Questions - OCR MEI - A-Level Maths

OCR MEI Further Pure with Technology Specimen Q3

20 marks Challenging +1.2

3 This question explores the family of differential equations $\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { 1 + a x + 2 y }$ for various values of the parameter $a$. Fig. 3 shows the tangent field in the case $a = 1$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{141c85ec-5749-4f24-9f6d-fe7a01567511-4_691_696_452_696} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure}

(A) Sketch the tangent field in the case $a = - 2$.
(B) Explain why the tangent field is not defined for the whole coordinate plane.
(C) Give an inequality which describes the region in which the tangent field is defined.
(D) Find a value of $a$ such that the region for which the tangent field is defined includes the entire $x$-axis.
(A) For the case $a = 1$, with $y = 1$ when $x = 0$, construct a spreadsheet for the Runge-Kutta method of order 2 with formulae as follows, where $\mathrm { f } ( x , y ) = \frac { \mathrm { d } y } { \mathrm {~d} x }$. $$\begin{aligned} k _ { 1 } & = h \mathrm { f } \left( x _ { n } , y _ { n } \right) \\ k _ { 2 } & = h \mathrm { f } \left( x _ { n } + h , y _ { n } + k _ { 1 } \right) \\ y _ { n + 1 } & = y _ { n } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right) \end{aligned}$$ State the formulae you have used in your spreadsheet.
(B) Use your spreadsheet to obtain the value of $y$ correct to 4 decimal places when $x = 1$ for
(A) For the case $a = 0$ find the analytical solution that passes through the point ( 0,1 ).
(B) Verify that the solution in part (iii) (A) is a solution to the differential equation.
(C) Use the solution in part (iii) (A) to find the value of $y$ correct to 4 decimal places when $x = 1$.
(A) Verify that $y = - \frac { a } { 2 } x + \frac { a ^ { 2 } } { 8 } - \frac { 1 } { 2 }$ is a solution for all cases when $a \leq 0$.
(B) Show that this is the only straight line solution in these cases. \section*{Copyright Information:} }{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the

OCR MEI FP2 2009 January Q2

18 marks Standard +0.3

Write down the modulus and argument of the complex number $\mathrm { e } ^ { \mathrm { j } \pi / 3 }$.
The triangle OAB in an Argand diagram is equilateral. O is the origin; A corresponds to the complex number $a = \sqrt { 2 } ( 1 + \mathrm { j } ) ; \mathrm { B }$ corresponds to the complex number $b$. Show A and the two possible positions for B in a sketch. Express $a$ in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$. Find the two possibilities for $b$ in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$.
Given that $z _ { 1 } = \sqrt { 2 } \mathrm { e } ^ { \mathrm { j } \pi / 3 }$, show that $z _ { 1 } ^ { 6 } = 8$. Write down, in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$, the other five complex numbers $z$ such that $z ^ { 6 } = 8$. Sketch all six complex numbers in a new Argand diagram. Let $w = z _ { 1 } \mathrm { e } ^ { - \mathrm { j } \pi / 12 }$.
Find $w$ in the form $x + \mathrm { j } y$, and mark this complex number on your Argand diagram.
Find $w ^ { 6 }$, expressing your answer in as simple a form as possible.

OCR MEI FP2 2013 January Q2

18 marks Challenging +1.3

1. Show that $$1 + \mathrm { e } ^ { \mathrm { j } 2 \theta } = 2 \cos \theta ( \cos \theta + \mathrm { j } \sin \theta )$$
2. The series $C$ and $S$ are defined as follows. $$\begin{aligned} & C = 1 + \binom { n } { 1 } \cos 2 \theta + \binom { n } { 2 } \cos 4 \theta + \ldots + \cos 2 n \theta \\ & S = \binom { n } { 1 } \sin 2 \theta + \binom { n } { 2 } \sin 4 \theta + \ldots + \sin 2 n \theta \end{aligned}$$ By considering $C + \mathrm { j } S$, show that $$C = 2 ^ { n } \cos ^ { n } \theta \cos n \theta$$ and find a corresponding expression for $S$.
1. Express $\mathrm { e } ^ { \mathrm { j } 2 \pi / 3 }$ in the form $x + \mathrm { j } y$, where the real numbers $x$ and $y$ should be given exactly.
2. An equilateral triangle in the Argand diagram has its centre at the origin. One vertex of the triangle is at the point representing $2 + 4 \mathrm { j }$. Obtain the complex numbers representing the other two vertices, giving your answers in the form $x + \mathrm { j } y$, where the real numbers $x$ and $y$ should be given exactly.
3. Show that the length of a side of the triangle is $2 \sqrt { 15 }$.

