OCR H240/01 (Pure Mathematics) 2019 June

1 In this question you must show detailed reasoning. Solve the inequality \(10 x ^ { 2 } + x - 2 > 0\).

2 The point \(A\) is such that the magnitude of \(\overrightarrow { O A }\) is 8 and the direction of \(\overrightarrow { O A }\) is \(240 ^ { \circ }\).

1. 1. Show the point \(A\) on the axes provided in the Printed Answer Booklet.
   2. Find the position vector of point \(A\). Give your answer in terms of \(\mathbf { i }\) and \(\mathbf { j }\). The point \(B\) has position vector \(6 \mathbf { i }\).
2. Find the exact area of triangle \(A O B\). The point \(C\) is such that \(O A B C\) is a parallelogram.
3. Find the position vector of \(C\). Give your answer in terms of \(\mathbf { i }\) and \(\mathbf { j }\).

3 The function f is defined by \(\mathrm { f } ( x ) = ( x - 3 ) ^ { 2 } - 17\) for \(x \geqslant k\), where \(k\) is a constant.

1. Given that \(\mathrm { f } ^ { - 1 } ( x )\) exists, state the least possible value of \(k\).
2. Evaluate \(\mathrm { ff } ( 5 )\).
3. Solve the equation \(\mathrm { f } ( x ) = x\).
4. Explain why your solution to part (c) is also the solution to the equation \(\mathrm { f } ( x ) = \mathrm { f } ^ { - 1 } ( x )\).

4 Sam starts a job with an annual salary of \(\pounds 16000\). It is promised that the salary will go up by the same amount every year. In the second year Sam is paid \(\pounds 17200\).

1. Find Sam's salary in the tenth year.
2. Find the number of complete years needed for Sam's total salary to first exceed \(\pounds 500000\).
3. Comment on how realistic this model may be in the long term.

5 A curve has equation \(x ^ { 3 } - 3 x ^ { 2 } y + y ^ { 2 } + 1 = 0\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 x y - 3 x ^ { 2 } } { 2 y - 3 x ^ { 2 } }\).
2. Find the equation of the normal to the curve at the point ( 1,2 ).

6 Let \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x\). Use differentiation from first principles to show that \(\mathrm { f } ^ { \prime } ( x ) = 6 x ^ { 2 } + 3\).

7 In this question you must show detailed reasoning. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } \ldots\) is defined by \(u _ { n } = 25 \times 0.6 ^ { n }\).
Use an algebraic method to find the smallest value of \(N\) such that \(\sum _ { n = 1 } ^ { \infty } u _ { n } - \sum _ { n = 1 } ^ { N } u _ { n } < 10 ^ { - 4 }\).

8 A cylindrical tank is initially full of water. There is a small hole at the base of the tank out of which the water leaks. The height of water in the tank is \(x \mathrm {~m}\) at time \(t\) seconds. The rate of change of the height of water may be modelled by the assumption that it is proportional to the square root of the height of water. When \(t = 100 , x = 0.64\) and, at this instant, the height is decreasing at a rate of \(0.0032 \mathrm {~ms} ^ { - 1 }\).

1. Show that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = - 0.004 \sqrt { x }\).
2. Find an expression for \(x\) in terms of \(t\).
3. Hence determine at what time, according to this model, the tank will be empty.

9

1. Express \(3 \cos 3 x + 7 \sin 3 x\) in the form \(R \cos ( 3 x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
2. Give full details of a sequence of three transformations needed to transform the curve \(y = \cos x\) to the curve \(y = 3 \cos 3 x + 7 \sin 3 x\).
3. Determine the greatest value of \(3 \cos 3 x + 7 \sin 3 x\) as \(x\) varies and give the smallest positive value of \(x\) for which it occurs.
4. Determine the least value of \(3 \cos 3 x + 7 \sin 3 x\) as \(x\) varies and give the smallest positive value of \(x\) for which it occurs.

10
\includegraphics[max width=\textwidth, alt={}, center]{05bec6d6-b526-4b6f-86f3-39aa38cbf5c6-6_405_661_251_703} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 6 cm .
The angle \(A O B\) is \(\theta\) radians.
The area of the segment bounded by the chord \(A B\) and the \(\operatorname { arc } A B\) is \(7.2 \mathrm {~cm} ^ { 2 }\).

1. Show that \(\theta = 0.4 + \sin \theta\).
2. Let \(\mathrm { F } ( \theta ) = 0.4 + \sin \theta\). By considering the value of \(\mathrm { F } ^ { \prime } ( \theta )\) where \(\theta = 1.2\), explain why using an iterative method based on the equation in part (a) will converge to the root, assuming that 1.2 is sufficiently close to the root.
3. Use the iterative formula \(\theta _ { n + 1 } = 0.4 + \sin \theta _ { n }\) with a starting value of 1.2 to find the value of \(\theta\) correct to 4 significant figures.
   You should show the result of each iteration.
4. Use a change of sign method to show that the value of \(\theta\) found in part (c) is correct to 4 significant figures.

11
\includegraphics[max width=\textwidth, alt={}, center]{05bec6d6-b526-4b6f-86f3-39aa38cbf5c6-7_540_734_260_667} The diagram shows part of the curve \(y = \ln ( x - 4 )\).

1. Use integration by parts to show that \(\int \ln ( x - 4 ) \mathrm { d } x = ( x - 4 ) \ln | x - 4 | - x + c\).
2. State the equation of the vertical asymptote to the curve \(y = \ln ( x - 4 )\).
3. Find the total area enclosed by the curve \(y = \ln ( x - 4 )\), the \(x\)-axis and the lines \(x = 4.5\) and \(x = 7\). Give your answer in the form \(a \ln 3 + b \ln 2 + c\) where \(a , b\) and \(c\) are constants to be found.

12 A curve has equation \(y = a ^ { 3 x ^ { 2 } }\), where \(a\) is a constant greater than 1 .

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x a ^ { 3 x ^ { 2 } } \ln a\).
2. The tangent at the point \(\left( 1 , a ^ { 3 } \right)\) passes through the point \(\left( \frac { 1 } { 2 } , 0 \right)\). Find the value of \(a\), giving your answer in an exact form.
3. By considering \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) show that the curve is convex for all values of \(x\). \section*{OCR} \section*{Oxford Cambridge and RSA}