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OCR MEI Paper 3 2018 June Q10
10 marks Challenging +1.2
10 Point A has position vector \(\left( \begin{array} { l } a \\ b \\ 0 \end{array} \right)\) where \(a\) and \(b\) can vary, point B has position vector \(\left( \begin{array} { l } 4 \\ 2 \\ 0 \end{array} \right)\) and point C has position vector \(\left( \begin{array} { l } 2 \\ 4 \\ 2 \end{array} \right)\). ABC is an isosceles triangle with \(\mathrm { AC } = \mathrm { AB }\).
  1. Show that \(a - b + 1 = 0\).
  2. Determine the position vector of A such that triangle ABC has minimum area. Answer all the questions.
    Section B (15 marks) The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.
OCR MEI Paper 3 2018 June Q11
2 marks Moderate -0.8
11 Line 8 states that \(\frac { a + b } { 2 } \geqslant \sqrt { a b }\) for \(a\), \(b \geqslant 0\). Explain why the result cannot be extended to apply in each of the following cases.
  1. One of the numbers \(a\) and \(b\) is positive and the other is negative.
  2. Both numbers \(a\) and \(b\) are negative.
OCR MEI Paper 3 2018 June Q12
3 marks Standard +0.3
12 Lines 5 and 6 outline the stages in a proof that \(\frac { a + b } { 2 } \geqslant \sqrt { a b }\). Starting from \(( a - b ) ^ { 2 } \geqslant 0\), give a detailed proof of the inequality of arithmetic and geometric means.
OCR MEI Paper 3 2018 June Q13
3 marks Moderate -0.5
13 Consider a geometric sequence in which all the terms are positive real numbers. Show that, for any three consecutive terms of this sequence, the middle one is the geometric mean of the other two.
OCR MEI Paper 3 2018 June Q14
4 marks Standard +0.8
14
  1. In Fig. C1.3, angle CBD \(= \theta\). Show that angle CDA is also \(\theta\), as given in line 23 .
  2. Prove that \(h = \sqrt { a b }\), as given in line 24 .
OCR MEI Paper 3 2018 June Q15
3 marks Challenging +1.2
15 It is given in lines \(31 - 32\) that the square has the smallest perimeter of all rectangles with the same area. Using this fact, prove by contradiction that among rectangles of a given perimeter, \(4 L\), the square with side \(L\) has the largest area. \section*{END OF QUESTION PAPER}
OCR MEI Paper 3 2022 June Q1
2 marks Easy -1.8
1 A curve for which \(y\) is inversely proportional to \(x\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{c30a926b-d832-46f5-aa65-0066ef482c3d-4_824_1125_561_242} Find the equation of the curve.
OCR MEI Paper 3 2022 June Q2
6 marks Moderate -0.3
2 The function \(\mathrm { f } ( x ) = \sqrt { x }\) is defined on the domain \(x \geqslant 0\).
The function \(\mathrm { g } ( x ) = 25 - x ^ { 2 }\) is defined on the domain \(\mathbb { R }\).
  1. Write down an expression for \(\mathrm { fg } ( x )\).
    1. Find the domain of \(\mathrm { fg } ( x )\).
    2. Find the range of \(\mathrm { fg } ( x )\).
OCR MEI Paper 3 2022 June Q3
4 marks Moderate -0.3
3 An infinite sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by \(a _ { \mathrm { n } } = \frac { \mathrm { n } } { \mathrm { n } + 1 }\), for all positive integers \(n\).
  1. Find the limit of the sequence.
  2. Prove that this is an increasing sequence.
OCR MEI Paper 3 2022 June Q4
5 marks Standard +0.3
4 In this question you must show detailed reasoning.
Determine the exact solutions of the equation \(2 \cos ^ { 2 } x = 3 \sin x\) for \(0 \leqslant x \leqslant 2 \pi\).
OCR MEI Paper 3 2022 June Q5
7 marks Standard +0.8
5 A curve is defined implicitly by the equation \(2 x ^ { 2 } + 3 x y + y ^ { 2 } + 2 = 0\).
  1. Show that \(\frac { d y } { d x } = - \frac { 4 x + 3 y } { 3 x + 2 y }\).
  2. In this question you must show detailed reasoning. Find the coordinates of the stationary points of the curve.
OCR MEI Paper 3 2022 June Q6
8 marks Moderate -0.3
6 A hot drink is cooling. The temperature of the drink at time \(t\) minutes is \(T ^ { \circ } \mathrm { C }\).
The rate of decrease in temperature of the drink is proportional to \(( T - 20 )\).