OCR MEI C1 2009 January Q11

14 marks Moderate -0.3

Show that the equation of the circle with AB as diameter may be written as $$( x - 5 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 40$$
Find the coordinates of the points of intersection of this circle with the $y$-axis. Give your answer in the form $a \pm \sqrt { b }$.
Find the equation of the tangent to the circle at B . Hence find the coordinates of the points of intersection of this tangent with the axes.

OCR MEI C1 Q10

12 marks Moderate -0.3

Write down the equations of the circles A and B .
Find the $x$ coordinates of the points where the two curves intersect.
Find the $y$ coordinates of these points, giving your answers in surd form.

OCR MEI C2 2006 January Q11

11 marks Standard +0.3

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Find, in exact form, the range of values of $x$ for which $x ^ { 3 } - 6 x + 2$ is a decreasing function.
Find the equation of the tangent to the curve at the point $( - 1,7 )$. Find also the coordinates of the point where this tangent crosses the curve again.

OCR MEI C2 2011 January Q11

11 marks Moderate -0.3

Use calculus to find $\int _ { 1 } ^ { 3 } \left( x ^ { 3 } - 3 x ^ { 2 } - x + 3 \right) \mathrm { d } x$ and state what this represents.
Find the $x$-coordinates of the turning points of the curve $y = x ^ { 3 } - 3 x ^ { 2 } - x + 3$, giving your answers in surd form. Hence state the set of values of $x$ for which $y = x ^ { 3 } - 3 x ^ { 2 } - x + 3$ is a decreasing function.

OCR MEI C2 2009 June Q10

12 marks Moderate -0.5

On the insert, complete the table and plot $h$ against $\log _ { 10 } t$, drawing by eye a line of best fit.
Use your graph to find an equation for $h$ in terms of $\log _ { 10 } t$ for this model.
Find the height of the tree at age 100 years, as predicted by this model.
Find the age of the tree when it reaches a height of 29 m , according to this model.
Comment on the suitability of the model when the tree is very young.

OCR MEI C2 Q11

12 marks Moderate -0.8

The speed-time graph on the insert sheet provides the axes and the first two points plotted. Plot the remainder of these points and join them with a smooth curve. The area between this curve and the $t$-axis represents the distance travelled by the car in this time.
Using the trapezium rule with 6 values of $t$ estimate the area under the curve to give the distance travelled. Illustrate on your graph the area found.
John's teacher suggests that the equation of the curve could be $v = 6 t - \frac { 1 } { 2 } t ^ { 2 }$. Find, by calculus, the area between this curve and the $t$ axis.
Plot this curve on your graph. Comment on whether the estimates obtained in parts (ii) and (iii) are overestimates or underestimates. 12 Fig. 12 shows a window. The base and sides are parts of a rectangle with dimensions $2 x$ metres horizontally by $y$ metres vertically. The top is a semicircle of radius $x$ metres. The perimeter of the window is 10 metres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{73d1c02b-1b7b-426d-a171-c762597cfed4-4_428_433_1638_766} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure}

Express $y$ as a function of $x$.
Find the total area, $A \mathrm {~m} ^ { 2 }$, in terms of $x$ and $y$. Use your answer to part (i) to show that this simplifies to $$A = 10 x - 2 x ^ { 2 } - \frac { 1 } { 2 } \pi x ^ { 2 }$$
Prove that for the maximum value of $A$, $y = x$ exactly.
\section*{MEI STRUCTURED MATHEMATICS } \section*{CONCEPTS FOR ADVANCED MATHEMATICS, C2} \section*{Practice Paper C2-B
Insert sheet for question 11}

OCR MEI C2 Q3

12 marks Standard +0.3

Express $\mathrm { f } ( x )$ in factorised form.
Show that the equation of the curve may be written as $y = x ^ { 3 } + 5 x ^ { 2 } - 4 x - 20$.
Use calculus to show that, correct to 1 decimal place, the $x$-coordinate of the minimum point on the curve is 0.4 . Find also the coordinates of the maximum point on the curve, giving your answers correct to 1 decimal place.
State, correct to 1 decimal place, the coordinates of the maximum point on the curve $y = \mathrm { f } ( 2 x )$.

