OCR H240/03 (Pure Mathematics and Mechanics) 2023 June

1 Using logarithms, solve the equation
\(4 ^ { 2 x + 1 } = 5 ^ { x }\),
giving your answer correct to \(\mathbf { 3 }\) significant figures.

2

1. Express \(3 \sin x - 4 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to \(\mathbf { 4 }\) significant figures.
2. Hence solve the equation \(3 \sin x - 4 \cos x = 2\) for \(0 ^ { \circ } < x < 90 ^ { \circ }\), giving your answer correct to 3 significant figures.

3 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + p x + q\), where \(p\) and \(q\) are constants.

1. 1. Given that \(\mathrm { f } ^ { \prime } ( 2 ) = 13\), find the value of \(p\).
   2. Given also that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\), find the value of \(q\). The curve \(y = \mathrm { f } ( x )\) is translated by the vector \(\binom { 2 } { - 3 }\).
2. Using the values from part (a), determine the equation of the curve after it has been translated. Give your answer in the form \(y = x ^ { 3 } + a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are integers to be found.

4 A circle \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + k = 0\).

1. Find the set of possible values of \(k\).
2. It is given that \(k = - 46\). Determine the coordinates of the two points on \(C\) at which the gradient of the tangent is \(\frac { 1 } { 2 }\).

5 A mathematics department is designing a new emblem to place on the walls outside its classrooms. The design for the emblem is shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{977ffad6-2440-46bf-9f17-0f30817d2ddf-05_453_1200_358_242} The emblem is modelled by the region between the \(x\)-axis and the curve with parametric equations \(x = 1 + 0.2 t - \cos t , \quad y = k \sin ^ { 2 } t\), where \(k\) is a positive constant and \(0 \leqslant t \leqslant \pi\). Lengths are in metres and the area of the emblem must be \(1 \mathrm {~m} ^ { 2 }\).

1. Show that \(k \int _ { 0 } ^ { \pi } \left( 0.2 + \sin t - 0.2 \cos ^ { 2 } t - \sin t \cos ^ { 2 } t \right) \mathrm { d } t = 1\).
2. Determine the exact value of \(k\).

6 The first, third and fourth terms of an arithmetic progression are \(u _ { 1 } , u _ { 3 }\) and \(u _ { 4 }\) respectively, where \(u _ { 1 } = 2 \sin \theta , \quad u _ { 3 } = - \sqrt { 3 } \cos \theta , \quad u _ { 4 } = \frac { 7 } { 2 } \sin \theta\), and \(\frac { 1 } { 2 } \pi < \theta < \pi\).

1. Determine the exact value of \(\theta\).
2. Hence determine the value of \(\sum _ { r = 1 } ^ { 100 } u _ { r }\).

7 A car \(C\) is moving horizontally in a straight line with velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t\) seconds, where \(v > 0\) and \(t \geqslant 0\). The acceleration, \(a \mathrm {~ms} ^ { - 2 }\), of \(C\) is modelled by the equation
\(a = v \left( \frac { 8 t } { 7 + 4 t ^ { 2 } } - \frac { 1 } { 2 } \right)\).

1. In this question you must show detailed reasoning. Find the times when the acceleration of \(C\) is zero. At \(t = 0\) the velocity of \(C\) is \(17.5 \mathrm {~ms} ^ { - 1 }\) and at \(t = T\) the velocity of \(C\) is \(5 \mathrm {~ms} ^ { - 1 }\).
2. By setting up and solving a differential equation, show that \(T\) satisfies the equation $$T = 2 \ln \left( \frac { 7 + 4 T ^ { 2 } } { 2 } \right)$$
3. Use an iterative formula, based on the equation in part (b), to find the value of \(T\), giving your answer correct to \(\mathbf { 4 }\) significant figures. Use an initial value of 11.25 and show the result of each step of the iteration process.
4. The diagram below shows the velocity-time graph for the motion of \(C\).
   \includegraphics[max width=\textwidth, alt={}, center]{977ffad6-2440-46bf-9f17-0f30817d2ddf-06_751_878_1372_322} Find the time taken for \(C\) to decelerate from travelling at its maximum speed until it is travelling at \(5 \mathrm {~ms} ^ { - 1 }\).

8 A particle \(P\) moves with constant acceleration \(( 3 \mathbf { i } - 2 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\). At time \(t = 4\) seconds, \(P\) has velocity \(6 \mathbf { i } \mathrm {~ms} ^ { - 1 }\). Determine the speed of \(P\) at time \(t = 0\) seconds.

