OCR Mechanics 1 (Mechanics 1) 2018 December

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

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

3
\includegraphics[max width=\textwidth, alt={}, center]{918c616a-a0c7-4779-8d3c-84ddf1fa36d6-04_695_714_1087_248} 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. Given that the point \(( 1 , - 5 )\) lies on the circle, find the exact area of the circle.

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

5
\includegraphics[max width=\textwidth, alt={}, center]{918c616a-a0c7-4779-8d3c-84ddf1fa36d6-05_444_757_548_251} The functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined for \(x \geqslant 0\) by \(\mathrm { f } ( x ) = \frac { x } { x ^ { 2 } + 3 }\) and \(\mathrm { g } ( x ) = \mathrm { e } ^ { - 2 x }\). The diagram shows the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\). The equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) has exactly one real root \(\alpha\).

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

6
\includegraphics[max width=\textwidth, alt={}, center]{918c616a-a0c7-4779-8d3c-84ddf1fa36d6-06_544_1232_251_260} The diagram shows the curve with parametric equations \(x = \ln \left( t ^ { 2 } - 4 \right) , \quad y = \frac { 4 } { t ^ { 2 } } , \quad\) 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
   \(\int _ { a } ^ { b } \frac { 8 } { t \left( t ^ { 2 } - 4 \right) } \mathrm { d } t\),
   where \(a\) and \(b\) are constants to be determined.
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.
3. Find a cartesian equation of the curve in the form \(y = \mathrm { f } ( x )\).

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

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

8 A uniform ladder \(A B\), of weight 150 N and length 4 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 N is standing on the ladder at a distance \(x \mathrm {~m}\) from \(A\).

1. Show that the magnitude of the frictional force exerted by the ground on the ladder is \(\frac { 25 } { 2 } ( 2 + 5 x ) \mathrm { N }\). The coefficient of friction between the ladder and the ground is \(\frac { 1 } { 4 }\).
2. Find the greatest value of \(x\) for which equilibrium is possible.

9 A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\), where \(v = 2 t ^ { 4 } + k t ^ { 2 } - 4\). The acceleration of \(P\) when \(t = 2\) is \(28 \mathrm {~ms} ^ { - 2 }\).

1. Show that \(k = - 9\).
2. Show that the velocity of \(P\) has its minimum value when \(t = 1.5\). When \(t = 1 , P\) is at the point \(( - 6.4125,0 )\).
3. Find the distance of \(P\) from the origin \(O\) when \(P\) is moving with minimum velocity.
   \includegraphics[max width=\textwidth, alt={}, center]{918c616a-a0c7-4779-8d3c-84ddf1fa36d6-08_698_1009_260_246}
   \(A\) and \(B\) are points at the upper and lower ends, respectively, of a line of greatest slope on a plane inclined at \(30 ^ { \circ }\) to the horizontal. The distance \(A B\) is \(20 \mathrm {~m} . M\) 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 \(4.2 \mathrm {~ms} ^ { - 1 }\) from \(A\) down the line of greatest slope (see diagram). \(P\) moves down the plane and reaches \(B\) with speed \(12.6 \mathrm {~ms} ^ { - 1 }\). The coefficient of friction between \(P\) and the rough part of the plane is \(\frac { \sqrt { 3 } } { 6 }\).
4. Find the distance \(A M\).
5. 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 \(B\).

11 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. \(H = \frac { V ^ { 2 } } { 2 g } \sin ^ { 2 } \alpha\),
   2. \(R = \frac { V ^ { 2 } } { g } \sin 2 \alpha\).
2. Hence show that \(16 H ^ { 2 } - 8 R _ { 0 } H + R ^ { 2 } = 0\), where \(R _ { 0 }\) is the maximum range for the given speed of projection.
3. Given that \(R _ { 0 } = 200 \mathrm {~m}\) and \(R = 192 \mathrm {~m}\), find
   1. the two possible values of the greatest height of \(B\),
   2. the corresponding values of the angle of projection.
4. State one limitation of the model that could affect your answers to part (iii). \section*{OCR} Oxford Cambridge and RSA