OCR MEI C2 Q12

4 marks Easy -1.2

$y = \mathrm { f } ( x - 2 )$,
$y = 3 \mathrm { f } ( x )$.

OCR MEI C2 Q6

5 marks Moderate -0.8

On the copy of Fig. 5, draw by eye a tangent to the curve at the point where $x = 2$. Hence find an estimate of the gradient of $y = 2 ^ { x }$ when $x = 2$.
Calculate the $y$-values on the curve when $x = 1.8$ and $x = 2.2$. Hence calculate another approximation to the gradient of $y = 2 ^ { x }$ when $x = 2$.

OCR MEI C2 Q11

5 marks Moderate -0.8

Solve the equation $\cos x = 0.4$ for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.
Describe the transformation which maps the graph of $y = \cos x$ onto the graph of $y = \cos 2 x$.

OCR MEI C3 Q3

4 marks Easy -1.2

$\quad y = 2 \mathrm { f } ( x )$,
$y = \mathrm { f } ( 2 x )$.

OCR MEI C4 2010 June Q5

8 marks Standard +0.3

Verify that $\overrightarrow { \mathrm { AB } } = \left( \begin{array} { l } 300 \\ 100 \\ 100 \end{array} \right)$ and find the length of the pipeline.
Write down a vector equation of the line AB , and calculate the angle it makes with the vertical. A thin flat layer of hard rock runs through the mountain. The equation of the plane containing this layer is $x + 2 y + 3 z = 320$.
Find the coordinates of the point where the pipeline meets the layer of rock.
By calculating the angle between the line AB and the normal to the plane of the layer, find the angle at which the pipeline cuts through the layer. 8 Part of the track of a roller-coaster is modelled by a curve with the parametric equations $$x = 2 \theta - \sin \theta , \quad y = 4 \cos \theta \quad \text { for } 0 \leqslant \theta \leqslant 2 \pi$$ This is shown in Fig. 8. B is a minimum point, and BC is vertical. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c149cb5-7392-4219-b285-486f4694aa6f-4_602_1447_488_351} \caption{Fig. 8}
\end{figure}

Find the values of the parameter at A and B . Hence show that the ratio of the lengths OA and AC is $( \pi - 1 ) : ( \pi + 1 )$.
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $\theta$. Find the gradient of the track at A .
Show that, when the gradient of the track is $1 , \theta$ satisfies the equation $$\cos \theta - 4 \sin \theta = 2 .$$
Express $\cos \theta - 4 \sin \theta$ in the form $R \cos ( \theta + \alpha )$. Hence solve the equation $\cos \theta - 4 \sin \theta = 2$ for $0 \leqslant \theta \leqslant 2 \pi$. {www.ocr.org.uk}) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the \section*{ADVANCED GCE
MATHEMATICS (MEI)} 4754B
Applications of Advanced Mathematics (C4) Paper B: Comprehension \section*{Candidates answer on the Question Paper} OCR Supplied Materials:
\section*{Other Materials Required:}
Wednesday 9 June 2010 Afternoon \includegraphics[max width=\textwidth, alt={}, center]{5c149cb5-7392-4219-b285-486f4694aa6f-5_264_456_881_1361} 1 The train journey from Swansea to London is 307 km and that by road is 300 km . Carry out the calculations performed on the First Great Western website to estimate how much lower the carbon dioxide emissions are when travelling by rail rather than road.
2 The equation of the curve in Fig. 3 is $$y = \frac { 1 } { 10 ^ { 4 } } \left( x ^ { 3 } - 100 x ^ { 2 } - 10000 x + 2100100 \right)$$ Calculate the speed at which the car has its lowest carbon dioxide emissions and the value of its emissions at that speed.
[0pt] [An answer obtained from the graph will be given no marks.]

3

In line 109 the carbon dioxide emissions for a particular train journey from Exeter to London are estimated to be 3.7 tonnes. Obtain this figure.
The text then goes on to state that the emissions per extra passenger on this journey are less than $\frac { 1 } { 2 } \mathrm {~kg}$. Justify this figure.

