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OCR MEI AS Paper 1 2018 June Q11
11 The intensity of the sun's radiation, \(y\) watts per square metre, and the average distance from the sun, \(x\) astronomical units, are shown in Fig. 11 for the planets Mercury and Jupiter. \begin{table}[h]
\(x\)\(y\)
Mercury0.307514400
Jupiter4.95055.8
\captionsetup{labelformat=empty} \caption{Fig. 11}
\end{table} The intensity \(y\) is proportional to a power of the distance \(x\).
  1. Write down an equation for \(y\) in terms of \(x\) and two constants.
  2. Show that the equation can be written in the form \(\ln y = a + b \ln x\).
  3. In the Printed Answer Booklet, complete the table for \(\ln x\) and \(\ln y\) correct to 4 significant figures.
  4. Use the values from part (iii) to find \(a\) and \(b\).
  5. Hence rewrite your equation from part (i) for \(y\) in terms of \(x\), using suitable numerical values for the constants.
  6. Sketch a graph of the equation found in part (v).
  7. Earth is 1 astronomical unit from the sun. Find the intensity of the sun's radiation for Earth.
OCR MEI AS Paper 1 2019 June Q3
3 Given that \(k\) is an integer, express \(\frac { 3 \sqrt { 2 } - k } { \sqrt { 8 } + 1 }\) in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are rational expressions in terms of \(k\).
OCR MEI AS Paper 1 2019 June Q4
4 A triangle ABC has sides \(\mathrm { AB } = 5 \mathrm {~cm} , \mathrm { AC } = 9 \mathrm {~cm}\) and \(\mathrm { BC } = 10 \mathrm {~cm}\).
  1. Find the cosine of angle BAC, giving your answer as a fraction in its lowest terms.
  2. Find the exact area of the triangle.
OCR MEI AS Paper 1 2019 June Q5
5 In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertically upwards respectively. A particle has mass 2.5 kg .
  1. Write the weight of the particle as a vector. The particle moves under the action of its weight and two external forces ( \(3 \mathbf { i } - 2 \mathbf { j }\) ) N and \(( - \mathbf { i } + 18 \mathbf { j } ) N\).
  2. Find the acceleration of the particle, giving your answer in vector form.
OCR MEI AS Paper 1 2019 June Q6
6 Fig. 6 shows a train consisting of an engine of mass 80 tonnes pulling two trucks each of mass 25 tonnes. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0b1c272a-f0f4-4931-be89-5d045804a7af-4_189_1262_938_246} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} The engine exerts a driving force of \(D \mathrm {~N}\) and experiences a resistance to motion of 2000 N . Each truck experiences a resistance of 600 N . The train travels in a straight line on a level track with an acceleration of \(0.1 \mathrm {~ms} ^ { - 2 }\).
  1. Complete the force diagram in the Printed Answer Booklet to show all the forces acting on the engine and each of the trucks.
  2. Calculate the value of \(D\).
  3. The tension in the coupling between the engine and truck A is larger than that in the coupling between the trucks. Determine how much larger.
OCR MEI AS Paper 1 2019 June Q7
7 In this question you must show detailed reasoning.
  1. Nigel is asked to determine whether \(( x + 7 )\) is a factor of \(x ^ { 3 } - 37 x + 84\). He substitutes \(x = 7\) and calculates \(7 ^ { 3 } - 37 \times 7 + 84\). This comes to 168 , so Nigel concludes that ( \(x + 7\) ) is not a factor. Nigel's conclusion is wrong.
    • Explain why Nigel's argument is not valid.
    • Show that \(( x + 7 )\) is a factor of \(x ^ { 3 } - 37 x + 84\).
    • Sketch the graph of \(y = x ^ { 3 } - 37 x + 84\), indicating the coordinates of the points at which the curve crosses the coordinate axes.
    • The graph in part (b) is translated by \(\binom { 1 } { 0 }\). Find the equation of the translated graph, giving your answer in the form \(y = x ^ { 3 } + a x ^ { 2 } + b x + c\) where \(a , b\) and \(c\) are integers.
OCR MEI AS Paper 1 2019 June Q10
10 In this question you must show detailed reasoning.
  1. Sketch the gradient function for the curve \(y = 24 x - 3 x ^ { 2 } - x ^ { 3 }\).
  2. Determine the set of values of \(x\) for which \(24 x - 3 x ^ { 2 } - x ^ { 3 }\) is decreasing.
OCR MEI AS Paper 1 2019 June Q11
11 David puts a block of ice into a cool-box. He wishes to model the mass \(m \mathrm {~kg}\) of the remaining block of ice at time \(t\) hours later. He finds that when \(t = 5 , m = 2.1\), and when \(t = 50 , m = 0.21\).
  1. David at first guesses that the mass may be inversely proportional to time. Show that this model fits his measurements.
