Questions — OCR (4907 questions)

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OCR PURE Q8
4 marks Easy -1.8
8 A random sample of 10 students from a college was chosen. They were asked how much time, \(x\) hours, they spent studying, and how much money, \(\pounds y\), they earned, in a typical week during term time. The results are shown in the scatter diagram. \includegraphics[max width=\textwidth, alt={}, center]{c4cc2cd8-46bf-448f-b223-92378984bfde-5_544_741_555_242}
  1. Comment on the relationship shown by the diagram between hours spent studying and money earned, during term time, by these 10 students. The coordinates of the points in the diagram are \(( 18,23 ) , ( 20,21 ) , ( 23,20 ) , ( 25,19 ) , ( 25,21 )\), \(( 27,18 ) , ( 32,16 ) , ( 38,17 ) , ( 40,16 )\) and \(( 41,23 )\).
  2. Find the mean and standard deviation of the number of hours spent per week studying during term time by these 10 students.
OCR PURE Q9
10 marks Standard +0.3
9 Last year, market research showed that \(8 \%\) of adults living in a certain town used a particular local coffee shop. Following an advertising campaign, it was expected that this proportion would increase. In order to test whether this had happened, a random sample of 150 adults in the town was chosen. The random variable \(X\) denotes the number of these 150 adults who said that they used the local coffee shop.
    1. Assuming that the proportion of adults using the local coffee shop is unchanged from the previous year, state a suitable binomial distribution with which to model the variable \(X\).
    2. The probabilities given by this model are the terms of the binomial expansion of an expression of the form \(( a + b ) ^ { n }\). Write down this expression, using appropriate values of \(a , b\) and \(n\). It was found that 18 of these 150 adults said that they use the local coffee shop.
  2. Test, at the 5\% significance level, whether the proportion of adults in the town who use the local coffee shop has increased. It was later discovered by a statistician that the random sample of 150 adults had been chosen from shoppers in the town on a Friday and a Saturday.
  3. Explain why this suggests that the assumptions made when using a binomial model for \(X\) may not be valid in this context.
OCR PURE Q10
6 marks Easy -1.8
10 The table shows the increases, between 2001 and 2011, in the percentages of employees travelling to work by various methods, in the Local Authorities (LAs) in the North East region of the UK.
Geography codeLocal authorityWork mainly at or from homeUnderground, metro, light rail or tramBus, minibus or coachDriving a car or vanPassenger in a car or vanOn foot
E06000047County Durham0.74\%0.05\%-1.50\%4.58\%-2.99\%-0.97\%
E06000005Darlington0.26\%-0.01\%-3.25\%3.06\%-1.28\%0.99\%
E08000020Gateshead-0.01\%-0.01\%-2.28\%4.62\%-2.35\%-0.18\%
E06000001Hartlepool0.03\%-0.04\%-1.62\%4.80\%-2.38\%-0.26\%
E06000002Middlesbrough-0.34\%-0.01\%-2.32\%2.19\%-1.33\%0.67\%
E08000021Newcastle upon Tyne0.10\%-0.23\%-0.67\%-0.48\%-1.51\%1.75\%
E08000022North Tyneside0.05\%0.54\%-1.18\%3.30\%-2.21\%-0.60\%
E06000048Northumberland1.39\%-0.08\%-0.95\%3.50\%-2.37\%-1.44\%
E06000003Redcar and Cleveland-0.02\%-0.01\%-2.09\%4.20\%-2.06\%-0.49\%
E08000023South Tyneside-0.36\%2.03\%-3.05\%4.50\%-2.41\%-0.51\%
E06000004Stockton-on-Tees0.14\%0.03\%-2.02\%3.52\%-2.01\%-0.15\%
E08000024Sunderland0.17\%1.48\%-3.11\%4.89\%-2.21\%-0.52\%
Increase in percentage of employees travelling to work by various methods
The first two digits of the Geography code give the type of each of the LAs:
06: Unitary authority
07: Non-metropolitan district
08: Metropolitan borough
  1. In what type of LA are the largest increases in percentages of people travelling by underground, metro, light rail or tram?
  2. Identify two main changes in the pattern of travel to work in the North East region between 2001 and 2011. Now assume the following.
OCR PURE Q11
6 marks Standard +0.3
11 Alex models the number of goals that a local team will score in any match as follows.
Number of goals01234
More
than 4
Probability\(\frac { 3 } { 25 }\)\(\frac { 1 } { 5 }\)\(\frac { 8 } { 25 }\)\(\frac { 7 } { 25 }\)\(\frac { 2 } { 25 }\)0
The number of goals scored in any match is independent of the number of goals scored in any other match.
