OCR H240/03 (Pure Mathematics and Mechanics) 2018 June

1 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 8 x - 2 y - 7 = 0\).
Find

1. the coordinates of \(C\),
2. the radius of the circle.

2 Solve the equation \(| 2 x - 1 | = | x + 3 |\).

3 In this question you must show detailed reasoning. A gardener is planning the design for a rectangular flower bed. The requirements are:

• the length of the flower bed is to be 3 m longer than the width,
• the length of the flower bed must be at least 14.5 m ,
• the area of the flower bed must be less than \(180 \mathrm {~m} ^ { 2 }\).

The width of the flower bed is \(x \mathrm {~m}\).
By writing down and solving appropriate inequalities in \(x\), determine the set of possible values for the width of the flower bed.

4 In this question you must show detailed reasoning.
The functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = x ^ { 3 } \quad \text { and } \quad \mathrm { g } ( x ) = x ^ { 2 } + 2 .$$

1. Write down expressions for
   (a) \(\mathrm { fg } ( x )\),
   (b) \(\operatorname { gf } ( x )\).
2. Hence find the values of \(x\) for which \(\mathrm { fg } ( x ) - \mathrm { gf } ( x ) = 24\).

5

1. Use the trapezium rule, with two strips of equal width, to show that $$\int _ { 0 } ^ { 4 } \frac { 1 } { 2 + \sqrt { x } } \mathrm {~d} x \approx \frac { 11 } { 4 } - \sqrt { 2 }$$
2. Use the substitution \(x = u ^ { 2 }\) to find the exact value of $$\int _ { 0 } ^ { 4 } \frac { 1 } { 2 + \sqrt { x } } \mathrm {~d} x$$
3. Using your answers to parts (i) and (ii), show that $$\ln 2 \approx k + \frac { \sqrt { 2 } } { 4 }$$ where \(k\) is a rational number to be determined.

6 It is given that the angle \(\theta\) satisfies the equation \(\sin \left( 2 \theta + \frac { 1 } { 4 } \pi \right) = 3 \cos \left( 2 \theta + \frac { 1 } { 4 } \pi \right)\).

1. Show that \(\tan 2 \theta = \frac { 1 } { 2 }\).
2. Hence find, in surd form, the exact value of \(\tan \theta\), given that \(\theta\) is an obtuse angle.

7 The gradient of the curve \(y = \mathrm { f } ( x )\) is given by the differential equation $$( 2 x - 1 ) ^ { 3 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y ^ { 2 } = 0$$ and the curve passes through the point \(( 1,1 )\). By solving this differential equation show that $$f ( x ) = \frac { a x ^ { 2 } - a x + 1 } { b x ^ { 2 } - b x + 1 }$$ where \(a\) and \(b\) are integers to be determined.

8 In this question \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) denote unit vectors which are horizontal and vertically upwards respectively.
A particle of mass 5 kg , initially at rest at the point with position vector \(\binom { 2 } { 45 } \mathrm {~m}\), is acted on by gravity and also by two forces \(\binom { 15 } { - 8 } \mathrm {~N}\) and \(\binom { - 7 } { - 2 } \mathrm {~N}\).

1. Find the acceleration vector of the particle.
2. Find the position vector of the particle after 10 seconds.

9 A uniform plank \(A B\) has weight 100 N and length 4 m . The plank rests horizontally in equilibrium on two smooth supports \(C\) and \(D\), where \(A C = x \mathrm {~m}\) and \(C D = 0.5 \mathrm {~m}\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{d5ab20c8-afd5-473e-8238-96762bd3786d-6_181_1271_1101_395} The magnitude of the reaction of the support on the plank at \(C\) is 75 N . Modelling the plank as a rigid rod, find

1. the magnitude of the reaction of the support on the plank at \(D\),
2. the value of \(x\). A stone block, which is modelled as a particle, is now placed at the end of the plank at \(B\) and the plank is on the point of tilting about \(D\).
3. Find the weight of the stone block.
4. Explain the limitation of modelling
   (a) the stone block as a particle,
   (b) the plank as a rigid rod.

10 Three forces, of magnitudes \(4 \mathrm {~N} , 6 \mathrm {~N}\) and \(P \mathrm {~N}\), act at a point in the directions shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{d5ab20c8-afd5-473e-8238-96762bd3786d-7_604_601_306_724} The forces are in equilibrium.

1. Show that \(\theta = 41.4 ^ { \circ }\), correct to 3 significant figures.
2. Hence find the value of \(P\). The force of magnitude 4 N is now removed and the force of magnitude 6 N is replaced by a force of magnitude 3 N acting in the same direction.
3. Find
   (a) the magnitude of the resultant of the two remaining forces,
   (b) the direction of the resultant of the two remaining forces.

11 The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of a car at time \(t \mathrm {~s}\), during the first 20 s of its journey, is given by \(v = k t + 0.03 t ^ { 2 }\), where \(k\) is a constant. When \(t = 20\) the acceleration of the car is \(1.3 \mathrm {~ms} ^ { - 2 }\). For \(t > 20\) the car continues its journey with constant acceleration \(1.3 \mathrm {~ms} ^ { - 2 }\) until its speed reaches \(25 \mathrm {~ms} ^ { - 1 }\).

1. Find the value of \(k\).
2. Find the total distance the car has travelled when its speed reaches \(25 \mathrm {~ms} ^ { - 1 }\).

12 One end of a light inextensible string is attached to a particle \(A\) of mass \(m \mathrm {~kg}\). The other end of the string is attached to a second particle \(B\) of mass \(\lambda m \mathrm {~kg}\), where \(\lambda\) is a constant. Particle \(A\) is in contact with a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. 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. The particle \(B\) hangs freely below \(P\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{d5ab20c8-afd5-473e-8238-96762bd3786d-8_405_670_493_685} The coefficient of friction between \(A\) and the plane is \(\mu\).

1. It is given that \(A\) is on the point of moving down the plane.
   (a) Find the exact value of \(\mu\) when \(\lambda = \frac { 1 } { 4 }\).
   (b) Show that the value of \(\lambda\) must be less than \(\frac { 1 } { 2 }\).
2. Given instead that \(\lambda = 2\) and that the acceleration of \(A\) is \(\frac { 1 } { 4 } g \mathrm {~ms} ^ { - 2 }\), find the exact value of \(\mu\). \section*{END OF QUESTION PAPER}