  2. Explain why this model
    1. is not suitable for small values of \(t\),
    2. cannot be used to find the time for the block to melt completely. David instead proposes a linear model \(m = a t + b\), where \(a\) and \(b\) are constants.
  3. Find the values of the constants for which the model fits the mass of the block when \(t = 5\) and \(t = 50\).
  4. Interpret these values of \(a\) and \(b\).
  5. Find the time according to this model for the block of ice to melt completely.
OCR MEI AS Paper 1 2022 June Q1
1 Rationalise the denominator of the fraction \(\frac { 2 + \sqrt { n } } { 3 + \sqrt { n } }\), where \(n\) is a positive integer.
OCR MEI AS Paper 1 2022 June Q2
2
  1. Determine the value of \(\frac { 100 ! } { 98 ! }\).
  2. Find the coefficient of \(x ^ { 98 }\) in the expansion of \(( 1 + x ) ^ { 100 }\).
OCR MEI AS Paper 1 2022 June Q3
3 The velocity-time graph for the motion of a particle is shown below. The velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t \mathrm {~s}\) is given by \(\mathrm { v } = - \mathrm { t } ^ { 2 } + 6 \mathrm { t } - 6\) where \(0 \leqslant t \leqslant 5\).
\includegraphics[max width=\textwidth, alt={}, center]{7af62e61-c67f-4d05-b6b9-c1a110345812-3_860_979_1082_239}
  1. Find the times at which the velocity is \(2 \mathrm {~ms} ^ { - 1 }\).
  2. Write down the greatest speed of the particle.
OCR MEI AS Paper 1 2022 June Q4
4 The quadratic function \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = x ^ { 2 } - 3 x + 2\).
  1. Write \(\mathrm { f } ( x )\) in the form \(( \mathrm { x } + \mathrm { a } ) ^ { 2 } + \mathrm { b }\), where \(a\) and \(b\) are constants.
  2. Write down the coordinates of the minimum point on the graph of \(y = f ( x )\).
  3. Describe fully the transformation that maps the graph of \(y = f ( x )\) onto the graph of \(y = ( x + 1 ) ^ { 2 } - \frac { 1 } { 4 }\).
OCR MEI AS Paper 1 2022 June Q5
5 Part of the graph of \(y = f ( x )\) is shown below. The graph is the image of \(y = \tan x ^ { \circ }\) after a stretch in the \(x\)-direction.
\includegraphics[max width=\textwidth, alt={}, center]{7af62e61-c67f-4d05-b6b9-c1a110345812-4_791_1022_1014_244}
  1. Find the equation of the graph.
  2. Write down the period of the function \(\mathrm { f } ( x )\).
  3. In this question you must show detailed reasoning. Find all the roots of the equation \(\mathrm { f } ( x ) = 1\) for \(0 ^ { \circ } \leqslant x ^ { \circ } \leqslant 360 ^ { \circ }\).
OCR MEI AS Paper 1 2022 June Q6
6 The gradient of a curve is given by the equation \(\frac { d y } { d x } = 6 x ^ { 2 } - 20 x + 6\). The curve passes through the point \(( 2,6 )\).
  1. Find the equation of the curve.
  2. Verify that the equation of the curve can be written as \(y = 2 ( x + 1 ) ( x - 3 ) ^ { 2 }\).
  3. Sketch the curve, indicating the points where the curve meets the axes.
OCR MEI AS Paper 1 2022 June Q7
7 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. A canal narrowboat of mass 9 tonnes is pulled by two ropes. The tensions in the ropes are \(( 450 \mathbf { i } + 20 \mathbf { j } ) \mathbf { N }\) and \(( 420 \mathbf { i } - 20 \mathbf { j } ) \mathbf { N }\). The boat experiences a resistance to motion \(\mathbf { R }\) of magnitude 300 N .
  1. Explain what it means to model the boat as a particle. The boat is travelling in a straight line due east.
  2. Find the equation of motion of the boat.
  3. Find the acceleration of the boat giving your answer as a vector.
OCR MEI AS Paper 1 2022 June Q8
8 A team of volunteers donates cakes for sale at a charity stall. The number of cakes that can be sold depends on the price. A model for this is \(\mathrm { y } = 190 - 70 \mathrm { x }\), where \(y\) cakes can be sold when the price of a cake is \(\pounds\) x.
  1. Find how many cakes could be given away for free according to this model. The number of volunteers who are willing to donate cakes goes up as the price goes up. If the cakes sell for \(\pounds 1.20\) they will donate 50 cakes, but if they sell for \(\pounds 2.40\) they will donate 140 cakes. They use the linear model \(\mathrm { y } = \mathrm { mx } + \mathrm { c }\) to relate the number of cakes donated, \(y\), to the price of a cake, \(\pounds x\).
  2. Find the values of the constants \(m\) and \(c\) for which this linear model fits the two data points.
  3. Explain why the model is not suitable for very low prices.
  4. The team would like to sell all the cakes that they donate. Find the set of possible prices that the cakes could have to achieve this.