  1. Alex chooses 3 matches at random. Use the model to determine the probability of each of the following.
    1. The team will score a total of exactly 1 goal in the 3 matches.
    2. The numbers of goals scored in the first 2 of the 3 matches will be equal, but the number of goals scored in the 3rd match will be different. During the first 10 matches this season, the team scores a total of 31 goals.
  2. Without carrying out a formal test, explain briefly whether this casts doubt on the validity of Alex's model. \section*{END OF QUESTION PAPER}
OCR PURE Q1
6 marks Standard +0.3
1 In triangle \(A B C , A B = 20 \mathrm {~cm}\) and angle \(B = 45 ^ { \circ }\).
  1. Given that \(A C = 16 \mathrm {~cm}\), find the two possible values for angle \(C\), correct to 1 decimal place.
  2. Given instead that the area of the triangle is \(75 \sqrt { 2 } \mathrm {~cm} ^ { 2 }\), find \(B C\).
OCR PURE Q2
4 marks Moderate -0.8
2
  1. The curve \(y = \frac { 2 } { 3 + x }\) is translated by four units in the positive \(x\)-direction. State the equation of the curve after it has been translated.
  2. Describe fully the single transformation that transforms the curve \(y = \frac { 2 } { 3 + x }\) to \(y = \frac { 5 } { 3 + x }\).
OCR PURE Q3
3 marks Standard +0.3
3 In each of the following cases choose one of the statements $$P \Rightarrow Q \quad P \Leftarrow Q \quad P \Leftrightarrow Q$$ to describe the relationship between \(P\) and \(Q\).
  1. \(P : y = 3 x ^ { 5 } - 4 x ^ { 2 } + 12 x\) \(Q : \frac { \mathrm { d } y } { \mathrm {~d} x } = 15 x ^ { 4 } - 8 x + 12\)
  2. \(\quad P : x ^ { 5 } - 32 = 0\) where \(x\) is real \(Q : x = 2\)
  3. \(\quad P : \ln y < 0\) \(Q : y < 1\)
OCR PURE Q4
6 marks Moderate -0.3
4
  1. Express \(4 x ^ { 2 } - 12 x + 11\) in the form \(a ( x + b ) ^ { 2 } + c\).
  2. State the number of real roots of the equation \(4 x ^ { 2 } - 12 x + 11 = 0\).
  3. Explain fully how the value of \(r\) is related to the number of real roots of the equation \(p ( x + q ) ^ { 2 } + r = 0\) where \(p , q\) and \(r\) are real constants and \(p > 0\).
OCR PURE Q5
5 marks Standard +0.3
5 In this question you must show detailed reasoning.
The line \(x + 5 y = k\) is a tangent to the curve \(x ^ { 2 } - 4 y = 10\). Find the value of the constant \(k\).
OCR PURE Q6
9 marks Standard +0.3
6 A pan of water is heated until it reaches \(100 ^ { \circ } \mathrm { C }\). Once the water reaches \(100 ^ { \circ } \mathrm { C }\), the heat is switched off and the temperature \(T ^ { \circ } \mathrm { C }\) of the water decreases. The temperature of the water is modelled by the equation $$T = 25 + a \mathrm { e } ^ { - k t }$$ where \(t\) denotes the time, in minutes, after the heat is switched off and \(a\) and \(k\) are positive constants.
  1. Write down the value of \(a\).
  2. Explain what the value of 25 represents in the equation \(T = 25 + a \mathrm { e } ^ { - k t }\). When the heat is switched off, the initial rate of decrease of the temperature of the water is \(15 ^ { \circ } \mathrm { C }\) per minute.
  3. Calculate the value of \(k\).
  4. Find the time taken for the temperature of the water to drop from \(100 ^ { \circ } \mathrm { C }\) to \(45 ^ { \circ } \mathrm { C }\).
  5. A second pan of water is heated, but the heat is turned off when the water is at a temperature of less than \(100 ^ { \circ } \mathrm { C }\). Suggest how the equation for the temperature as the water cools would be modified by this.