$\_\_\_\_$
$\_\_\_\_$ 4 The daily number of trains, $n$, on a line in another country may be modelled by the function defined below, where $P$ is the annual number of passengers. $$\begin{aligned} & n = 10 \text { for } 0 \leqslant P < 10 ^ { 6 } \\ & n = 11 \text { for } 10 ^ { 6 } \leqslant P < 1.5 \times 10 ^ { 6 } \\ & n = 12 \text { for } 1.5 \times 10 ^ { 6 } \leqslant P < 2 \times 10 ^ { 6 } \\ & n = 13 \text { for } 2 \times 10 ^ { 6 } \leqslant P < 2.5 \times 10 ^ { 6 } \\ & n = 14 \text { for } 2.5 \times 10 ^ { 6 } \leqslant P < 3 \times 10 ^ { 6 } \\ & \ldots \text { and so on } \ldots \end{aligned}$$

Sketch the graph of $n$ against $P$.
Describe, in words, the relationship between the daily number of trains and the annual number of passengers.

\includegraphics[max width=\textwidth, alt={}, center]{5c149cb5-7392-4219-b285-486f4694aa6f-7_716_1249_1011_440}
$\_\_\_\_$

OCR MEI S1 2005 June Q5

6 marks Moderate -0.8

On the insert, complete the table giving the lowest common multiples of all pairs of integers between 1 and 6 .
[0pt] [1]
\multirow{2}{*}{} Second integer
1 2 3 4 5 6
\multirow{6}{*}{First integer} 1 1 2 3 4 5 6
2 2 2 6 4 10 6
3 3 6 3 12 15 6
4 4 4 12 12
5 5 10 15
6 6 6 6 12
Two fair dice are thrown and the lowest common multiple of the two scores is found.
Use the table to find the probabilities of the following events.
(A) The lowest common multiple is greater than 6 .
(B) The lowest common multiple is a multiple of 5 .
(C) The lowest common multiple is both greater than 6 and a multiple of 5 .
Use your answers to part (ii) to show that the events "the lowest common multiple is greater than 6 " and "the lowest common multiple is a multiple of 5 " are not independent.

OCR MEI S1 Q3

6 marks Easy -1.8

On die insert, complete the lable giving due lowest common multiples of all pairs of integers between 1 and 6 .

Second integer

\cline { 2 - 8 } \multicolumn{2}{|c|}{}

1

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6

\multirow{5}{*}{

First

integer

}

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5

6

\cline { 2 - 8 }

2

6

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6

\cline { 2 - 8 }

3

6

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12

15

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\cline { 2 - 8 }

4

12

\cline { 2 - 8 }

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\cline { 2 - 8 }

6

12

Two fair dice are thrown and the lowest common multiple of the two scores is found.

Use the table to find the probabilities of the following events.
(A) The lowest common multiple is greater than 6 .
(B) The lowest common multiple is a multiple of 5 .
(C) The lowest common multiple is both greater than 6 and a multiple of 5.
Use your answers to part (ii) to show that the events "the lowest common multiple is greater than 6 " and "the lowest common multiple is a multiple of 5 " are not independent.

OCR MEI M1 2008 June Q7

17 marks Moderate -0.3

What information in the question indicates that the tension in the string section CB is also 60 N ?
Show that the string sections AC and CB are equally inclined to the horizontal (so that $\alpha = \beta$ in Fig. 7.1).
Calculate the angle of the string sections AC and CB to the horizontal. In a different situation the same box is supported by two separate light strings, PC and QC, that are tied to the box at C . There is also a horizontal force of 10 N acting at C . This force and the angles between these strings and the horizontal are shown in Fig. 7.2. The box is in equilibrium. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{170edb27-324e-44df-8dc1-7d8fbad680fe-4_323_503_1649_822} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
\end{figure}
Calculate the tensions in the two strings.

OCR MEI M1 2013 June Q7

18 marks Moderate -0.3

Represent the forces acting on the object as a fully labelled triangle of forces.
Find $F$ and $\theta$. Show that the distance between the object and the vertical section of rope A is 3 m . Abi then pulls harder and the object moves upwards. Bob adjusts the tension in rope B so that the object moves along a vertical line. Fig. 7.2 shows the situation when the object is part of the way up. The tension in rope A is $S \mathrm {~N}$ and the tension in rope B is $T \mathrm {~N}$. The ropes make angles $\alpha$ and $\beta$ with the vertical as shown in the diagram. Abi and Bob are taking a rest and holding the object stationary and in equilibrium. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{83e69140-4abf-4713-85da-922ce7530e47-5_383_360_534_854} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
\end{figure}
Give the equations, involving $S , T , \alpha$ and $\beta$, for equilibrium in the vertical and horizontal directions.
Find the values of $S$ and $T$ when $\alpha = 8.5 ^ { \circ }$ and $\beta = 35 ^ { \circ }$.
Abi's mass is 40 kg . Explain why it is not possible for her to raise the object to a position in which $\alpha = 60 ^ { \circ }$.

