OCR PURE

1 The quadratic equation \(k x ^ { 2 } + 3 x + k = 0\) has no real roots. Determine the set of possible values of \(k\).

2 In this question you must show detailed reasoning. Solve the equation \(x \sqrt { 5 } + 32 = x \sqrt { 45 } + 2 x\). Give your answer in the form \(a \sqrt { 5 } + b\), where \(a\) and \(b\) are integers to be determined.

3 A Ferris wheel at a fairground rotates in a vertical plane. The height above the ground of a seat on the wheel is \(h\) metres at time \(t\) seconds after the seat is at its lowest point. The height is given by the equation \(h = 15 - 14 \cos ( k t ) ^ { \circ }\), where \(k\) is a positive constant.

1. 1. Write down the greatest height of a seat above the ground.
   2. Write down the least height of a seat above the ground.
2. Given that a seat first returns to its lowest point after 150 seconds, calculate the value of \(k\).
3. Determine for how long a seat is 20 metres or more above the ground during one complete revolution. Give your answer correct to the nearest tenth of a second.

4

1. Find and simplify the first three terms in the expansion, in ascending powers of \(x\), of \(\left( 2 + \frac { 1 } { 3 } k x \right) ^ { 6 }\), where \(k\) is a constant.
2. In the expansion of \(( 3 - 4 x ) \left( 2 + \frac { 1 } { 3 } k x \right) ^ { 6 }\), the constant term is equal to the coefficient of \(x ^ { 2 }\). Determine the exact value of \(k\), given that \(k\) is positive.

5
\includegraphics[max width=\textwidth, alt={}, center]{8c0b68bd-2257-4994-b444-def0b3f64334-4_591_547_262_242} The diagram shows the graphs of \(y = 2 ^ { 3 x }\) and \(y = 2 ^ { 3 x + 2 }\). The graph of \(y = 2 ^ { 3 x }\) can be transformed to the graph of \(y = 2 ^ { 3 x + 2 }\) by means of a stretch.

1. Give details of the stretch. The point \(A\) lies on \(y = 2 ^ { 3 x }\) and the point \(B\) lies on \(y = 2 ^ { 3 x + 2 }\). The line segment \(A B\) is parallel to the \(y\)-axis and the difference between the \(y\)-coordinates of \(A\) and \(B\) is 36 .
2. Determine the \(x\)-coordinate of \(A\). Give your answer in the form \(m \log _ { 2 } n\) where \(m\) and \(n\) are constants to be determined.

6 The vertices of triangle \(A B C\) are \(A ( - 3,1 ) , B ( 5,0 )\) and \(C ( 9,7 )\).

1. Show that \(A B = B C\).
2. Show that angle \(A B C\) is not a right angle.
3. Find the coordinates of the midpoint of \(A C\).
4. Determine the equation of the line of symmetry of the triangle, giving your answer in the form \(p x + q y = r\), where \(p , q\) and \(r\) are integers to be determined.
5. Write down an equation of the circle with centre \(A\) which passes through \(B\). This circle intersects the line of symmetry of the triangle at \(B\) and at a second point.
6. Find the coordinates of this second point.

7
\includegraphics[max width=\textwidth, alt={}, center]{8c0b68bd-2257-4994-b444-def0b3f64334-5_944_938_260_244} The diagram shows the curve \(C\) with equation \(y = 4 x ^ { 2 } - 10 x + 7\) and two straight lines, \(l _ { 1 }\) and \(l _ { 2 }\). The line \(l _ { 1 }\) is the normal to \(C\) at the point \(\left( \frac { 1 } { 2 } , 3 \right)\). The line \(l _ { 2 }\) is the normal to \(C\) at the minimum point of \(C\).

1. Determine the equation of \(l _ { 1 }\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be determined. The shaded region shown in the diagram is bounded by \(C , l _ { 1 }\) and \(l _ { 2 }\).
2. Determine the inequalities that define the shaded region, including its boundaries.

8 In this question you must show detailed reasoning. Given that \(\int _ { 4 } ^ { a } \left( \frac { 4 } { \sqrt { x } } + 3 \right) \mathrm { d } x = 7\), find the value of \(a\).

9 A cyclist travels along a straight horizontal road between house \(A\) and house \(B\). The cyclist starts from rest at \(A\) and moves with constant acceleration for 20 seconds, reaching a velocity of \(15 \mathrm {~ms} ^ { - 1 }\). The cyclist then moves at this constant velocity before decelerating at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), coming to rest at \(B\).

1. Find the time, in seconds, for which the cyclist is decelerating.
2. Sketch a velocity-time graph for the motion of the cyclist between \(A\) and \(B\). [Your sketch need not be drawn to scale; numerical values need not be shown.] The total distance between \(A\) and \(B\) is 1950 m .
3. Find the time, in seconds, for which the cyclist is moving at constant velocity.

10 A particle \(P\) is moving in a straight line. At time \(t\) seconds, where \(t \geqslant 0 , P\) has velocity \(v \mathrm {~ms} ^ { - 1 }\) and acceleration \(a \mathrm {~ms} ^ { - 2 }\) where \(a = 4 t - 9\). It is given that \(v = 2\) when \(t = 1\).

1. Find an expression for \(v\) in terms of \(t\). The particle \(P\) is instantaneously at rest when \(t = t _ { 1 }\) and \(t = t _ { 2 }\), where \(t _ { 1 } < t _ { 2 }\).
2. Find the values of \(t _ { 1 }\) and \(t _ { 2 }\).
3. Determine the total distance travelled by \(P\) between times \(t = 0\) and \(t = t _ { 2 }\).

11 Two balls \(P\) and \(Q\) have masses 0.6 kg and 0.4 kg respectively. The balls are attached to the ends of a string. The string passes over a pulley which is fixed at the edge of a rough horizontal surface. Ball \(P\) is held at rest on the surface 2 m from the pulley. Ball \(Q\) hangs vertically below the pulley. Ball \(Q\) is attached to a third ball \(R\) of mass \(m \mathrm {~kg}\) by another string and \(R\) hangs vertically below \(Q\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{8c0b68bd-2257-4994-b444-def0b3f64334-7_419_945_493_246} The system is released from rest with the strings taut. Ball \(P\) moves towards the pulley with acceleration \(3.5 \mathrm {~ms} ^ { - 2 }\) and a constant frictional force of magnitude 4.5 N opposes the motion of \(P\). The balls are modelled as particles, the pulley is modelled as being small and smooth, and the strings are modelled as being light and inextensible.

1. By considering the motion of \(P\), find the tension in the string connecting \(P\) and \(Q\).
2. Hence determine the value of \(m\). Give your answer correct to \(\mathbf { 3 }\) significant figures. When the balls have been in motion for 0.4 seconds the string connecting \(Q\) and \(R\) breaks.
3. Show that, according to the model, \(P\) does not reach the pulley. It is given that in fact ball \(P\) does reach the pulley.
4. Identify one factor in the modelling that could account for this difference.