OCR C1 (Core Mathematics 1) 2012 June

1 Simplify \(( x - 5 ) \left( x ^ { 2 } + 3 \right) - ( x + 4 ) ( x - 1 )\).

2 Express each of the following in the form \(7 ^ { k }\) :

1. \(\sqrt [ 4 ] { 7 }\),
2. \(\frac { 1 } { 7 \sqrt { 7 } }\),
3. \(7 ^ { 4 } \times 49 ^ { 10 }\).

3

1. Find the gradient of the line \(l\) which has equation \(3 x - 5 y - 20 = 0\).
2. The line \(l\) crosses the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\). Find the coordinates of the mid-point of \(P Q\).

4

1. Express \(2 x ^ { 2 } - 20 x + 49\) in the form \(p ( x - q ) ^ { 2 } + r\).
2. State the coordinates of the vertex of the curve \(y = 2 x ^ { 2 } - 20 x + 49\).

5

1. Sketch the curve \(y = \sqrt { x }\).
2. Describe the transformation that transforms the curve \(y = \sqrt { x }\) to the curve \(y = \sqrt { x - 4 }\).
3. The curve \(y = \sqrt { x }\) is stretched by a scale factor of 5 parallel to the \(x\)-axis. State the equation of the transformed curve.

6 Find the equation of the normal to the curve \(y = \frac { 6 } { x ^ { 2 } } - 5\) at the point on the curve where \(x = 2\). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.

7 Solve the equation \(x - 6 x ^ { \frac { 1 } { 2 } } + 2 = 0\), giving your answers in the form \(p \pm q \sqrt { r }\), where \(p , q\) and \(r\) are integers.

8

1. Find the coordinates of the stationary point on the curve \(y = x ^ { 4 } + 32 x\).
2. Determine whether this stationary point is a maximum or a minimum.
3. For what values of \(x\) does \(x ^ { 4 } + 32 x\) increase as \(x\) increases?

9

1. A rectangular tile has length \(4 x \mathrm {~cm}\) and width \(( x + 3 ) \mathrm { cm }\). The area of the rectangle is less than \(112 \mathrm {~cm} ^ { 2 }\). By writing down and solving an inequality, determine the set of possible values of \(x\).
2. A second rectangular tile of length \(4 y \mathrm {~cm}\) and width \(( y + 3 ) \mathrm { cm }\) has a rectangle of length \(2 y \mathrm {~cm}\) and width \(y \mathrm {~cm}\) removed from one corner as shown in the diagram.
   \includegraphics[max width=\textwidth, alt={}, center]{ae6cdd3c-0df9-4fec-b4bd-2237b585c766-3_358_757_479_662} Given that the perimeter of this tile is between 20 cm and 54 cm , determine the set of possible values of \(y\).

10 A circle has equation \(( x - 5 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 25\).

1. Find the coordinates of the centre \(C\) and the length of the diameter.
2. Find the equation of the line which passes through \(C\) and the point \(P ( 7,2 )\).
3. Calculate the length of \(C P\) and hence determine whether \(P\) lies inside or outside the circle.
4. Determine algebraically whether the line with equation \(y = 2 x\) meets the circle. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}