OCR H240/01 (Pure Mathematics) 2020 November

1

1. For a small angle \(\theta\), where \(\theta\) is in radians, show that \(2 \cos \theta + ( 1 - \tan \theta ) ^ { 2 } \approx 3 - 2 \theta\).
2. Hence determine an approximate solution to \(2 \cos \theta + ( 1 - \tan \theta ) ^ { 2 } = 28 \sin \theta\).

2 Simplify fully.

1. \(\sqrt { 12 a } \times \sqrt { 3 a ^ { 5 } }\)
2. \(\left( 64 b ^ { 3 } \right) ^ { \frac { 1 } { 3 } } \times \left( 4 b ^ { 4 } \right) ^ { - \frac { 1 } { 2 } }\)
3. \(7 \times 9 ^ { 3 c } - 4 \times 27 ^ { 2 c }\)

3 A cylindrical metal tin of radius \(r \mathrm {~cm}\) is closed at both ends. It has a volume of \(16000 \pi \mathrm {~cm} ^ { 3 }\).

1. Show that its total surface area, \(A \mathrm {~cm} ^ { 2 }\), is given by \(A = 2 \pi r ^ { 2 } + 32000 \pi r ^ { - 1 }\).
2. Use calculus to determine the minimum total surface area of the tin. You should justify that it is a minimum.

4 Prove by contradiction that there is no greatest multiple of 5 .

5
\includegraphics[max width=\textwidth, alt={}, center]{febe231d-200a-4957-b41b-de5b9be98b0a-5_424_583_255_244} The diagram shows points \(A\) and \(B\), which have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) with respect to an origin \(O\). \(P\) is the point on \(O B\) such that \(O P : P B = 3 : 1\) and \(Q\) is the midpoint of \(A B\).

1. Find \(\overrightarrow { P Q }\) in terms of \(\mathbf { a }\) and \(\mathbf { b }\). The line \(O A\) is extended to a point \(R\), so that \(P Q R\) is a straight line.
2. Explain why \(\overrightarrow { P R } = k ( 2 \mathbf { a } - \mathbf { b } )\), where \(k\) is a constant.
3. Hence determine the ratio \(O A : A R\).

6 A mobile phone company records their annual sales on \(31 ^ { \text {st } }\) December every year.
Paul thinks that the annual sales, \(S\) million, can be modelled by the equation \(S = a b ^ { t }\), where \(a\) and \(b\) are both positive constants and \(t\) is the number of years since \(31 ^ { \text {st } }\) December 2015. Paul tests his theory by using the annual sales figures from \(31 ^ { \text {st } }\) December 2015 to \(31 { } ^ { \text {st } }\) December 2019. He plots these results on a graph, with \(t\) on the horizontal axis and \(\log _ { 10 } S\) on the vertical axis.

1. Explain why, if Paul's model is correct, the results should lie on a straight line of best fit on his graph. The results lie on a straight line of best fit which has a gradient of 0.146 and an intercept on the vertical axis of 0.583 .
2. Use these values to obtain estimates for \(a\) and \(b\), correct to 2 significant figures.
3. Use this model to predict the year in which, on the \(31 { } ^ { \text {st } }\) December, the annual sales would first be recorded as greater than 200 million.
4. Give a reason why this prediction may not be reliable.

7 Two students, Anna and Ben, are starting a revision programme. They will both revise for 30 minutes on Day 1. Anna will increase her revision time by 15 minutes for every subsequent day. Ben will increase his revision time by \(10 \%\) for every subsequent day.

1. Verify that on Day 10 Anna does 94 minutes more revision than Ben, correct to the nearest minute. Let Day \(X\) be the first day on which Ben does more revision than Anna.
2. Show that \(X\) satisfies the inequality \(X > \log _ { 1.1 } ( 0.5 X + 0.5 ) + 1\).
3. Use the iterative formula \(x _ { n + 1 } = \log _ { 1.1 } \left( 0.5 x _ { n } + 0.5 \right) + 1\) with \(x _ { 1 } = 10\) to find the value of \(X\). You should show the result of each iteration.
4. 1. Give a reason why Anna's revision programme may not be realistic.
   2. Give a different reason why Ben's revision programme may not be realistic.

8

1. Differentiate \(\left( 2 + 3 x ^ { 2 } \right) \mathrm { e } ^ { 2 x }\) with respect to \(x\).
2. Hence show that \(\left( 2 + 3 x ^ { 2 } \right) \mathrm { e } ^ { 2 x }\) is increasing for all values of \(x\).

9
\includegraphics[max width=\textwidth, alt={}, center]{febe231d-200a-4957-b41b-de5b9be98b0a-6_391_606_1672_244} The diagram shows the graph of \(y = | 2 x - 3 |\).

1. State the coordinates of the points of intersection with the axes.
2. Given that the graphs of \(y = | 2 x - 3 |\) and \(y = a x + 2\) have two distinct points of intersection, determine
   1. the set of possible values of \(a\),
   2. the \(x\)-coordinates of the points of intersection of these graphs, giving your answers in terms of \(a\).

10
\includegraphics[max width=\textwidth, alt={}, center]{febe231d-200a-4957-b41b-de5b9be98b0a-7_352_545_258_239} The diagram shows the curve \(y = \sin \left( \frac { 1 } { 2 } \sqrt { x - 1 } \right)\), for \(1 \leqslant x \leqslant 2\).

1. Use rectangles of width 0.25 to find upper and lower bounds for \(\int _ { 1 } ^ { 2 } \sin \left( \frac { 1 } { 2 } \sqrt { x - 1 } \right) \mathrm { d } x\). Give your answers correct to 3 significant figures.
2. 1. Use the substitution \(t = \sqrt { x - 1 }\) to show that \(\int \sin \left( \frac { 1 } { 2 } \sqrt { x - 1 } \right) \mathrm { d } x = \int 2 t \sin \left( \frac { 1 } { 2 } t \right) \mathrm { d } t\).
   2. Hence show that \(\int _ { 1 } ^ { 2 } \sin \left( \frac { 1 } { 2 } \sqrt { x - 1 } \right) \mathrm { d } x = 8 \sin \frac { 1 } { 2 } - 4 \cos \frac { 1 } { 2 }\).

1. 1. Show that the \(x\)-coordinate of \(A\) satisfies the equation \(\left( m ^ { 2 } + 1 \right) x ^ { 2 } - 10 ( m + 1 ) x + 40 = 0\).
   2. Hence determine the equation of the tangent to the circle at \(A\) which passes through \(P\). [4] A second tangent is drawn from \(P\) to meet the circle at a second point \(B\). The equation of this tangent is of the form \(y = n x + 2\), where \(n\) is a constant less than 1 .
2. Determine the exact value of \(\tan A P B\).

12 Find the general solution of the differential equation
\(\left( 2 x ^ { 3 } - 3 x ^ { 2 } - 11 x + 6 \right) \frac { \mathrm { d } y } { \mathrm {~d} x } = y ( 20 x - 35 )\).
Give your answer in the form \(y = \mathrm { f } ( x )\). \section*{END OF QUESTION PAPER} \section*{OCR
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