  1. Write down a differential equation to describe the temperature of the drink as a function of time.
  2. When \(t = 0\), the temperature of the drink is \(90 ^ { \circ } \mathrm { C }\) and the temperature is decreasing at a rate of \(4.9 ^ { \circ } \mathrm { C }\) per minute. Determine how long it takes for the drink to cool from \(90 ^ { \circ } \mathrm { C }\) to \(40 ^ { \circ } \mathrm { C }\).
OCR MEI Paper 3 2022 June Q7
12 marks Standard +0.8
7 A student is trying to find the binomial expansion of \(\sqrt { 1 - x ^ { 3 } }\).
She gets the first three terms as \(1 - \frac { x ^ { 3 } } { 2 } + \frac { x ^ { 6 } } { 8 }\).
She draws the graphs of the curves \(y = \sqrt { 1 - x ^ { 3 } } , y = 1 - \frac { x ^ { 3 } } { 2 }\) and \(y = 1 - \frac { x ^ { 3 } } { 2 } + \frac { x ^ { 6 } } { 8 }\) using software. \includegraphics[max width=\textwidth, alt={}, center]{c30a926b-d832-46f5-aa65-0066ef482c3d-6_901_1265_516_248}
  1. Explain why \(1 - \frac { x ^ { 3 } } { 2 } + \frac { x ^ { 6 } } { 8 } \geqslant 1 - \frac { x ^ { 3 } } { 2 }\) for all values of \(x\).
  2. Explain why the graphs suggest that the student has made a mistake in the binomial expansion.
  3. Find the first four terms in the binomial expansion of \(\sqrt { 1 - x ^ { 3 } }\).
  4. State the set of values of \(x\) for which the binomial expansion in part (c) is valid.
  5. Sketch the curve \(y = 2.5 \sqrt { 1 - x ^ { 3 } }\) on the grid in the Printed Answer Booklet. \section*{(f) In this question you must show detailed reasoning.} The end of a bus shelter is modelled by the area between the curve \(\mathrm { y } = 2.5 \sqrt { 1 - x ^ { 3 } }\), the lines \(x = - 0.75 , x = 0.75\) and the \(x\)-axis. Lengths are in metres. Calculate, using your answer to part (c), an approximation for the area of the end of the bus shelter as given by this model.
OCR MEI Paper 3 2022 June Q8
16 marks Standard +0.8
8 The curves \(\mathrm { y } = \mathrm { h } ( \mathrm { x } )\) and \(\mathrm { y } = \mathrm { h } ^ { - 1 } ( \mathrm { x } )\), where \(\mathrm { h } ( x ) = x ^ { 3 } - 8\), are shown below.
The curve \(\mathrm { y } = \mathrm { h } ( \mathrm { x } )\) crosses the \(x\)-axis at B and the \(y\)-axis at A.
The curve \(\mathrm { y } = \mathrm { h } ^ { - 1 } ( \mathrm { x } )\) crosses the \(x\)-axis at D and the \(y\)-axis at C . \includegraphics[max width=\textwidth, alt={}, center]{c30a926b-d832-46f5-aa65-0066ef482c3d-7_826_819_520_255}
  1. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).
  2. Determine the coordinates of A, B, C and D.
  3. Determine the equation of the perpendicular bisector of AB . Give your answer in the form \(\mathrm { y } = \mathrm { mx } + c\), where \(m\) and \(c\) are constants to be determined.
  4. Points A , B , C and D lie on a circle. Determine the equation of the circle. Give your answer in the form \(( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }\), where \(a\), \(b\) and \(r ^ { 2 }\) are constants to be determined.
OCR MEI Paper 3 2022 June Q9
2 marks Easy -1.8
9 Show that \(\mathrm { y } = \mathrm { x }\) has the same gradient as \(\mathrm { y } = \sin \mathrm { x }\) when \(\mathrm { x } = 0\), as stated in line 5 .
OCR MEI Paper 3 2022 June Q11
3 marks Moderate -0.5
11 Show that, for the angle \(45 ^ { \circ }\), the formula \(\sin \theta \approx \frac { 4 \theta ( 180 - \theta ) } { 40500 - \theta ( 180 - \theta ) }\) given in line 28 gives the same approximation for the sine of the angle as the formula \(\sin x \approx \frac { 16 x ( \pi - x ) } { 5 \pi ^ { 2 } - 4 x ( \pi - x ) }\) given in line 23.
OCR MEI Paper 3 2022 June Q12
5 marks Standard +0.8
12
  1. Show that \(\cos x = \sin \left( x + \frac { \pi } { 2 } \right)\).
  2. Hence show that \(\sin x \approx \frac { 16 x ( \pi - x ) } { 5 \pi ^ { 2 } - 4 x ( \pi - x ) }\) gives the approximation \(\cos x \approx \frac { \pi ^ { 2 } - 4 x ^ { 2 } } { \pi ^ { 2 } + x ^ { 2 } }\), as stated in line 31. \section*{END OF QUESTION PAPER} }{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.