OCR MEI AS Paper 1 2022 June Q9
9 A tractor of mass 1800 kg uses a towbar to pull a trailer of mass 1000 kg on a level field. The tractor and trailer experience resistances to motion of 1600 N and 800 N respectively. The tractor provides a driving force of 6600 N .
  1. Draw a force diagram showing all the horizontal forces acting on the tractor and trailer.
  2. Find the tension in the towbar.
OCR MEI AS Paper 1 2022 June Q10
10 A triangle has vertices \(A ( 1,4 ) , B ( 7,0 )\) and \(C ( - 4 , - 1 )\).
  1. Show that the equation of the line AC is \(\mathrm { y } = \mathrm { x } + 3\). M is the midpoint of AB . The line AC intersects the \(x\)-axis at D .
  2. Determine the angle DMA.
OCR MEI AS Paper 1 2022 June Q11
11 A sports car accelerates along a straight road from rest. After 5 s its velocity is \(9 \mathrm {~ms} ^ { - 1 }\). In model A, the acceleration is assumed to be constant.
  1. Calculate the distance travelled by the car in the first 5 seconds according to model A . In model B , the velocity \(v\) in \(\mathrm { ms } ^ { - 1 }\) is given by \(\mathrm { v } = 0.05 \mathrm { t } ^ { 3 } + \mathrm { kt }\), where \(t\) is the time in seconds after the start and \(k\) is a constant.
  2. Find the value of \(k\) which gives the correct value of \(v\) when \(t = 5\).
  3. Using this value of \(k\) in model B , calculate the acceleration of the car when \(t = 5\). The car travels 16 m in the first 5 seconds.
  4. Show that model B, with the value of \(k\) found in part (b), better fits this information than model A does.
OCR MEI AS Paper 1 2022 June Q12
12 Below is a faulty argument that appears to show that the gradient of the curve \(y = x ^ { 2 }\) at the point \(( 3,9 )\) is 1 . Consider the chord joining \(( 3,9 )\) to the point \(\left( 3 + h , ( 3 + h ) ^ { 2 } \right)\)
The gradient is \(\frac { ( 3 + h ) ^ { 2 } - 9 } { h } = \frac { 6 h + h ^ { 2 } } { h }\)
When \(h = 0\) the gradient is \(\frac { 0 } { 0 }\) so the gradient of the curve is 1
  1. Identify a fault in the argument.
  2. Write a valid first principles argument leading to the correct value for the gradient at (3, 9).
  3. Find the equation of the normal to the curve at the point ( 3,9 ).
OCR MEI AS Paper 1 2023 June Q1
1 A particle moves along a straight line. Its velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t\) s is given by \(\mathbf { v } = 2 \mathbf { t } + 0.6 \mathbf { t } ^ { 2 }\).
Find an expression for the acceleration of the particle at time \(t\).
OCR MEI AS Paper 1 2023 June Q2
2 The height of the first part of a rollercoaster track is \(h \mathrm {~m}\) at a horizontal distance of \(x \mathrm {~m}\) from the start. A student models this using the equation \(h = 17 + 15 \cos 6 x\), for \(0 \leqslant x \leqslant 40\), using the values of \(h\) given when their calculator is set to work in degrees.
  1. Find the height that the student's model predicts when the horizontal distance from the start is 40 m .
  2. The student argues that the model predicts that the rollercoaster track will achieve a maximum height of 32 m more than once because the cosine function is periodic. Comment on the validity of the student's argument.
OCR MEI AS Paper 1 2023 June Q3
3 The points \(A\) and \(B\) have position vectors \(\binom { 2 } { - 1 }\) and \(\binom { 5 } { 4 }\) respectively. The vector \(\overrightarrow { \mathrm { AC } }\) is \(\binom { - 2 } { 2 }\).
  1. Write down the position vector of C as a column vector.
  2. Show that B is equidistant from A and C .
OCR MEI AS Paper 1 2023 June Q4
4 In this question you must show detailed reasoning.
Solve the equation \(6 \cos ^ { 2 } x + \sin x = 5\), giving all the roots in the interval \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
OCR MEI AS Paper 1 2023 June Q5
5 The graph shows displacement \(s m\) against time \(t \mathrm {~s}\) for a model of the motion of a bead moving along a straight wire. The points \(( 0,4 ) , ( 2,7 ) , ( 5,7 )\) and \(( 9 , - 7 )\) are the endpoints of the line segments.
\includegraphics[max width=\textwidth, alt={}, center]{1d1e41f3-a834-4230-b6e1-4b0be9450d30-4_741_1301_404_239}
  1. Find an expression for the displacement of the bead for \(0 \leqslant t \leqslant 2\).
  2. Sketch the velocity-time graph for this model.
  3. Explain why the model may not be suitable at \(t = 2\) and \(t = 5\).