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

Use logarithms to solve the equation \(2^{3x-1} = 3^{x+4}\), giving your answer correct to 3 significant figures. [3]

In this question you must show detailed reasoning. Find the values of \(x\) for which the gradient of the curve \(y = \frac{2}{3}x^3 + \frac{5}{2}x^2 - 3x + 7\) is positive. Give your answer in set notation. [5]

\includegraphics{figure_3} The diagram shows a circle with centre \((a, -a)\) that passes through the origin.

1. Write down an equation for the circle in terms of \(a\). [2]
2. Given that the point \((1, -5)\) lies on the circle, find the exact area of the circle. [3]

The first three terms of an arithmetic series are \(9p\), \(8p - 3\), \(5p\) respectively, where \(p\) is a constant. Given that the sum of the first \(n\) terms of this series is \(-1512\), find the value of \(n\). [6]

\includegraphics{figure_5} The functions f(x) and g(x) are defined for \(x \geqslant 0\) by \(\text{f}(x) = \frac{x}{x^2 + 3}\) and \(\text{g}(x) = \text{e}^{-2x}\). The diagram shows the curves \(y = \text{f}(x)\) and \(y = \text{g}(x)\). The equation \(\text{f}(x) = \text{g}(x)\) has exactly one real root \(\alpha\).

1. Show that \(\alpha\) satisfies the equation \(\text{h}(x) = 0\), where \(\text{h}(x) = x^2 + 3 - x\text{e}^{2x}\). [2]
2. Hence show that a Newton-Raphson iterative formula for finding \(\alpha\) can be written in the form $$x_{n+1} = \frac{x_n^2(1 - 2\text{e}^{2x_n}) - 3}{2x_n - (1 + 2x_n)\text{e}^{2x_n}}.$$ [5]
3. Use this iterative formula, with a suitable initial value, to find \(\alpha\) correct to 3 decimal places. Show the result of each iteration. [3]
4. In this question you must show detailed reasoning. Find the exact value of \(x\) for which \(\text{fg}(x) = \frac{2}{13}\). [6]

\includegraphics{figure_6} The diagram shows the curve with parametric equations \(x = \ln(t^2 - 4)\), \(y = \frac{4}{t}\), where \(t > 2\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = \ln 5\) and \(x = \ln 12\).

1. Show that the area of \(R\) is given by $$\int_a^b \frac{8}{t(t^2 - 4)} dt,$$ where \(a\) and \(b\) are constants to be determined. [4]
2. In this question you must show detailed reasoning. Hence find the exact area of \(R\), giving your answer in the form \(\ln k\), where \(k\) is a constant to be determined. [8]
3. Find a cartesian equation of the curve in the form \(y = \text{f}(x)\). [3]

A particle \(P\) moves with constant acceleration \((3\mathbf{i} - 5\mathbf{j})\text{m s}^{-2}\). At time \(t = 0\) seconds \(P\) is at the origin. At time \(t = 4\) seconds \(P\) has velocity \((2\mathbf{i} + 4\mathbf{j})\text{m s}^{-1}\).

1. Find the displacement vector of \(P\) at time \(t = 4\) seconds. [2]
2. Find the speed of \(P\) at time \(t = 0\) seconds. [4]

A uniform ladder \(AB\), of weight \(150\text{N}\) and length \(4\text{m}\), rests in equilibrium with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth vertical wall. The ladder is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = 3\). A man of weight \(750\text{N}\) is standing on the ladder at a distance \(x\text{m}\) from \(A\).

1. Show that the magnitude of the frictional force exerted by the ground on the ladder is \(\frac{75}{2}(2 + 5x)\text{N}\). [4]

The coefficient of friction between the ladder and the ground is \(\frac{1}{4}\).

1. Find the greatest value of \(x\) for which equilibrium is possible. [3]

A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v\text{m s}^{-1}\), where \(v = 2t^4 + kt^2 - 4\). The acceleration of \(P\) when \(t = 2\) is \(28\text{m s}^{-2}\).

1. Show that \(k = -9\). [3]
2. Show that the velocity of \(P\) has its minimum value when \(t = 1.5\). [3]

When \(t = 1\), \(P\) is at the point \((-6.4125, 0)\).

1. Find the distance of \(P\) from the origin \(O\) when \(P\) is moving with minimum velocity. [4]

\includegraphics{figure_10} \(A\) and \(B\) are points at the upper and lower ends, respectively, of a line of greatest slope on a plane inclined at \(30°\) to the horizontal. The distance \(AB\) is \(20\text{m}\). \(M\) is a point on the plane between \(A\) and \(B\). The surface of the plane is smooth between \(A\) and \(M\), and rough between \(M\) and \(B\). A particle \(P\) is projected with speed \(4.2\text{m s}^{-1}\) from \(A\) down the line of greatest slope (see diagram). \(P\) moves down the plane and reaches \(B\) with speed \(12.6\text{m s}^{-1}\). The coefficient of friction between \(P\) and the rough part of the plane is \(\frac{\sqrt{3}}{6}\).

1. Find the distance \(AM\). [8]
2. Find the angle between the contact force and the downward direction of the line of greatest slope when \(P\) is in motion between \(M\) and \(B\). [3]

A ball \(B\) is projected with speed \(V\) at an angle \(\alpha\) above the horizontal from a point \(O\) on horizontal ground. The greatest height of \(B\) above \(O\) is \(H\) and the horizontal range of \(B\) is \(R\). The ball is modelled as a particle moving freely under gravity.

1. Show that
   1. \(H = \frac{V^2}{2g}\sin^2 \alpha\), [2]
   2. \(R = \frac{V^2}{g}\sin 2\alpha\). [3]
2. Hence show that \(16H^2 - 8R_0 H + R^2 = 0\), where \(R_0\) is the maximum range for the given speed of projection. [5]
3. Given that \(R_0 = 200\text{m}\) and \(R = 192\text{m}\), find
   1. the two possible values of the greatest height of \(B\), [2]
   2. the corresponding values of the angle of projection. [3]
4. State one limitation of the model that could affect your answers to part (iii). [1]