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OCR MEI Paper 3 2023 June Q1
3 marks Easy -1.2
1 In this question you must show detailed reasoning.
The obtuse angle \(\theta\) is such that \(\sin \theta = \frac { 2 } { \sqrt { 13 } }\).
Find the exact value of \(\cos \theta\).
OCR MEI Paper 3 2023 June Q2
4 marks Moderate -0.8
2 The straight line \(y = 5 - 2 x\) is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{20639e13-01cc-4d96-b694-fb3cf1828f4d-04_705_773_881_239}
  1. On the copy of the diagram in the Printed Answer Booklet, sketch the graph of \(y = | 5 - 2 x |\).
  2. Solve the inequality \(| 5 - 2 x | < 3\).
OCR MEI Paper 3 2023 June Q3
3 marks Moderate -0.3
3 In this question you must show detailed reasoning.
Find the value of \(k\) such that \(\frac { 1 } { \sqrt { 5 } + \sqrt { 6 } } + \frac { 1 } { \sqrt { 6 } + \sqrt { 7 } } = \frac { k } { \sqrt { 5 } + \sqrt { 7 } }\).
OCR MEI Paper 3 2023 June Q4
4 marks Moderate -0.8
4 In this question you must show detailed reasoning.
Find the coordinates of the points where the curve \(y = x ^ { 3 } - 2 x ^ { 2 } - 5 x + 6\) crosses the \(x\)-axis.
OCR MEI Paper 3 2023 June Q5
8 marks Moderate -0.8
5 In this question you must show detailed reasoning.
This question is about the curve \(y = x ^ { 3 } - 5 x ^ { 2 } + 6 x\).
  1. Find the equation of the tangent, \(T\), to the curve at the point ( 0,0 ).
  2. Find the equation of the normal, \(N\), to the curve at the point ( 1,2 ).
  3. Find the coordinates of the point of intersection of \(T\) and \(N\).
OCR MEI Paper 3 2023 June Q6
10 marks Standard +0.3
6
  1. Quadrilateral KLMN has vertices \(\mathrm { K } ( - 4,1 ) , \mathrm { L } ( 5 , - 1 ) , \mathrm { M } ( 6,2 )\) and \(\mathrm { N } ( 2,5 )\), as shown in Fig. 6.1. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 6.1} \includegraphics[alt={},max width=\textwidth]{20639e13-01cc-4d96-b694-fb3cf1828f4d-06_567_1004_404_319}
    \end{figure}
    1. Find the coordinates of the following midpoints.
      • P , the midpoint of KL
      • Q, the midpoint of LM
      • R, the midpoint of MN
      • S, the midpoint of NK
      • Verify that PQRS is a parallelogram.
      • TVWX is a quadrilateral as shown in Fig. 6.2.
      Points A and B divide side TV into 3 equal parts. Points C and D divide side VW into 3 equal parts. Points E and F divide side WX into 3 equal parts. Points G and H divide side TX into 3 equal parts. \(\overrightarrow { \mathrm { TA } } = \mathbf { a } , \quad \overrightarrow { \mathrm { TH } } = \mathbf { b } , \quad \overrightarrow { \mathrm { VC } } = \mathbf { c }\). \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Fig. 6.2} \includegraphics[alt={},max width=\textwidth]{20639e13-01cc-4d96-b694-fb3cf1828f4d-06_577_671_1877_319}
      \end{figure}
      1. Show that \(\overrightarrow { \mathrm { WX } } = k ( - \mathbf { a } + \mathbf { b } - \mathbf { c } )\), where \(k\) is a constant to be determined.
      2. Verify that AH is parallel to DE .
      3. Verify that BC is parallel to GF .
OCR MEI Paper 3 2023 June Q7
6 marks Standard +0.8
7 A wire, 10 cm long, is bent to form the perimeter of a sector of a circle, as shown in the diagram. The radius is \(r \mathrm {~cm}\) and the angle at the centre is \(\theta\) radians. \includegraphics[max width=\textwidth, alt={}, center]{20639e13-01cc-4d96-b694-fb3cf1828f4d-07_323_204_342_242} Determine the maximum possible area of the sector, showing that it is a maximum.
OCR MEI Paper 3 2023 June Q8
7 marks Challenging +1.2
8 A circle with centre \(A\) and radius 8 cm and a circle with centre \(C\) and radius 12 cm intersect at points B and D . Quadrilateral \(A B C D\) has area \(60 \mathrm {~cm} ^ { 2 }\).
Determine the two possible values for the length AC.