OCR H240/01 (Pure Mathematics) 2021 November

1 Determine the set of values of \(k\) such that the equation \(x ^ { 2 } + 4 x + ( k + 3 ) = 0\) has two distinct real roots.

2 Alex is comparing the cost of mobile phone contracts. Contract \(\boldsymbol { A }\) has a set-up cost of \(\pounds 40\) and then costs 4 p per minute. Contract \(\boldsymbol { B }\) has no set-up cost, does not charge for the first 100 minutes and then costs 6 p per minute.

1. Find an expression for the cost of each of the contracts in terms of \(m\), where \(m\) is the number of minutes for which the phone is used and \(m > 100\).
2. Hence find the value of \(m\) for which both contracts would cost the same.

3 It is given that \(x\) is proportional to the product of the square of \(y\) and the positive square root of \(z\). When \(y = 2\) and \(z = 9 , x = 30\).

1. Write an equation for \(x\) in terms of \(y\) and \(z\).
2. Find the value of \(x\) when \(y = 3\) and \(z = 25\).

4 In this question you must show detailed reasoning.
The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 11 x + 6\).

1. Use the factor theorem to show that \(( 2 x - 1 )\) is a factor of \(\mathrm { f } ( x )\).
2. Express \(\mathrm { f } ( x )\) in fully factorised form.
3. Hence solve the equation \(2 \times 8 ^ { y } - 3 \times 4 ^ { y } - 11 \times 2 ^ { y } + 6 = 0\).

5

1. The graph of the function \(y = \mathrm { f } ( x )\) passes through the point \(P\) with coordinates (2, 6), and is a one-one function. State the coordinates of the point corresponding to \(P\) on each of the following curves.
   1. \(\quad y = \mathrm { f } ( x ) + 3\)
   2. \(\quad y = 2 \mathrm { f } ( 3 x - 1 )\)
   3. \(y = \mathrm { f } ^ { - 1 } ( x )\)
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{6b766f5c-8533-4e0c-bb10-0d9949dc777b-5_494_739_806_333} The diagram shows part of the graph of \(y = \mathrm { g } ^ { \prime } ( x )\). This is the graph of the gradient function of \(y = \mathrm { g } ( x )\). The graph intersects the \(x\)-axis at \(x = - 2\) and \(x = 4\).
   1. State the \(x\)-coordinate of any stationary points on the graph of \(y = \mathrm { g } ( x )\).
   2. State the set of values of \(x\) for which \(y = \mathrm { g } ( x )\) is a decreasing function.
   3. State the \(x\)-coordinate of any points of inflection on the graph of \(y = \mathrm { g } ( x )\).

6

1. Find the first three terms in the expansion of \(( 8 - 3 x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\).
2. State the range of values of \(x\) for which the expansion in part (a) is valid.
3. Find the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { ( 8 - 3 x ) ^ { \frac { 1 } { 3 } } } { ( 1 + 2 x ) ^ { 2 } }\).

7 The curve \(y = \left( x ^ { 2 } - 2 \right) \ln x\) has one stationary point which is close to \(x = 1\).

1. Show that the \(x\)-coordinate of this stationary point satisfies the equation \(2 x ^ { 2 } \ln x + x ^ { 2 } - 2 = 0\).
2. Show that the Newton-Raphson iterative formula for finding the root of the equation in part (a) can be written in the form \(x _ { n + 1 } = \frac { 2 x _ { n } ^ { 2 } \ln x _ { n } + 3 x _ { n } ^ { 2 } + 2 } { 4 x _ { n } \left( \ln x _ { n } + 1 \right) }\).
3. Apply the Newton-Raphson formula with initial value \(x _ { 1 } = 1\) to find \(x _ { 2 }\) and \(x _ { 3 }\).
4. Find the coordinates of this stationary point, giving each coordinate correct to \(\mathbf { 3 }\) decimal places.

8 Functions f and g are defined for \(0 \leqslant x \leqslant 2 \pi\) by \(\mathrm { f } ( x ) = 2 \tan x\) and \(\mathrm { g } ( x ) = \sec x\).

