OCR PURE 2066

1 In this question you must show detailed reasoning. Solve the equation \(x ( 3 - \sqrt { 5 } ) = 24\), giving your answer in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are positive integers.

2

1. Express \(5 x ^ { 2 } - 20 x + 3\) in the form \(p ( x + q ) ^ { 2 } + r\), where \(p , q\) and \(r\) are integers.
2. State the coordinates of the minimum point of the curve \(y = 5 x ^ { 2 } - 20 x + 3\).
3. State the equation of the normal to the curve \(y = 5 x ^ { 2 } - 20 x + 3\) at its minimum point.

3

1. Sketch the curve \(y = - \frac { 1 } { x ^ { 2 } }\).
2. The curve \(y = - \frac { 1 } { x ^ { 2 } }\) is translated by 2 units in the positive \(x\)-direction. State the equation of the curve after it has been translated.
3. The curve \(y = - \frac { 1 } { x ^ { 2 } }\) is stretched parallel to the \(y\)-axis with scale factor \(\frac { 1 } { 2 }\) and, as a result, the point \(\left( \frac { 1 } { 2 } , - 4 \right)\) on the curve is transformed to the point \(P\). State the coordinates of \(P\).

4

1. Find and simplify the first three terms in the expansion of \(( 2 - 5 x ) ^ { 5 }\) in ascending powers of \(x\).
2. In the expansion of \(( 1 + a x ) ^ { 2 } ( 2 - 5 x ) ^ { 5 }\), the coefficient of \(x\) is 48 . Find the value of \(a\).

5 Points \(A , B , C\) and \(D\) have position vectors \(\mathbf { a } = \binom { 1 } { 2 } , \mathbf { b } = \binom { 3 } { 5 } , \mathbf { c } = \binom { 7 } { 4 }\) and \(\mathbf { d } = \binom { 4 } { k }\).

1. Find the value of \(k\) for which \(D\) is the midpoint of \(A C\).
2. Find the two values of \(k\) for which \(| \overrightarrow { A D } | = \sqrt { 13 }\).
3. Find one value of \(k\) for which the four points form a trapezium.

1. Show that the equation \(6 \cos ^ { 2 } \theta = \tan \theta \cos \theta + 4\)
   can be expressed in the form \(6 \sin ^ { 2 } \theta + \sin \theta - 2 = 0\).
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{d6430776-0b87-4e5e-8f78-c6228ee163d5-4_446_1150_1119_338} The diagram shows parts of the curves \(y = 6 \cos ^ { 2 } \theta\) and \(y = \tan \theta \cos \theta + 4\), where \(\theta\) is in degrees. Solve the inequality \(6 \cos ^ { 2 } \theta > \tan \theta \cos \theta + 4\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

7
\includegraphics[max width=\textwidth, alt={}, center]{d6430776-0b87-4e5e-8f78-c6228ee163d5-5_647_741_260_260} The diagram shows part of the curve \(y = ( 5 - x ) ( x - 1 )\) and the line \(x = a\).
Given that the total area of the regions shaded in the diagram is 19 units \({ } ^ { 2 }\), determine the exact value of \(a\).

8

1. Show that the equation \(2 \log _ { 2 } x = \log _ { 2 } ( k x - 1 ) + 3\), where \(k\) is a constant, can be expressed in the form \(x ^ { 2 } - 8 k x + 8 = 0\).
2. Given that the equation \(2 \log _ { 2 } x = \log _ { 2 } ( k x - 1 ) + 3\) has only one real root, find the value of this root.

9 Three forces \(\binom { 7 } { - 6 } \mathrm {~N} , \binom { 2 } { 5 } \mathrm {~N}\) and \(\mathbf { F N }\) act on a particle.
Given that the particle is in equilibrium under the action of these three forces, calculate \(\mathbf { F }\).

10
\includegraphics[max width=\textwidth, alt={}, center]{d6430776-0b87-4e5e-8f78-c6228ee163d5-6_670_1106_797_258} The diagram shows the velocity-time graph modelling the velocity of a car as it approaches, and drives through, a residential area. The velocity of the car, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds for the time interval \(0 \leqslant t \leqslant 5\) is modelled by the equation \(v = p t ^ { 2 } + q t + r\), where \(p , q\) and \(r\) are constants. It is given that the acceleration of the car is zero at \(t = 5\) and the speed of the car then remains constant.

1. Determine the values of \(p , q\) and \(r\).
2. Calculate the distance travelled by the car from \(t = 2\) to \(t = 10\).

11 Two small balls \(P\) and \(Q\) have masses 3 kg and 2 kg respectively. The balls are attached to the ends of a string. \(P\) is held at rest on a rough horizontal surface. The string passes over a pulley which is fixed at the edge of the surface. \(Q\) hangs vertically below the pulley at a height of 2 m above a horizontal floor.
\includegraphics[max width=\textwidth, alt={}, center]{d6430776-0b87-4e5e-8f78-c6228ee163d5-7_346_906_445_255} The system is initially at rest with the string taut. A horizontal force of magnitude 40 N acts on \(P\) as shown in the diagram.
\(P\) is released and moves directly away from the pulley. A constant frictional force of magnitude 8 N opposes the motion of \(P\). It is given that \(P\) does not leave the horizontal surface and that \(Q\) does not reach the pulley in the subsequent motion. The balls are modelled as particles, the pulley is modelled as being small and smooth, and the string is modelled as being light and inextensible.

1. Show that the magnitude of the acceleration of each particle is \(2.48 \mathrm {~ms} ^ { - 2 }\).
2. Find the tension in the string. When the balls have been in motion for 0.5 seconds, the string breaks.
3. Find the additional time that elapses until \(Q\) hits the floor.
4. Find the speed of \(Q\) as it hits the floor.
5. Write down the magnitude of the normal reaction force acting on \(Q\) when \(Q\) has come to rest on the floor.
6. State one improvement that could be made to the model. \section*{OCR} Oxford Cambridge and RSA