OCR H240/03 (Pure Mathematics and Mechanics) 2017 Specimen

1. If \(|x| = 3\), find the possible values of \(|2x - 1|\). [3]
2. Find the set of values of \(x\) for which \(|2x - 1| > x + 1\). Give your answer in set notation. [4]

1. Use the trapezium rule, with four strips each of width 0.25, to find an approximate value for \(\int_0^1 \frac{1}{\sqrt{1+x^2}} dx\). [3]
2. Explain how the trapezium rule might be used to give a better approximation to the integral given in part (a). [1]

In this question you must show detailed reasoning. Given that \(5\sin 2x = 3\cos x\), where \(0° < x < 90°\), find the exact value of \(\sin x\). [4]

For a small angle \(\theta\), where \(\theta\) is in radians, show that \(1 + \cos \theta - 3\cos^2 \theta \approx -1 + \frac{3}{2}\theta^2\). [4]

1. Find the first three terms in the expansion of \((1 + px)^{\frac{1}{3}}\) in ascending powers of \(x\). [3]
2. The expansion of \((1 + qx)(1 + px)^{\frac{1}{3}}\) is \(1 + x - \frac{2}{9}x^2 + ...\) Find the possible values of the constants \(p\) and \(q\). [5]

A curve has equation \(y = x^2 + kx - 4x^{-1}\) where \(k\) is a constant. Given that the curve has a minimum point when \(x = -2\)

• find the value of \(k\)
• show that the curve has a point of inflection which is not a stationary point. [7]

1. Find \(\int 5x^3\sqrt{x^2 + 1} dx\). [5]
2. Find \(\int \theta \tan^2 \theta d\theta\). You may use the result \(\int \tan \theta d\theta = \ln|\sec \theta| + c\). [5]

In this question you must show detailed reasoning. The diagram shows triangle \(ABC\). \includegraphics{figure_8} The angles \(CAB\) and \(ABC\) are each \(45°\), and angle \(ACB = 90°\). The points \(D\) and \(E\) lie on \(AC\) and \(AB\) respectively. \(AE = DE = 1\), \(DB = 2\). Angle \(BED = 90°\), angle \(EBD = 30°\) and angle \(DBC = 15°\).

1. Show that \(BC = \frac{\sqrt{2} + \sqrt{6}}{2}\). [3]
2. By considering triangle \(BCD\), show that \(\sin 15° = \frac{\sqrt{6} - \sqrt{2}}{4}\). [3]

Two forces, of magnitudes 2 N and 5 N, act on a particle in the directions shown in the diagram below. \includegraphics{figure_9}

1. Calculate the magnitude of the resultant force on the particle. [3]
2. Calculate the angle between this resultant force and the force of magnitude 5 N. [1]

A body of mass 20 kg is on a rough plane inclined at angle \(\alpha\) to the horizontal. The body is held at rest on the plane by the action of a force of magnitude \(P\) N. The force is acting up the plane in a direction parallel to a line of greatest slope of the plane. The coefficient of friction between the body and the plane is \(\mu\).

1. When \(P = 100\), the body is on the point of sliding down the plane. Show that \(g \sin \alpha = g\mu \cos \alpha + 5\). [4]
2. When \(P\) is increased to 150, the body is on the point of sliding up the plane. Use this, and your answer to part (a), to find an expression for \(\alpha\) in terms of \(g\). [3]

In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the directions east and north respectively. A particle of mass 0.12 kg is moving so that its position vector \(\mathbf{r}\) metres at time \(t\) seconds is given by \(\mathbf{r} = 2t^2\mathbf{i} + (5t^2 - 4t)\mathbf{j}\).

1. Show that when \(t = 0.7\) the bearing on which the particle is moving is approximately \(044°\). [3]
2. Find the magnitude of the resultant force acting on the particle at the instant when \(t = 0.7\). [4]
3. Determine the times at which the particle is moving on a bearing of \(045°\). [2]

A girl is practising netball. She throws the ball from a height of 1.5 m above horizontal ground and aims to get the ball through a hoop. The hoop is 2.5 m vertically above the ground and is 6 m horizontally from the point of projection. The situation is modelled as follows.

• The initial velocity of the ball has magnitude \(U\) m s\(^{-1}\).
• The angle of projection is \(40°\).
• The ball is modelled as a particle.
• The hoop is modelled as a point.

This is shown on the diagram below. \includegraphics{figure_12}

1. For \(U = 10\), find
   1. the greatest height above the ground reached by the ball [5]
   2. the distance between the ball and the hoop when the ball is vertically above the hoop. [4]
2. Calculate the value of \(U\) which allows her to hit the hoop. [3]
3. How appropriate is this model for predicting the path of the ball when it is thrown by the girl? [1]
4. Suggest one improvement that might be made to this model. [1]

Particle \(A\), of mass \(m\) kg, lies on the plane \(\Pi_1\) inclined at an angle of \(\tan^{-1} \frac{4}{3}\) to the horizontal. Particle \(B\), of \(4m\) kg, lies on the plane \(\Pi_2\) inclined at an angle of \(\tan^{-1} \frac{4}{3}\) to the horizontal. The particles are attached to the ends of a light inextensible string which passes over a smooth pulley at \(P\). The coefficient of friction between particle \(A\) and \(\Pi_1\) is \(\frac{1}{4}\) and plane \(\Pi_2\) is smooth. Particle \(A\) is initially held at rest such that the string is taut and lies in a line of greatest slope of each plane. This is shown on the diagram below. \includegraphics{figure_13}

1. Show that when \(A\) is released it accelerates towards the pulley at \(\frac{7g}{15}\) m s\(^{-2}\). [6]
2. Assuming that \(A\) does not reach the pulley, show that it has moved a distance of \(\frac{1}{4}\) m when its speed is \(\sqrt{\frac{7g}{30}}\) m s\(^{-1}\). [2]

A uniform ladder \(AB\) of mass 35 kg and length 7 m rests with its end \(A\) on rough horizontal ground and its end \(B\) against a rough vertical wall. The ladder is inclined at an angle of \(45°\) to the horizontal. A man of mass 70 kg is standing on the ladder at a point \(C\), which is \(x\) metres from \(A\). The coefficient of friction between the ladder and the wall is \(\frac{1}{4}\) and the coefficient of friction between the ladder and the ground is \(\frac{1}{2}\). The system is in limiting equilibrium. Find \(x\). [8]