OCR H240/03 (Pure Mathematics and Mechanics) 2019 June

1
\includegraphics[max width=\textwidth, alt={}, center]{7d1b7598-8f97-43a0-8366-efa8192d549e-04_239_867_504_255} The diagram shows triangle \(A B C\), with \(A C = 13.5 \mathrm {~cm} , B C = 8.3 \mathrm {~cm}\) and angle \(A B C = 32 ^ { \circ }\).
Find angle \(C A B\).

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

1. Find
   1. the coordinates of \(C\),
   2. the radius of the circle.
2. Determine the set of values of \(k\) for which the line \(y = k x - 3\) does not intersect or touch the circle.

3

1. In this question you must show detailed reasoning.
   Solve the inequality \(| x - 2 | \leqslant | 2 x - 6 |\).
2. Give full details of a sequence of two transformations needed to transform the graph of \(y = | x - 2 |\) to the graph of \(y = | 2 x - 6 |\).

4
\includegraphics[max width=\textwidth, alt={}, center]{7d1b7598-8f97-43a0-8366-efa8192d549e-05_456_634_260_251} The diagram shows the part of the curve \(y = 3 x \sin 2 x\) for which \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
The maximum point on the curve is denoted by \(P\).

1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(\tan 2 x + 2 x = 0\).
2. Use the Newton-Raphson method, with a suitable initial value, to find the \(x\)-coordinate of \(P\), giving your answer correct to 4 decimal places. Show the result of each iteration.
3. The trapezium rule, with four strips of equal width, is used to find an approximation to \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } 3 x \sin 2 x \mathrm {~d} x\). Show that the result can be expressed as \(k \pi ^ { 2 } ( \sqrt { 2 } + 1 )\), where \(k\) is a rational number to be determined.
4. 1. Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } 3 x \sin 2 x \mathrm {~d} x\).
   2. Hence determine whether using the trapezium rule with four strips of equal width gives an under- or over-estimate for the area of the region enclosed by the curve \(y = 3 x \sin 2 x\) and the \(x\)-axis for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
   3. Explain briefly why it is not easy to tell from the diagram alone whether the trapezium rule with four strips of equal width gives an under- or over-estimate for the area of the region in this case.

1. Prove that \(( \cot \theta + \operatorname { cosec } \theta ) ^ { 2 } = \frac { 1 + \cos \theta } { 1 - \cos \theta }\).
2. Hence solve, for \(0 < \theta < 2 \pi , 3 ( \cot \theta + \operatorname { cosec } \theta ) ^ { 2 } = 2 \sec \theta\).
   \includegraphics[max width=\textwidth, alt={}]{7d1b7598-8f97-43a0-8366-efa8192d549e-06_574_695_306_258}
   The diagram shows part of the curve \(y = \frac { 2 x - 1 } { ( 2 x + 3 ) ( x + 1 ) ^ { 2 } }\).
   Find the exact area of the shaded region, giving your answer in the form \(p + q \ln r\), where \(p\) and \(q\) are positive integers and \(r\) is a positive rational number.

7 A cyclist starting from rest accelerates uniformly at \(1.5 \mathrm {~ms} ^ { - 2 }\) for 4 s and then travels at constant speed.

1. Sketch a velocity-time graph to represent the first 10 seconds of the cyclist's motion.
2. Calculate the distance travelled by the cyclist in the first 10 seconds.

8 A particle \(P\) projected from a point \(O\) on horizontal ground hits the ground after 2.4 seconds. The horizontal component of the initial velocity of \(P\) is \(\frac { 5 } { 3 } d \mathrm {~ms} ^ { - 1 }\).

1. Find, in terms of \(d\), the horizontal distance of \(P\) from \(O\) when it hits the ground.
2. Find the vertical component of the initial velocity of \(P\).
   \(P\) just clears a vertical wall which is situated at a horizontal distance \(d \mathrm {~m}\) from \(O\).
3. Find the height of the wall. The speed of \(P\) as it passes over the wall is \(16 \mathrm {~ms} ^ { - 1 }\).
4. Find the value of \(d\) correct to 3 significant figures.

9
\includegraphics[max width=\textwidth, alt={}, center]{7d1b7598-8f97-43a0-8366-efa8192d549e-08_362_1191_262_438} The diagram shows a small block \(B\), of mass 0.2 kg , and a particle \(P\), of mass 0.5 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 block can move on the horizontal surface, which is rough. The particle can move on the inclined plane, which is smooth and which makes an angle of \(\theta\) with the horizontal where \(\tan \theta = \frac { 3 } { 4 }\). The system is released from rest. In the first 0.4 seconds of the motion \(P\) moves 0.3 m down the plane and \(B\) does not reach the pulley.

1. Find the tension in the string during the first 0.4 seconds of the motion.
2. Calculate the coefficient of friction between \(B\) and the horizontal surface.

10 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the directions east and north respectively.
A particle \(R\) of mass 2 kg is moving on a smooth horizontal surface under the action of a single horizontal force \(\mathbf { F }\) N. At time \(t\) seconds, the velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) of \(R\), relative to a fixed origin \(O\), is given by \(\mathbf { v } = \left( p t ^ { 2 } - 3 t \right) \mathbf { i } + ( 8 t + q ) \mathbf { j }\), where \(p\) and \(q\) are constants and \(p < 0\).

1. Given that when \(t = 0.5\) the magnitude of \(\mathbf { F }\) is 20 , find the value of \(p\). When \(t = 0 , R\) is at the point with position vector \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m }\).
2. Find, in terms of \(q\), an expression for the displacement vector of \(R\) at time \(t\). When \(t = 1 , R\) is at a point on the line \(L\), where \(L\) passes through \(O\) and the point with position vector \(2 \mathbf { i } - 8 \mathbf { j }\).
3. Find the value of \(q\).
   \includegraphics[max width=\textwidth, alt={}, center]{7d1b7598-8f97-43a0-8366-efa8192d549e-09_544_1297_251_255} The diagram shows a ladder \(A B\), of length \(2 a\) and mass \(m\), resting in equilibrium on a vertical wall of height \(h\). The ladder is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The end \(A\) is in contact with horizontal ground. An object of mass \(2 m\) is placed on the ladder at a point \(C\) where \(A C = d\). The ladder is modelled as uniform, the ground is modelled as being rough, and the vertical wall is modelled as being smooth.
4. Show that the normal contact force between the ladder and the wall is \(\frac { m g ( a + 2 d ) \sqrt { 3 } } { 4 h }\). It is given that the equilibrium is limiting and the coefficient of friction between the ladder and the ground is \(\frac { 1 } { 8 } \sqrt { 3 }\).
5. Show that \(h = k ( a + 2 d )\), where \(k\) is a constant to be determined.
6. Hence find, in terms of \(a\), the greatest possible value of \(d\).
7. State one improvement that could be made to the model.