OCR MEI M2 2010 June Q2

18 marks Standard +0.3

Calculate the coordinates of the centre of mass of the stand. A small object of mass 5 kg is fixed to the rod AB at a distance of 40 cm from A .
Show that the coordinates of the centre of mass of the stand with the object are ( 22,68 ). The stand is tilted about the edge PQ until it is on the point of toppling. The angle through which the stand is tilted is called 'the angle of tilt'. This procedure is repeated about the edges QR and RS.
Making your method clear, determine which edge requires the smallest angle of tilt for the stand to topple. The small object is removed. A light string is attached to the stand at A and pulled at an angle of $50 ^ { \circ }$ to the downward vertical in the plane $\mathrm { O } x y$ in an attempt to tip the stand about the edge RS.
Assuming that the stand does not slide, find the tension in the string when the stand is about to turn about the edge RS.

OCR MEI M2 2016 June Q3

18 marks Standard +0.3

Use an energy method to find the magnitude of the frictional force acting on the block. Calculate the coefficient of friction between the block and the plane.
Calculate the power of the tension in the string when the block has a speed of $7 \mathrm {~ms} ^ { - 1 }$. Fig. 3.1 shows a thin planar uniform rigid rectangular sheet of metal, OPQR, of width 1.65 m and height 1.2 m . The mass of the sheet is $M \mathrm {~kg}$. The sides OP and PQ have thin rigid uniform reinforcements attached with masses $0.6 M \mathrm {~kg}$ and $0.4 M \mathrm {~kg}$, respectively. Fig. 3.1 also shows coordinate axes with origin at O . The sheet with its reinforcements is to be used as an inn sign.

Calculate the coordinates of the centre of mass of the inn sign. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_421_492_210_1334} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{figure} The inn sign has a weight of 300 N . It hangs in equilibrium with QR horizontal when vertical forces $Y _ { \mathrm { Q } } \mathrm { N }$ and $Y _ { \mathrm { R } } \mathrm { N }$ act at Q and R respectively.
Calculate the value of $Y _ { \mathrm { Q } }$ and show that $Y _ { \mathrm { R } } = 120$. The inn sign is hung from a framework, ABCD , by means of two light vertical inextensible wires attached to the sign at Q and R and the framework at B and C , as shown in Fig. 3.2. QR and BC are horizontal. The framework is made from light rigid rods $\mathrm { AB } , \mathrm { BC } , \mathrm { CA }$ and CD freely pin-jointed together at $\mathrm { A } , \mathrm { B }$ and C and to a vertical wall at A and D . Fig. 3.3 shows the dimensions of the framework in metres as well as the external forces $X _ { \mathrm { A } } \mathrm { N } , Y _ { \mathrm { A } } \mathrm { N }$ acting at A and $X _ { \mathrm { D } } \mathrm { N } , Y _ { \mathrm { D } } \mathrm { N }$ acting at D . You are given that $\sin \alpha = \frac { 5 } { 13 } , \cos \alpha = \frac { 12 } { 13 } , \sin \beta = \frac { 4 } { 5 }$ and $\cos \beta = \frac { 3 } { 5 }$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_543_526_1420_253} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_629_793_1343_964} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
\end{figure}
Mark on the diagram in your Printed Answer Book all the forces acting on the pin-joints at $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D , including those internal to the rods, when the inn sign is hanging from the framework.
Show that $X _ { \mathrm { D } } = 261$.
Calculate the forces internal to the rods $\mathrm { AB } , \mathrm { BC }$ and CD , stating whether each rod is in tension or thrust (compression). Calculate also the values of $Y _ { \mathrm { D } }$ and $Y _ { \mathrm { A } }$. [Your working in this part should correspond to your diagram in part (iii).]

OCR MEI D1 2006 January Q2

8 marks Easy -1.2

Complete the table in the insert showing the outcome of applying the algorithm to the following two lists: $$\begin{array} { l r l l l l l } \text { List 1: } & 2 , & 34 , & 35 , & 56 & & \\ \text { List 2: } & 13 , & 22 , & 34 , & 81 , & 90 , & 92 \end{array}$$
What does the algorithm achieve?
How many comparisons did you make in applying the algorithm?
If the number of elements in List 1 is $x$, and the number of elements in List 2 is $y$, what is the maximum number of comparisons that will have to be made in applying the algorithm, and what is the minimum number?

