OCR MEI AS Paper 1 (AS Paper 1) Specimen

1 Simplify \(\frac { \left( 2 x ^ { 2 } y \right) ^ { 3 } \times 4 x ^ { 3 } y ^ { 5 } } { 2 x y ^ { 10 } }\).

2 Find the coefficient of \(x ^ { 4 }\) in the binomial expansion of \(( x - 3 ) ^ { 5 }\).

3 Fig. 3 shows a particle of weight 8 N on a rough horizontal table.
The particle is being pulled by a horizontal force of 10 N .
It remains at rest in equilibrium. \begin{figure}[h] \end{figure}

1. What information given in the question, tells you that the forces shown in Fig. 3 cannot be the only forces acting on the particle?
2. The only other forces acting on the particle are due to the particle being on the table. State the types of these forces and their magnitudes.

4

1. Express \(x ^ { 2 } + 4 x + 7\) in the form \(( x + b ) ^ { 2 } + c\).
2. Explain why the minimum point on the curve \(y = ( x + b ) ^ { 2 } + c\) occurs when \(x = - b\).

5 Particle P moves on a straight line that contains the point O .
At time \(t\) seconds the displacement of P from O is \(s\) metres, where \(s = t ^ { 3 } - 3 t ^ { 2 } + 3\).

1. Determine the times when the particle has zero velocity.
2. Find the distances of P from O at the times when it has zero velocity.

6 Two points, \(A\) and \(B\), have position vectors \(\mathbf { a } = \mathbf { i } - 3 \mathbf { j }\) and \(\mathbf { b } = 4 \mathbf { i } + 3 \mathbf { j }\).
The point C lies on the line \(y = 1\). The lengths of the line segments AC and BC are equal. Determine the position vector of \(C\).

7 A car is usually driven along the whole of a 5 km stretch of road at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). On one occasion, during a period of 50 seconds, the speed of the car is as shown by the speed-time graph in Fig. 7.
The rest of the 5 km is travelled at \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \begin{figure}[h] \end{figure} How much more time than usual did the journey take on this occasion?
Show your working clearly.

8 A circle has equation \(( x - 2 ) ^ { 2 } + ( y + 3 ) ^ { 2 } = 25\).

1. Write down
   • the radius of the circle,
   • the coordinates of the centre of the circle.
   • Find, in exact form, the coordinates of the points of intersection of the circle with the \(y\)-axis.
   • Show that the point \(( 1,2 )\) lies outside the circle.
   • The point \(\mathrm { P } ( - 1,1 )\) lies on the circle. Find the equation of the tangent to the circle at P .

9 A biologist is investigating the growth of bacteria in a piece of bread.
He believes that the number, \(N\), of bacteria after \(t\) hours may be modelled by the relationship \(N = A \times 2 ^ { k t }\), where \(A\) and \(k\) are constants.

1. Show that, according to the model, the graph of \(\log _ { 10 } N\) against \(t\) is a straight line. Give, in terms of \(A\) and \(k\),
   • the gradient of the line
   • the intercept on the vertical axis.
   The biologist measures the number of bacteria at regular intervals over 22 hours and plots a graph of \(\log _ { 10 } N\) against \(t\). He finds that the graph is approximately a straight line with gradient 0.20 . The line crosses the vertical axis at 2.0 .
2. Find the values of \(A\) and \(k\).
3. Use the model to predict the number of bacteria after 24 hours.
4. Give a reason why the model may not be appropriate for large values of \(t\).

10

1. Sketch the graph of \(y = \frac { 1 } { x } + a\), where \(a\) is a positive constant.
   • State the equations of the horizontal and vertical asymptotes.
   • Give the coordinates of any points where the graph crosses the axes.
   • Find the equation of the normal to the curve \(y = \frac { 1 } { x } + 2\) at the point where \(x = 2\).
   • Find the coordinates of the point where this normal meets the curve again.

11 In this question you must show detailed reasoning.
Determine for what values of \(k\) the graphs \(y = 2 x ^ { 2 } - k x\) and \(y = x ^ { 2 } - k\) intersect.

12 A box hangs from a balloon by means of a light inelastic string. The string is always vertical. The mass of the box is 15 kg . Catherine initially models the situation by assuming that there is no air resistance to the motion of the box. Use Catherine's model to calculate the tension in the string if:

1. the box is held at rest by the tension in the string,
2. the box is instantaneously at rest and accelerating upwards at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
3. the box is moving downwards at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and accelerating upwards at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Catherine now carries out an experiment to find the magnitude of the air resistance on the box when it is moving.
   At a time when the box is accelerating downwards at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), she finds that the tension in the string is 140 N .
4. Calculate the magnitude of the air resistance at that time. Give, with a reason, the direction of motion of the box. \section*{END OF QUESTION PAPER}