OCR H240/01 (Pure Mathematics) 2023 June

1 In the triangle \(A B C\), the length \(A B = 6 \mathrm {~cm}\), the length \(A C = 15 \mathrm {~cm}\) and the angle \(B A C = 30 ^ { \circ }\).

1. Calculate the length \(B C\).
   \(D\) is the point on \(A C\) such that the length \(B D = 4 \mathrm {~cm}\).
2. Calculate the possible values of the angle \(A D B\).

2

1. 1. Show that \(\frac { 1 } { 3 - 2 \sqrt { x } } + \frac { 1 } { 3 + 2 \sqrt { x } }\) can be written in the form \(\frac { a } { b + c x }\), where \(a , b\) and \(c\) are constants to be determined.
   2. Hence solve the equation \(\frac { 1 } { 3 - 2 \sqrt { x } } + \frac { 1 } { 3 + 2 \sqrt { x } } = 2\).
2. In this question you must show detailed reasoning. Solve the equation \(2 ^ { 2 y } - 7 \times 2 ^ { y } - 8 = 0\).

3

1. Given that \(\mathrm { f } ( x ) = x ^ { 2 } + 2 x\), use differentiation from first principles to show that \(\mathrm { f } ^ { \prime } ( x ) = 2 x + 2\).
2. The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x + 2\) and the curve passes through the point \(( - 1,5 )\). Find the equation of the curve.

4 It is given that \(A B C D\) is a quadrilateral. The position vector of \(A\) is \(\mathbf { i } + \mathbf { j }\), and the position vector of \(B\) is \(3 \mathbf { i } + 5 \mathbf { j }\).

1. Find the length \(A B\).
2. The position vector of \(C\) is \(p \mathbf { i } + p \mathbf { j }\) where \(p\) is a constant greater than 1 . Given that the length \(A B\) is equal to the length \(B C\), determine the position vector of \(C\).
3. The point \(M\) is the midpoint of \(A C\). Given that \(\overrightarrow { M D } = 2 \overrightarrow { B M }\), determine the position vector of \(D\).
4. State the name of the quadrilateral \(A B C D\), giving a reason for your answer.

5

1. The function \(\mathrm { f } ( x )\) is defined for all values of \(x\) as \(\mathrm { f } ( x ) = | a x - b |\), where \(a\) and \(b\) are positive constants.
   1. The graph of \(y = \mathrm { f } ( x ) + c\), where \(c\) is a constant, has a vertex at \(( 3,1 )\) and crosses the \(y\)-axis at \(( 0,7 )\). Find the values of \(a , b\) and \(c\).
   2. Explain why \(\mathrm { f } ^ { - 1 } ( x )\) does not exist.
2. The function \(\mathrm { g } ( x )\) is defined for \(x \geqslant \frac { q } { p }\) as \(\mathrm { g } ( x ) = | p x - q |\), where \(p\) and \(q\) are positive constants.
   1. Find, in terms of \(p\) and \(q\), an expression for \(\mathrm { g } ^ { - 1 } ( x )\), stating the domain of \(\mathrm { g } ^ { - 1 } ( x )\).
   2. State the set of values of \(p\) for which the equation \(\mathrm { g } ( x ) = \mathrm { g } ^ { - 1 } ( x )\) has no solutions.

6 A curve has equation \(y = \mathrm { e } ^ { x ^ { 2 } + 3 x }\).

1. Determine the \(x\)-coordinates of any stationary points on the curve.
2. Show that the curve is convex for all values of \(x\).

7

1. Use the result \(\cos ( A + B ) = \cos A \cos B - \sin A \sin B\) to show that \(\cos ( A - B ) = \cos A \cos B + \sin A \sin B\). The function \(\mathrm { f } ( \theta )\) is defined as \(\cos \left( \theta + 30 ^ { \circ } \right) \cos \left( \theta - 30 ^ { \circ } \right)\), where \(\theta\) is in degrees.
2. Show that \(f ( \theta ) = \cos ^ { 2 } \theta - \frac { 1 } { 4 }\).
3. 1. Determine the following.
      • The maximum value of \(\mathrm { f } ( \theta )\)
4. The smallest positive value of \(\theta\) for which this maximum value occurs
   (ii) Determine the following.
5. The minimum value of \(\mathrm { f } ( \theta )\)
6. The smallest positive value of \(\theta\) for which this minimum value occurs

8

1. Find the first three terms in the expansion of \(( 4 + 3 x ) ^ { \frac { 3 } { 2 } }\) in ascending powers of \(x\).
2. State the range of values of \(x\) for which the expansion in part (a) is valid.
3. In the expansion of \(( 4 + 3 x ) ^ { \frac { 3 } { 2 } } ( 1 + a x ) ^ { 2 }\) the coefficient of \(x ^ { 2 }\) is \(\frac { 107 } { 16 }\). Determine the possible values of the constant \(a\).