9
\includegraphics[max width=\textwidth, alt={}, center]{977ffad6-2440-46bf-9f17-0f30817d2ddf-07_273_569_781_242} A block \(B\) of weight 10 N lies at rest in equilibrium on a rough plane inclined at \(\theta\) to the horizontal. A horizontal force of magnitude 2 N , acting above a line of greatest slope, is applied to \(B\) (see diagram).

1. Complete the diagram in the Printed Answer Booklet to show all the forces acting on \(B\). It is given that \(B\) remains at rest and the coefficient of friction between \(B\) and the plane is 0.8 .
2. Determine the greatest possible value of \(\tan \theta\).

10 A particle \(P\) of mass \(m \mathrm {~kg}\) is moving on a smooth horizontal surface under the action of two constant horizontal forces \(( - 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) and \(( a \mathbf { i } + b \mathbf { j } ) \mathrm { N }\). The resultant of these two forces is \(\mathbf { R } \mathrm { N }\). It is given that \(\mathbf { R }\) acts in a direction which is parallel to the vector \(- \mathbf { i } + 3 \mathbf { j }\).

1. Show that \(3 a + b = 10\). It is given that \(a = 6\) and that \(P\) moves with an acceleration of magnitude \(5 \sqrt { 10 } \mathrm {~ms} ^ { - 2 }\).
2. Determine the value of \(m\).
   \includegraphics[max width=\textwidth, alt={}, center]{977ffad6-2440-46bf-9f17-0f30817d2ddf-08_515_889_248_242} A uniform rod \(A B\), of weight 20 N and length 2.8 m , rests in equilibrium with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth wall inclined at \(55 ^ { \circ }\) to the horizontal. The rod, which rests in a vertical plane that is perpendicular to the wall, is inclined at \(30 ^ { \circ }\) to the horizontal (see diagram).
3. Show that the magnitude of the force acting on the rod at \(B\) is 9.56 N , correct to \(\mathbf { 3 }\) significant figures.
4. Determine the magnitude of the contact force between the rod and the ground. Give your answer correct to \(\mathbf { 3 }\) significant figures.

12 In this question you should take the acceleration due to gravity to be \(10 \mathrm {~ms ^ { - 2 }\).} A small ball \(P\) is projected from a point \(A\) with speed \(39 \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\theta\), where \(\sin \theta = \frac { 5 } { 13 }\) and \(\cos \theta = \frac { 12 } { 13 }\). Point \(A\) is 20 m vertically above a point \(B\) on horizontal ground. The ball first lands at a point \(C\) on the horizontal ground (see diagram). The ball \(P\) is modelled as a particle moving freely under gravity.

1. Find the maximum height of \(P\) above the ground during its motion. The time taken for \(P\) to travel from \(A\) to \(C\) is \(T\) seconds.
2. Determine the value of \(T\).
3. State one limitation of the model, other than air resistance or the wind, that could affect the answer to part (b). At the instant that \(P\) is projected, a second small ball \(Q\) is released from rest at \(B\) and moves towards \(C\) along the horizontal ground. At time \(t\) seconds, where \(t \geqslant 0\), the velocity \(v \mathrm {~ms} ^ { - 1 }\) of \(Q\) is given by
   \(v = k t ^ { 3 } + 6 t ^ { 2 } + \frac { 3 } { 2 } t\),
   where \(k\) is a positive constant.
4. Given that \(P\) and \(Q\) collide at \(C\), determine the acceleration of \(Q\) immediately before this collision.
   \includegraphics[max width=\textwidth, alt={}, center]{977ffad6-2440-46bf-9f17-0f30817d2ddf-10_607_803_303_246} The diagram shows a small block \(B\), of mass 2 kg , and a particle \(P\), of mass 4 kg , which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the intersection of a horizontal surface and an inclined plane. The particle can move on the inclined plane, which is rough, and which makes an angle of \(60 ^ { \circ }\) with the horizontal. The block can move on the horizontal surface, which is also rough. The system is released from rest, and in the subsequent motion \(P\) moves down the plane and \(B\) does not reach the pulley. It is given that the coefficient of friction between \(P\) and the inclined plane is twice the coefficient of friction between \(B\) and the horizontal surface.
5. Determine, in terms of \(g\), the tension in the string. When \(P\) is moving at \(2 \mathrm {~ms} ^ { - 1 }\) the string breaks. In the 0.5 seconds after the string breaks \(P\) moves 1.9 m down the plane.
6. Determine the deceleration of \(B\) after the string breaks. Give your answer correct to 3 significant figures.