1. 1. State the range of f .
   2. State the range of \(g\).
2. 1. Show that \(\operatorname { fg } ( 0.6 ) = 5.33\), correct to 3 significant figures.
   2. Explain why \(\mathrm { f } ^ { - 1 } \mathrm {~g} ( 0.6 )\) is not defined.
3. In this question you must show detailed reasoning. Solve the equation \(( \mathrm { f } ( x ) ) ^ { 2 } + 6 \mathrm {~g} ( x ) = 0\).

9 A particle moves in the \(x - y\) plane so that at time \(t\) seconds, where \(t \geqslant 0\), its coordinates are given by \(x = \mathrm { e } ^ { 2 t } - 4 \mathrm { e } ^ { t } + 3 , y = 2 \mathrm { e } ^ { - 3 t }\).

1. Explain why the path of the particle never crosses the \(x\)-axis.
2. Determine the exact values of \(t\) when the path of the particle intersects the \(y\)-axis.
3. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 2 \mathrm { e } ^ { 4 t } - \mathrm { e } ^ { 5 t } }\).
4. Hence find the coordinates of the particle when its path is parallel to the \(y\)-axis.

10

1. 
   \includegraphics[max width=\textwidth, alt={}, center]{6b766f5c-8533-4e0c-bb10-0d9949dc777b-7_599_780_267_328} The diagram shows triangle \(A B C\). The perpendicular from \(C\) to \(A B\) meets \(A B\) at \(D\). Angle \(A C D = x\), angle \(D C B = y\), length \(B C = a\) and length \(A C = b\).
   1. Explain why the length of \(C D\) can be written as \(a \cos y\).
   2. Show that the area of the triangle \(A D C\) is given by \(\frac { 1 } { 2 } a b \sin x \cos y\).
   3. Hence, or otherwise, show that \(\sin ( x + y ) = \sin x \cos y + \cos x \sin y\).
2. Given that \(\sin \left( 30 ^ { \circ } + \alpha \right) = \cos \left( 45 ^ { \circ } - \alpha \right)\), show that \(\tan \alpha = 2 + \sqrt { 6 } - \sqrt { 3 } - \sqrt { 2 }\).

11

1. Use the substitution \(u ^ { 2 } = x ^ { 2 } + 3\) to show that \(\int \frac { 4 x ^ { 3 } } { \sqrt { x ^ { 2 } + 3 } } \mathrm {~d} x = \frac { 4 } { 3 } \left( x ^ { 2 } - 6 \right) \sqrt { x ^ { 2 } + 3 } + c\).
2. In this question you must show detailed reasoning.
   \includegraphics[max width=\textwidth, alt={}, center]{6b766f5c-8533-4e0c-bb10-0d9949dc777b-7_620_951_1836_317} The graph shows part of the curve \(y = \frac { 4 x ^ { 3 } } { \sqrt { x ^ { 2 } + 2 } }\).
   Find the exact area enclosed by the curve \(y = \frac { 4 x ^ { 3 } } { \sqrt { x ^ { 2 } + 3 } }\), the normal to this curve at the point \(( 1,2 )\) and the \(x\)-axis.

12 A cake is cooling so that, \(t\) minutes after it is removed from an oven, its temperature is \(\theta ^ { \circ } \mathrm { C }\). When the cake is removed from the oven, its temperature is \(160 ^ { \circ } \mathrm { C }\). After 10 minutes its temperature has fallen to \(125 ^ { \circ } \mathrm { C }\).

1. In a simple model, the rate of decrease of the temperature of the cake is assumed to be constant.
   1. Write down a differential equation for this model.
   2. Solve this differential equation to find \(\theta\) in terms of \(t\).
   3. State one limitation of this model.
2. In a revised model, the rate of decrease of the temperature of the cake is proportional to the difference between the temperature of the cake and the temperature of the room. The temperature of the room is a constant \(20 ^ { \circ } \mathrm { C }\).
   1. Write down a differential equation for this revised model.
   2. Solve this differential equation to find \(\theta\) in terms of \(t\).
3. The cake can be decorated when its temperature is \(25 ^ { \circ } \mathrm { C }\). Find the difference in time between when the two models would predict that the cake can be decorated, giving your answer correct to the nearest minute. \section*{END OF QUESTION PAPER}