OCR MEI D2 2014 June Q2

16 marks Easy -1.2

Rachel thinks that the answer given in the newspaper article is not sensible. Give a verbal argument why Rachel might think that the batsman should be given out. Rachel tries to formalise her argument. She defines four simple propositions.
o: "The batsman is given out."
lb: "The batsman is given out (LBW)."
c: "The batsman is given out (caught)."
b: "The ball hit the bat."
An implication of the batsman not being out (LBW) is that the ball has hit the bat. Write this down in terms of Rachel's propositions.
Similarly, write down the implication of the batsman not being out (caught).
Using your answers to parts (ii) and (iii) write down the implication of a batsman being not out, in terms of $b$ and $\sim b$.
[0pt] [You may assume that if $\mathrm { w } \Rightarrow \mathrm { y }$ and $\mathrm { x } \Rightarrow \mathrm { z }$, then $( \mathrm { w } \wedge \mathrm { x } ) \Rightarrow ( \mathrm { y } \wedge \mathrm { z } )$. ]
By writing down the contrapositive of your implication from part (iv), produce an implication which supports Rachel's argument.
(b) A classroom rule has been broken by either Anja, Bobby, Catherine or Dimitria, or by a subset of those four. The teacher knows that Dimitria could not have done it on her own. Let $a$ be the proposition "Anja is guilty", and similarly for $b , c$ and $d$.
Express the teacher's knowledge as a compound proposition. Evidence emerges that Bobby and Catherine were elsewhere at the time, so they cannot be guilty. This can be expressed as the compound proposition $\sim ( b \vee c )$.
Construct a truth table to show the truth values of the compound proposition given by the conjunction of the two compound propositions, one from part (i) and one given above.
What does your truth table tell you about who is guilty? 3 Three products, A, B and C are to be made.
Three supplements are included in each product. Product A has 10 g per kg of supplement $\mathrm { X } , 5 \mathrm {~g}$ per kg of supplement Y and 5 g per kg of supplement Z . Product B has 5 g per kg of supplement $\mathrm { X } , 5 \mathrm {~g}$ per kg of supplement Y and 3 g per kg of supplement Z .
Product C has 12 g per kg of supplement $\mathrm { X } , 7 \mathrm {~g}$ per kg of supplement Y and 5 g per kg of supplement Z .
There are 12 kg of supplement X available, 12 kg of supplement Y , and 9 kg of supplement Z .
Product A will sell at $\pounds 7$ per kg and costs $\pounds 3$ per kg to produce. Product B will sell at $\pounds 5$ per kg and costs $\pounds 2$ per kg to produce. Product C will sell at $\pounds 4$ per kg and costs $\pounds 3$ per kg to produce. The profit is to be maximised.
Explain how the initial feasible tableau shown in Fig. 3 models this problem. \begin{table}[h]
1(v)
1(vi)
1

2(a)(i)
\end{table}

OCR MEI AS Paper 1 2019 June Q8

7 marks Moderate -0.8

The model gives the correct velocity of $25.6 \mathrm {~ms} ^ { - 1 }$ at time 8 s . Show that $k = 0.1$. A second model for the motion uses constant acceleration.
Find the value of the acceleration which gives the correct velocity of $25.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at time 8 s .
Show that these two models give the same value for the displacement in the first 8 s .

OCR MEI AS Paper 2 2024 June Q11

6 marks Moderate -0.8

Verify that the curve cuts the $x$-axis at $x = 4$ and at $x = 9$. The curve does not cut or touch the $x$-axis at any other points.
Determine the exact area bounded by the curve and the $x$-axis.

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OCR MEI M1 2013 June Q7

OCR MEI M2 2010 June Q2

OCR MEI M2 2016 June Q3

OCR MEI D1 2006 January Q2

OCR MEI D2 2014 June Q2

OCR MEI AS Paper 1 2019 June Q8

OCR MEI AS Paper 2 2024 June Q11

\multirow{2}{*}{}		Second integer
		1	2	3	4	5	6
\multirow{6}{*}{First integer}	1	1	2	3	4	5	6
	2	2	2	6	4	10	6
	3	3	6	3	12	15	6
	4	4	4	12			12
	5	5	10	15
	6	6	6	6	12

1(v)
1(vi)





1

2(a)(i)