrm {~N}\) and \(P \mathrm {~N}\), act at a point in the directions shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{d5ab20c8-afd5-473e-8238-96762bd3786d-7_604_601_306_724} The forces are in equilibrium.
  1. Show that \(\theta = 41.4 ^ { \circ }\), correct to 3 significant figures.
  2. Hence find the value of \(P\). The force of magnitude 4 N is now removed and the force of magnitude 6 N is replaced by a force of magnitude 3 N acting in the same direction.
  3. Find
    1. the magnitude of the resultant of the two remaining forces,
    2. the direction of the resultant of the two remaining forces.
OCR H240/03 2018 June Q11
10 marks Moderate -0.3
11 The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of a car at time \(t \mathrm {~s}\), during the first 20 s of its journey, is given by \(v = k t + 0.03 t ^ { 2 }\), where \(k\) is a constant. When \(t = 20\) the acceleration of the car is \(1.3 \mathrm {~ms} ^ { - 2 }\). For \(t > 20\) the car continues its journey with constant acceleration \(1.3 \mathrm {~ms} ^ { - 2 }\) until its speed reaches \(25 \mathrm {~ms} ^ { - 1 }\).
  1. Find the value of \(k\).
  2. Find the total distance the car has travelled when its speed reaches \(25 \mathrm {~ms} ^ { - 1 }\).
OCR H240/03 2018 June Q12
14 marks Standard +0.3
12 One end of a light inextensible string is attached to a particle \(A\) of mass \(m \mathrm {~kg}\). The other end of the string is attached to a second particle \(B\) of mass \(\lambda m \mathrm {~kg}\), where \(\lambda\) is a constant. Particle \(A\) is in contact with a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. The string is taut and passes over a small smooth pulley \(P\) at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. The particle \(B\) hangs freely below \(P\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{d5ab20c8-afd5-473e-8238-96762bd3786d-8_405_670_493_685} The coefficient of friction between \(A\) and the plane is \(\mu\).
  1. It is given that \(A\) is on the point of moving down the plane.
    1. Find the exact value of \(\mu\) when \(\lambda = \frac { 1 } { 4 }\).
    2. Show that the value of \(\lambda\) must be less than \(\frac { 1 } { 2 }\).
    3. Given instead that \(\lambda = 2\) and that the acceleration of \(A\) is \(\frac { 1 } { 4 } g \mathrm {~ms} ^ { - 2 }\), find the exact value of \(\mu\). \section*{END OF QUESTION PAPER}
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.