OCR MEI Paper 3 (Paper 3) 2022 June

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.

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 \(\mathrm { fg } ( x )\).
2. 1. Find the domain of \(\mathrm { fg } ( x )\).
   2. Find the range of \(\mathrm { fg } ( x )\).

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.

4 In this question you must show detailed reasoning.
Determine the exact solutions of the equation \(2 \cos ^ { 2 } x = 3 \sin x\) for \(0 \leqslant x \leqslant 2 \pi\).

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.

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 \(40 ^ { \circ } \mathrm { C }\).

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.

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.

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 .

10 In this question you must show detailed reasoning. Fig. C2.2 indicates that the curve \(\mathrm { y } = \frac { 4 \mathrm { x } ( \pi - \mathrm { x } ) } { \pi ^ { 2 } } - \sin \mathrm { x }\) has a stationary point near \(x = 3\).

• Verify that the \(x\)-coordinate of this stationary point is between 2.6 and 2.7.
• Show that this stationary point is a maximum turning point.

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.

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} OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{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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