OCR MEI Paper 3 (Paper 3) 2021 November

1

1. Express \(x ^ { 2 } + 8 x + 2\) in the form \(( x + a ) ^ { 2 } + b\).
2. Write down the coordinates of the turning point of the curve \(y = x ^ { 2 } + 8 x + 2\).
3. State the transformation(s) which map(s) the curve \(y = x ^ { 2 }\) onto the curve \(y = x ^ { 2 } + 8 x + 2\).

2 Solve the equation \(\sin 2 x = 0.3\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\). Give your answer(s) correct to \(\mathbf { 1 }\) decimal place.

3

1. Determine, in terms of \(k\), the coordinates of the point where the lines with the following equations intersect. $$\begin{array} { r } x + y = k
   2 x - y = 1 \end{array}$$
2. Determine, in terms of \(k\), the coordinates of the points where the line \(\mathrm { x } + \mathrm { y } = \mathrm { k }\) crosses the curve \(y = x ^ { 2 } + k\).

4 The diagram shows points \(A\) and \(B\) on the curve \(y = \left( \frac { x } { 4 } \right) ^ { - x }\).
The \(x\)-coordinate of A is 1 and the \(x\)-coordinate of B is 1.1 .
\includegraphics[max width=\textwidth, alt={}, center]{a0d9573f-8273-4562-a2d3-07f15d9da1af-4_522_707_1758_278}

1. Find the gradient of chord AB . Give your answer correct to 2 decimal places.
2. Give the \(x\)-coordinate of a point C on the curve such that the gradient of chord AC is a better approximation to the gradient of the tangent to the curve at A .

5

1. The diagram shows the curve \(\mathrm { y } = \mathrm { e } ^ { \mathrm { x } }\).
   \includegraphics[max width=\textwidth, alt={}, center]{a0d9573f-8273-4562-a2d3-07f15d9da1af-5_574_682_315_328} On the axes in the Printed Answer Booklet, sketch graphs of
   1. \(\frac { \mathrm { dy } } { \mathrm { dx } }\) against \(x\),
   2. \(\frac { \mathrm { dy } } { \mathrm { dx } }\) against \(y\).
2. Wolves were introduced to Yellowstone National Park in 1995. The population of wolves, \(y\), is modelled by the equation
   \(y = A e ^ { k t }\),
   where \(A\) and \(k\) are constants and \(t\) is the number of years after 1995.
   1. Give a reason why this model might be suitable for the population of wolves.
   2. When \(t = 0 , y = 21\) and when \(t = 1 , y = 51\). Find values of \(A\) and \(k\) consistent with the data.
   3. Give a reason why the model will not be a good predictor of wolf populations many years after 1995.

6 In this question you must show detailed reasoning.
Show that \(\sum _ { r = 1 } ^ { 3 } \frac { 1 } { \sqrt { r + 1 } + \sqrt { r } } = 1\).

7 Determine \(\int x \cos 2 x \mathrm {~d} x\).

8 For a particular value of \(a\), the curve \(\mathrm { y } = \frac { \mathrm { a } } { \mathrm { x } ^ { 2 } }\) passes through the point \(( 3,1 )\).
Find the coordinates of all the other points on the curve where both the \(x\)-coordinate and the \(y\)-coordinate are integers.

9 The diagram shows the curve \(\mathrm { y } = 3 - \sqrt { \mathrm { x } }\).
\includegraphics[max width=\textwidth, alt={}, center]{a0d9573f-8273-4562-a2d3-07f15d9da1af-6_810_1008_1155_283}

1. Draw the line \(\mathrm { y } = 5 \mathrm { x } - 1\) on the copy of the diagram in the Printed Answer Booklet.
2. In this question you must show detailed reasoning. Determine the exact area of the region bounded by the curve \(y = 3 - \sqrt { x }\), the lines \(y = 5 x - 1\) and \(x = 4\) and the \(x\)-axis.

10

1. Express \(\frac { 1 } { ( 4 x + 1 ) ( x + 1 ) }\) in partial fractions.
2. A curve passes through the point \(( 0,2 )\) and satisfies the differential equation \(\frac { d y } { d x } = \frac { y } { ( 4 x + 1 ) ( x + 1 ) }\),
   for \(x > - \frac { 1 } { 4 }\).
   Show by integration that \(\mathrm { y } = \mathrm { A } \left( \frac { 4 \mathrm { x } + 1 } { \mathrm { x } + 1 } \right) ^ { \mathrm { B } }\) where \(A\) and \(B\) are constants to be determined.

11 In this question you must show detailed reasoning. The diagram shows triangle ABC , with \(\mathrm { BC } = 8 \mathrm {~cm}\) and angle \(\mathrm { BAC } = 45 ^ { \circ }\).
The point D on AC is such that \(\mathrm { DC } = 5 \mathrm {~cm}\) and \(\mathrm { BD } = 7 \mathrm {~cm}\).
\includegraphics[max width=\textwidth, alt={}, center]{a0d9573f-8273-4562-a2d3-07f15d9da1af-7_684_553_1119_258} Determine the exact length of AB .

12 Show that \(\beta = \arctan \left( \frac { 1 } { 3 } \right)\), as given in line 15 .

13

1. Use triangle ABE in Fig. C 2 to show that \(\arctan x + \arctan \left( \frac { 1 } { x } \right) = \frac { \pi } { 2 }\), as given in line 29 .
2. Sketch the graph of \(\mathrm { y } = \arctan \mathrm { x }\).
3. What property of the arctan function ensures that \(\mathrm { y } > \frac { 1 } { \mathrm { x } } \Rightarrow \arctan y > \arctan \left( \frac { 1 } { \mathrm { x } } \right)\), as given in line 30 ?

14

1. Show that $$\arctan \left( \frac { 1 } { n + 1 } \right) + \arctan \left( \frac { 1 } { n ^ { 2 } + n + 1 } \right) = \arctan \left( \frac { 1 } { n } \right) \Rightarrow \arctan \left( \frac { 1 } { 2 } \right) + \arctan \left( \frac { 1 } { 3 } \right) = \arctan 1 .$$
2. Use the arctan addition formula in line 23 to show that $$\arctan \left( \frac { 1 } { n + 1 } \right) + \arctan \left( \frac { 1 } { n ^ { 2 } + n + 1 } \right) = \arctan \left( \frac { 1 } { n } \right) , \text { as given in line } 39 .$$

15 Prove that \(\arctan 1 + \arctan 2 + \arctan 3 = \pi\), as given in line 41 . \section*{END OF QUESTION PAPER} \section*{OCR
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