OCR PURE Q7
8 marks Standard +0.3
7
  1. Show that the equation $$2 \sin x \tan x = \cos x + 5$$ can be expressed in the form $$3 \cos ^ { 2 } x + 5 \cos x - 2 = 0$$
  2. Hence solve the equation $$2 \sin 2 \theta \tan 2 \theta = \cos 2 \theta + 5$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(180 ^ { \circ }\), correct to 1 decimal place.
OCR PURE Q9
4 marks Standard +0.3
9 In this question the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the directions east and north respectively.
A model ship of mass 2 kg is moving so that its acceleration vector \(\mathbf { a m s } ^ { - 2 }\) at time \(t\) seconds is given by \(\mathbf { a } = 3 ( 2 t - 5 ) \mathbf { i } + 4 \mathbf { j }\). When \(t = T\), the magnitude of the horizontal force acting on the ship is 10 N . Find the possible values of \(T\).
OCR PURE Q10
8 marks Standard +0.8
10 Particles \(P\) and \(Q\), of masses 3 kg and 5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The system is held at rest with the string taut. The hanging parts of the string are vertical and \(P\) and \(Q\) are above a horizontal plane (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{efde7b10-b4f3-469f-ba91-b765a16ea835-6_428_208_932_932}
  1. Find the tension in the string immediately after the particles are released. After descending \(2.5 \mathrm {~m} , Q\) strikes the plane and is immediately brought to rest. It is given that \(P\) does not reach the pulley in the subsequent motion.
  2. Find the distance travelled by \(P\) between the instant when \(Q\) strikes the plane and the instant when the string becomes taut again.
OCR PURE Q11
13 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{efde7b10-b4f3-469f-ba91-b765a16ea835-7_127_1147_260_459} A particle \(P\) is moving along a straight line with constant acceleration. Initially the particle is at \(O\). After 9 s , \(P\) is at a point \(A\), where \(O A = 18 \mathrm {~m}\) (see diagram) and the velocity of \(P\) at \(A\) is \(8 \mathrm {~ms} ^ { - 1 }\) in the direction \(\overrightarrow { O A }\).
  1. (a) Show that the initial speed of \(P\) is \(4 \mathrm {~ms} ^ { - 1 }\).
    (b) Find the acceleration of \(P\). \(B\) is a point on the line such that \(O B = 10 \mathrm {~m}\), as shown in the diagram.
  2. Show that \(P\) is never at point \(B\). A second particle \(Q\) moves along the same straight line, but has variable acceleration. Initially \(Q\) is at \(O\), and the displacement of \(Q\) from \(O\) at time \(t\) seconds is given by $$x = a t ^ { 3 } + b t ^ { 2 } + c t$$ where \(a\), \(b\) and \(c\) are constants.
    It is given that
    \section*{OCR} \section*{Oxford Cambridge and RSA}
OCR PURE Q2
6 marks Moderate -0.8
2
  1. Express \(5 x ^ { 2 } - 20 x + 3\) in the form \(p ( x + q ) ^ { 2 } + r\), where \(p , q\) and \(r\) are integers.
  2. State the coordinates of the minimum point of the curve \(y = 5 x ^ { 2 } - 20 x + 3\).
  3. State the equation of the normal to the curve \(y = 5 x ^ { 2 } - 20 x + 3\) at its minimum point.
OCR PURE Q3
5 marks Easy -1.3
3
  1. Sketch the curve \(y = - \frac { 1 } { x ^ { 2 } }\).
  2. The curve \(y = - \frac { 1 } { x ^ { 2 } }\) is translated by 2 units in the positive \(x\)-direction. State the equation of the curve after it has been translated.
  3. The curve \(y = - \frac { 1 } { x ^ { 2 } }\) is stretched parallel to the \(y\)-axis with scale factor \(\frac { 1 } { 2 }\) and, as a result, the point \(\left( \frac { 1 } { 2 } , - 4 \right)\) on the curve is transformed to the point \(P\). State the coordinates of \(P\).
OCR PURE Q4
6 marks Moderate -0.3
4
  1. Find and simplify the first three terms in the expansion of \(( 2 - 5 x ) ^ { 5 }\) in ascending powers of \(x\).
  2. In the expansion of \(( 1 + a x ) ^ { 2 } ( 2 - 5 x ) ^ { 5 }\), the coefficient of \(x\) is 48 . Find the value of \(a\).
OCR PURE Q5
6 marks Moderate -0.3
5 Points \(A , B , C\) and \(D\) have position vectors \(\mathbf { a } = \binom { 1 } { 2 } , \mathbf { b } = \binom { 3 } { 5 } , \mathbf { c } = \binom { 7 } { 4 }\) and \(\mathbf { d } = \binom { 4 } { k }\).
