OCR MEI Paper 1 (Paper 1) 2019 June

1 In this question you must show detailed reasoning. Show that \(\int _ { 4 } ^ { 9 } ( 2 x + \sqrt { x } ) \mathrm { d } x = \frac { 233 } { 3 }\).

2 Show that the line which passes through the points \(( 2 , - 4 )\) and \(( - 1,5 )\) does not intersect the line \(3 x + y = 10\).

3 The function \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = ( 1 - a x ) ^ { - 3 }\), where \(a\) is a non-zero constant. In the binomial expansion of \(\mathrm { f } ( x )\), the coefficients of \(x\) and \(x ^ { 2 }\) are equal.

1. Find the value of \(a\).
2. Using this value for \(a\),
   1. state the set of values of \(x\) for which the binomial expansion is valid,
   2. write down the quadratic function which approximates \(\mathrm { f } ( x )\) when \(x\) is small.

4 Fig. 4 shows a uniform beam of mass 4 kg and length 2.4 m resting on two supports P and Q . P is at one end of the beam and Q is 0.3 m from the other end.
Determine whether a person of mass 50 kg can tip the beam by standing on it. \begin{figure}[h] \end{figure}

5 A car of mass 1200 kg travels from rest along a straight horizontal road. The driving force is 4000 N and the total of all resistances to motion is 800 N .
Calculate the velocity of the car after 9 seconds.

6

1. Prove that \(\frac { \sin \theta } { 1 - \cos \theta } - \frac { 1 } { \sin \theta } = \cot \theta\).
2. Hence find the exact roots of the equation \(\frac { \sin \theta } { 1 - \cos \theta } - \frac { 1 } { \sin \theta } = 3 \tan \theta\) in the interval \(0 \leqslant \theta \leqslant \pi\). Answer all the questions.
   Section B (75 marks)

7 The velocity \(v \mathrm {~ms} ^ { - 1 }\) of a particle at time \(t \mathrm {~s}\) is given by
\(v = 0.5 t ( 7 - t )\). Determine whether the speed of the particle is increasing or decreasing when \(t = 8\).

8 An arithmetic series has first term 9300 and 10th term 3900.

1. Show that the 20th term of the series is negative.
2. The sum of the first \(n\) terms is denoted by \(S\). Find the greatest value of \(S\) as \(n\) varies.

9 A cannonball is fired from a point on horizontal ground at \(100 \mathrm {~ms} ^ { - 1 }\) at an angle of \(25 ^ { \circ }\) above the horizontal. Ignoring air resistance, calculate

1. the greatest height the cannonball reaches,
2. the range of the cannonball.

10

1. Express \(7 \cos x - 2 \sin x\) in the form \(R \cos ( x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 3 significant figures.
2. Give details of a sequence of two transformations which maps the curve \(y = \sec x\) onto the curve \(y = \frac { 1 } { 7 \cos x - 2 \sin x }\).

11 In this question, the unit vector \(\mathbf { i }\) is horizontal and the unit vector \(\mathbf { j }\) is vertically upwards. A particle of mass 0.8 kg moves under the action of its weight and two forces given by ( \(k \mathbf { i } + 5 \mathbf { j }\) ) N and \(( 4 \mathbf { i } + 3 \mathbf { j } ) \mathrm { N }\). The acceleration of the particle is vertically upwards.

1. Write down the value of \(k\). Initially the velocity of the particle is \(( 4 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
2. Find the velocity of the particle 10 seconds later.

12 Fig. 12 shows a curve C with parametric equations \(x = 4 t ^ { 2 } , y = 4 t\). The point P , with parameter \(t\), is a general point on the curve. Q is the point on the line \(x + 4 = 0\) such that PQ is parallel to the \(x\)-axis. R is the point \(( 4,0 )\). \begin{figure}[h] \end{figure}

1. Show algebraically that P is equidistant from Q and R .
2. Find a cartesian equation of C .

13 A 15 kg box is suspended in the air by a rope which makes an angle of \(30 ^ { \circ }\) with the vertical. The box is held in place by a string which is horizontal.

1. Draw a diagram showing the forces acting on the box.
2. Calculate the tension in the rope.
3. Calculate the tension in the string.

14 Fig. 14 shows a circle with centre O and radius \(r \mathrm {~cm}\). The chord AB is such that angle \(\mathrm { AOB } = x\) radians. The area of the shaded segment formed by AB is \(5 \%\) of the area of the circle. \begin{figure}[h] \end{figure}

1. Show that \(x - \sin x - \frac { 1 } { 10 } \pi = 0\). The Newton-Raphson method is to be used to find \(x\).
2. Write down the iterative formula to be used for the equation in part (a).
3. Use three iterations of the Newton-Raphson method with \(x _ { 0 } = 1.2\) to find the value of \(x\) to a suitable degree of accuracy.

15 A model for the motion of a small object falling through a thick fluid can be expressed using the differential equation
\(\frac { \mathrm { d } v } { \mathrm {~d} t } = 9.8 - k v\),
where \(v \mathrm {~ms} ^ { - 1 }\) is the velocity after \(t \mathrm {~s}\) and \(k\) is a positive constant.

1. Given that \(v = 0\) when \(t = 0\), solve the differential equation to find \(v\) in terms of \(t\) and \(k\).
2. Sketch the graph of \(v\) against \(t\). Experiments show that for large values of \(t\), the velocity tends to \(7 \mathrm {~ms} ^ { - 1 }\).
3. Find the value of \(k\).
4. Find the value of \(t\) for which \(v = 3.5\).

16 A particle of mass 2 kg slides down a plane inclined at \(20 ^ { \circ }\) to the horizontal. The particle has an initial velocity of \(1.4 \mathrm {~ms} ^ { - 1 }\) down the plane. Two models for the particle's motion are proposed. In model A the plane is taken to be smooth.

1. Calculate the time that model A predicts for the particle to slide the first 0.7 m .
2. Explain why model A is likely to underestimate the time taken. In model B the plane is taken to be rough, with a constant coefficient of friction between the particle and the plane.
3. Calculate the acceleration of the particle predicted by model B given that it takes 0.8 s to slide the first 0.7 m .
4. Find the coefficient of friction predicted by model B , giving your answer correct to 3 significant figures. \section*{END OF QUESTION PAPER}