OCR MEI C1 (Core Mathematics 1) 2013 January

1 Find the value of each of the following.

1. \(\left( \frac { 5 } { 3 } \right) ^ { - 2 }\)
2. \(81 ^ { \frac { 3 } { 4 } }\)

2 Simplify \(\frac { \left( 4 x ^ { 5 } y \right) ^ { 3 } } { \left( 2 x y ^ { 2 } \right) \times \left( 8 x ^ { 10 } y ^ { 4 } \right) }\).

3 A circle has diameter \(d\), circumference \(C\), and area \(A\). Starting with the standard formulae for a circle, show that \(C d = k A\), finding the numerical value of \(k\).

4 Solve the inequality \(5 x ^ { 2 } - 28 x - 12 \leqslant 0\).

5 You are given that \(\mathrm { f } ( x ) = x ^ { 2 } + k x + c\).
Given also that \(\mathrm { f } ( 2 ) = 0\) and \(\mathrm { f } ( - 3 ) = 35\), find the values of the constants \(k\) and \(c\).

6 The binomial expansion of \(\left( 2 x + \frac { 5 } { x } \right) ^ { 6 }\) has a term which is a constant. Find this term.

7

1. Express \(\sqrt { 48 } + \sqrt { 75 }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers.
2. Simplify \(\frac { 7 + 2 \sqrt { 5 } } { 7 + \sqrt { 5 } }\), expressing your answer in the form \(\frac { a + b \sqrt { 5 } } { c }\), where \(a , b\) and \(c\) are integers.

8 Rearrange the equation \(5 c + 9 t = a ( 2 c + t )\) to make \(c\) the subject.

9 You are given that \(\mathrm { f } ( x ) = ( x + 2 ) ^ { 2 } ( x - 3 )\).

1. Sketch the graph of \(y = \mathrm { f } ( x )\).
2. State the values of \(x\) which satisfy \(\mathrm { f } ( x + 3 ) = 0\).

10

1. Points A and B have coordinates \(( - 2,1 )\) and \(( 3,4 )\) respectively. Find the equation of the perpendicular bisector of AB and show that it may be written as \(5 x + 3 y = 10\).
2. Points C and D have coordinates \(( - 5,4 )\) and \(( 3,6 )\) respectively. The line through C and D has equation \(4 y = x + 21\). The point E is the intersection of CD and the perpendicular bisector of AB . Find the coordinates of point E .
3. Find the equation of the circle with centre E which passes through A and B . Show also that CD is a diameter of this circle.

11

1. Express \(x ^ { 2 } - 5 x + 6\) in the form \(( x - a ) ^ { 2 } - b\). Hence state the coordinates of the turning point of the curve \(y = x ^ { 2 } - 5 x + 6\).
2. Find the coordinates of the intersections of the curve \(y = x ^ { 2 } - 5 x + 6\) with the axes and sketch this curve.
3. Solve the simultaneous equations \(y = x ^ { 2 } - 5 x + 6\) and \(x + y = 2\). Hence show that the line \(x + y = 2\) is a tangent to the curve \(y = x ^ { 2 } - 5 x + 6\) at one of the points where the curve intersects the axes.

12 You are given that \(\mathrm { f } ( x ) = x ^ { 4 } - x ^ { 3 } + x ^ { 2 } + 9 x - 10\).

1. Show that \(x = 1\) is a root of \(\mathrm { f } ( x ) = 0\) and hence express \(\mathrm { f } ( x )\) as a product of a linear factor and a cubic factor.
2. Hence or otherwise find another root of \(\mathrm { f } ( x ) = 0\).
3. Factorise \(\mathrm { f } ( x )\), showing that it has only two linear factors. Show also that \(\mathrm { f } ( x ) = 0\) has only two real roots. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}