9 Conservationists are studying how the number of bees in a wildflower meadow varies according to the number of wildflower plants. The study takes place over a series of weeks in the summer. A model is suggested for the number of bees, \(B\), and the number of wildflower plants, \(F\), at time \(t\) weeks after the start of the study. In the model \(B = 20 + 2 t + \cos 3 t\) and \(F = 50 \mathrm { e } ^ { 0.1 t }\). The model assumes that \(B\) and \(F\) can be treated as continuous variables.

1. State the meaning of \(\frac { \mathrm { d } B } { \mathrm {~d} F }\).
2. Determine \(\frac { \mathrm { d } B } { \mathrm {~d} F }\) when \(t = 4\).
3. Suggest a reason why this model may not be valid for values of \(t\) greater than 12 .

10
\includegraphics[max width=\textwidth, alt={}, center]{9473b8f7-616a-485e-963b-696c6640ae6b-07_805_775_251_242} The diagram shows part of the curve \(\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { x } } { 4 x ^ { 2 } - 1 } + 2\). The equation \(\mathrm { f } ( x ) = 0\) has a positive root \(\alpha\) close to \(x = 0.3\).

1. Explain why using the sign change method with \(x = 0\) and \(x = 1\) will fail to locate \(\alpha\).
2. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as \(x = \frac { 1 } { 4 } \sqrt { \left( 4 - 2 \mathrm { e } ^ { x } \right) }\).
3. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { 4 } \sqrt { \left( 4 - 2 \mathrm { e } ^ { x _ { n } } \right) }\) with a starting value of \(x _ { 1 } = 0.3\) to find the value of \(\alpha\) correct to \(\mathbf { 4 }\) significant figures, showing the result of each iteration.
4. An alternative iterative formula is \(x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)\), where \(\mathrm { F } \left( x _ { n } \right) = \ln \left( 2 - 8 x _ { n } ^ { 2 } \right)\). By considering \(\mathrm { F } ^ { \prime } ( 0.3 )\) explain why this iterative formula will not find \(\alpha\).

11 The owners of an online shop believe that their sales can be modelled by \(S = a b ^ { t }\), where \(a\) and \(b\) are both positive constants, \(S\) is the number of items sold in a month and \(t\) is the number of complete months since starting their online shop. The sales for the first six months are recorded, and the values of \(\log _ { 10 } S\) are plotted against \(t\) in the graph below. The graph is repeated in the Printed Answer Booklet.
\includegraphics[max width=\textwidth, alt={}, center]{9473b8f7-616a-485e-963b-696c6640ae6b-08_1203_1408_552_244}

1. Explain why the graph suggests that the given model is appropriate. The owners believe that \(a = 120\) and \(b = 1.15\) are good estimates for the parameters in the model.
2. Show that the graph supports these estimates for the parameters.
3. Use the model \(S = 120 \times 1.15 ^ { t }\) to predict the number of items sold in the seventh month after opening.
4. 1. Use the model \(S = 120 \times 1.15 ^ { t }\) to predict the number of months after opening when the total number of items sold after opening will first exceed 70000 .
   2. Comment on how reliable this prediction may be.

12

1. Use the substitution \(u = \mathrm { e } ^ { x } - 2\) to show that $$\int \frac { 7 \mathrm { e } ^ { x } - 8 } { \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } } \mathrm {~d} x = \int \frac { 7 u + 6 } { u ^ { 2 } ( u + 2 ) } \mathrm { d } u$$
2. Hence show that $$\int _ { \ln 4 } ^ { \ln 6 } \frac { 7 \mathrm { e } ^ { x } - 8 } { \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } } \mathrm {~d} x = a + \ln b$$ where \(a\) and \(b\) are rational numbers to be determined. \section*{END OF QUESTION PAPER}