OCR H240/03 (Pure Mathematics and Mechanics) 2020 November

1 Triangle \(A B C\) has \(A B = 8.5 \mathrm {~cm} , B C = 6.2 \mathrm {~cm}\) and angle \(B = 35 ^ { \circ }\). Calculate the area of the triangle.

2 A sequence of transformations maps the curve \(y = \mathrm { e } ^ { x }\) to the curve \(y = \mathrm { e } ^ { 2 x + 3 }\). Give details of these transformations.

3 The functions f and g are defined for all real values of \(x\) by
\(f ( x ) = 2 x ^ { 2 } + 6 x\) and \(g ( x ) = 3 x + 2\).

1. Find the range of f .
2. Give a reason why f has no inverse.
3. Given that \(\mathrm { fg } ( - 2 ) = \mathrm { g } ^ { - 1 } ( a )\), where \(a\) is a constant, determine the value of \(a\).
4. Determine the set of values of \(x\) for which \(\mathrm { f } ( x ) > \mathrm { g } ( x )\). Give your answer in set notation.

4 A curve has equation \(y = 2 \ln ( k - 3 x ) + x ^ { 2 } - 3 x\), where \(k\) is a positive constant.

1. Given that the curve has a point of inflection where \(x = 1\), show that \(k = 6\). It is also given that the curve intersects the \(x\)-axis at exactly one point.
2. Show by calculation that the \(x\)-coordinate of this point lies between 0.5 and 1.5 .
3. Use the Newton-Raphson method, with initial value \(x _ { 0 } = 1\), to find the \(x\)-coordinate of the point where the curve intersects the \(x\)-axis, giving your answer correct to 5 decimal places. Show the result of each iteration to 6 decimal places.
4. By choosing suitable bounds, verify that your answer to part (c) is correct to 5 decimal places.

5
\includegraphics[max width=\textwidth, alt={}, center]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-05_339_869_262_244} The diagram shows the curve \(C\) with parametric equations
\(x = \frac { 3 } { t } , y = t ^ { 3 } \mathrm { e } ^ { - 2 t }\), where \(t > 0\).
The maximum point on \(C\) is denoted by \(P\).

1. Determine the exact coordinates of \(P\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 6\).
2. Show that the area of \(R\) is given by $$\int _ { a } ^ { b } 3 t \mathrm { e } ^ { - 2 t } \mathrm {~d} t ,$$ where \(a\) and \(b\) are constants to be determined.
3. Hence determine the exact area of \(R\).

6 In this question you must show detailed reasoning. The diagram shows the curve with equation \(4 x y = 2 \left( x ^ { 2 } + 4 y ^ { 2 } \right) - 9 x\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 x - 4 y - 9 } { 4 x - 16 y }\). At the point \(P\) on the curve the tangent to the curve is parallel to the \(y\)-axis and at the point \(Q\) on the curve the tangent to the curve is parallel to the \(x\)-axis.
2. Show that the distance \(P Q\) is \(k \sqrt { 5 }\), where \(k\) is a rational number to be determined.

7 A particle \(P\) moves with constant acceleration \(( - 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\). At time \(t = 0\) seconds, \(P\) is moving with velocity \(( 7 \mathbf { i } + 6 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).

1. Determine the speed of \(P\) when \(t = 3\).
2. Determine the change in displacement of \(P\) between \(t = 0\) and \(t = 3\).

8 A car is travelling on a straight horizontal road. The velocity of the car, \(v \mathrm {~ms} ^ { - 1 }\), at time \(t\) seconds as it travels past three points, \(P , Q\) and \(R\), is modelled by the equation
\(v = a t ^ { 2 } + b t + c\),
where \(a , b\) and \(c\) are constants.
The car passes \(P\) at time \(t = 0\) with velocity \(8 \mathrm {~ms} ^ { - 1 }\).

1. State the value of \(c\). The car passes \(Q\) at time \(t = 5\) and at that instant its deceleration is \(0.12 \mathrm {~ms} ^ { - 2 }\). The car passes \(R\) at time \(t = 18\) with velocity \(2.96 \mathrm {~ms} ^ { - 1 }\).
2. Determine the values of \(a\) and \(b\).
3. Find, to the nearest metre, the distance between points \(P\) and \(R\).
   \includegraphics[max width=\textwidth, alt={}, center]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-08_469_798_251_244} One end of a light inextensible string is attached to a particle \(A\) of mass 2 kg . The other end of the string is attached to a second particle \(B\) of mass 2.5 kg . Particle \(A\) is in contact with a rough plane inclined at \(\theta\) to the horizontal, where \(\cos \theta = \frac { 4 } { 5 }\). The string is taut and passes over a small smooth pulley \(P\) at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. Particle \(B\) hangs freely below \(P\) at a distance 1.5 m above horizontal ground, as shown in the diagram. The coefficient of friction between \(A\) and the plane is \(\mu\). The system is released from rest and in the subsequent motion \(B\) hits the ground before \(A\) reaches \(P\). The speed of \(B\) at the instant that it hits the ground is \(1.2 \mathrm {~ms} ^ { - 1 }\).

1. For the motion before \(B\) hits the ground, show that the acceleration of \(B\) is \(0.48 \mathrm {~ms} ^ { - 2 }\).
2. For the motion before \(B\) hits the ground, show that the tension in the string is 23.3 N .
3. Determine the value of \(\mu\). After \(B\) hits the ground, \(A\) continues to travel up the plane before coming to instantaneous rest before it reaches \(P\).
4. Determine the distance that \(A\) travels from the instant that \(B\) hits the ground until \(A\) comes to instantaneous rest.
   \includegraphics[max width=\textwidth, alt={}, center]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-09_917_784_244_242} The diagram shows a wall-mounted light. It consists of a rod \(A B\) of mass 0.25 kg and length 0.8 m which is freely hinged to a vertical wall at \(A\), and a lamp of mass 0.5 kg fixed at \(B\). The system is held in equilibrium by a chain \(C D\) whose end \(C\) is attached to the midpoint of \(A B\). The end \(D\) is fixed to the wall a distance 0.4 m vertically above \(A\). The rod \(A B\) makes an angle of \(60 ^ { \circ }\) with the downward vertical. The chain is modelled as a light inextensible string, the rod is modelled as uniform and the lamp is modelled as a particle.
5. By taking moments about \(A\), determine the tension in the chain.
6. 1. Determine the magnitude of the force exerted on the rod at \(A\).
   2. Calculate the direction of the force exerted on the rod at \(A\).
7. Suggest one improvement that could be made to the model to make it more realistic.

11
\includegraphics[max width=\textwidth, alt={}, center]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-10_474_853_264_242} A particle \(P\) moves freely under gravity in the plane of a fixed horizontal axis \(O x\), which lies on horizontal ground, and a fixed vertical axis \(O y . P\) is projected from \(O\) with a velocity whose components along \(O x\) and \(O y\) are \(U\) and \(V\), respectively. \(P\) returns to the ground at a point \(C\).

1. Determine, in terms of \(U , V\) and \(g\), the distance \(O C\).
   \includegraphics[max width=\textwidth, alt={}, center]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-10_478_851_1151_244}
   \(P\) passes through two points \(A\) and \(B\), each at a height \(h\) above the ground and a distance \(d\) apart, as shown in the diagram.
2. Write down the horizontal and vertical components of the velocity of \(P\) at \(A\).
3. Hence determine an expression for \(d\) in terms of \(U , V , g\) and \(h\).
4. Given that the direction of motion of \(P\) as it passes through \(A\) is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 1 } { 2 }\), determine an expression for \(V\) in terms of \(g , d\) and \(h\).