  1. Find the value of \(k\) for which \(D\) is the midpoint of \(A C\).
  2. Find the two values of \(k\) for which \(| \overrightarrow { A D } | = \sqrt { 13 }\).
  3. Find one value of \(k\) for which the four points form a trapezium.
OCR PURE Q7
8 marks Challenging +1.2
7 \includegraphics[max width=\textwidth, alt={}, center]{d6430776-0b87-4e5e-8f78-c6228ee163d5-5_647_741_260_260} The diagram shows part of the curve \(y = ( 5 - x ) ( x - 1 )\) and the line \(x = a\).
Given that the total area of the regions shaded in the diagram is 19 units \({ } ^ { 2 }\), determine the exact value of \(a\).
OCR PURE Q8
8 marks Standard +0.3
8
  1. Show that the equation \(2 \log _ { 2 } x = \log _ { 2 } ( k x - 1 ) + 3\), where \(k\) is a constant, can be expressed in the form \(x ^ { 2 } - 8 k x + 8 = 0\).
  2. Given that the equation \(2 \log _ { 2 } x = \log _ { 2 } ( k x - 1 ) + 3\) has only one real root, find the value of this root.
OCR PURE Q9
2 marks Moderate -0.8
9 Three forces \(\binom { 7 } { - 6 } \mathrm {~N} , \binom { 2 } { 5 } \mathrm {~N}\) and \(\mathbf { F N }\) act on a particle.
Given that the particle is in equilibrium under the action of these three forces, calculate \(\mathbf { F }\).
OCR PURE Q10
8 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{d6430776-0b87-4e5e-8f78-c6228ee163d5-6_670_1106_797_258} The diagram shows the velocity-time graph modelling the velocity of a car as it approaches, and drives through, a residential area. The velocity of the car, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds for the time interval \(0 \leqslant t \leqslant 5\) is modelled by the equation \(v = p t ^ { 2 } + q t + r\), where \(p , q\) and \(r\) are constants. It is given that the acceleration of the car is zero at \(t = 5\) and the speed of the car then remains constant.
  1. Determine the values of \(p , q\) and \(r\).
  2. Calculate the distance travelled by the car from \(t = 2\) to \(t = 10\).
OCR PURE Q11
16 marks Standard +0.3
11 Two small balls \(P\) and \(Q\) have masses 3 kg and 2 kg respectively. The balls are attached to the ends of a string. \(P\) is held at rest on a rough horizontal surface. The string passes over a pulley which is fixed at the edge of the surface. \(Q\) hangs vertically below the pulley at a height of 2 m above a horizontal floor. \includegraphics[max width=\textwidth, alt={}, center]{d6430776-0b87-4e5e-8f78-c6228ee163d5-7_346_906_445_255} The system is initially at rest with the string taut. A horizontal force of magnitude 40 N acts on \(P\) as shown in the diagram. \(P\) is released and moves directly away from the pulley. A constant frictional force of magnitude 8 N opposes the motion of \(P\). It is given that \(P\) does not leave the horizontal surface and that \(Q\) does not reach the pulley in the subsequent motion. The balls are modelled as particles, the pulley is modelled as being small and smooth, and the string is modelled as being light and inextensible.
  1. Show that the magnitude of the acceleration of each particle is \(2.48 \mathrm {~ms} ^ { - 2 }\).
  2. Find the tension in the string. When the balls have been in motion for 0.5 seconds, the string breaks.
  3. Find the additional time that elapses until \(Q\) hits the floor.
  4. Find the speed of \(Q\) as it hits the floor.
  5. Write down the magnitude of the normal reaction force acting on \(Q\) when \(Q\) has come to rest on the floor.
  6. State one improvement that could be made to the model. \section*{OCR} Oxford Cambridge and RSA
OCR PURE Q1
5 marks Moderate -0.8
1 The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 9.5 cm . The angle \(A O B\) is \(25 ^ { \circ }\).
  1. Calculate the length of the straight line \(A B\).
  2. Find the area of the segment shaded in the diagram.
OCR PURE Q2
5 marks Moderate -0.3
2 Two curves have equations \(y = \ln x\) and \(y = \frac { k } { x }\), where \(k\) is a positive constant.
  1. Sketch the curves on a single diagram.
  2. Explain how your diagram shows that the equation \(x \ln x - k = 0\) has exactly one real root.