OCR MEI C1 (Core Mathematics 1) 2009 June

1 A line has gradient - 4 and passes through the point (2,6). Find the coordinates of its points of intersection with the axes.

2 Make \(a\) the subject of the formula \(s = u t + \frac { 1 } { 2 } a t ^ { 2 }\).

3 When \(x ^ { 3 } - k x + 4\) is divided by \(x - 3\), the remainder is 1 . Use the remainder theorem to find the value of \(k\).

4 Solve the inequality \(x ( x - 6 ) > 0\).

5

1. Calculate \({ } ^ { 5 } \mathrm { C } _ { 3 }\).
2. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 + 2 x ) ^ { 5 }\).

6 Prove that, when \(n\) is an integer, \(n ^ { 3 } - n\) is always even.

7 Find the value of each of the following.

1. \(5 ^ { 2 } \times 5 ^ { - 2 }\)
2. \(100 ^ { \frac { 3 } { 2 } }\)

8

1. Simplify \(\frac { \sqrt { 48 } } { 2 \sqrt { 27 } }\).
2. Expand and simplify \(( 5 - 3 \sqrt { 2 } ) ^ { 2 }\).

11 \begin{figure}[h] \end{figure} Fig. 11 shows the line joining the points \(\mathrm { A } ( 0,3 )\) and \(\mathrm { B } ( 6,1 )\).

1. Find the equation of the line perpendicular to AB that passes through the origin, O .
2. Find the coordinates of the point where this perpendicular meets AB .
3. Show that the perpendicular distance of AB from the origin is \(\frac { 9 \sqrt { 10 } } { 10 }\).
4. Find the length of AB , expressing your answer in the form \(a \sqrt { 10 }\).
5. Find the area of triangle OAB .

12

1. You are given that \(\mathrm { f } ( x ) = ( x + 1 ) ( x - 2 ) ( x - 4 )\).
   (A) Show that \(\mathrm { f } ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8\).
   (B) Sketch the graph of \(y = \mathrm { f } ( x )\).
   (C) The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { 3 } { 0 }\). State an equation for the resulting graph. You need not simplify your answer.
   Find the coordinates of the point at which the resulting graph crosses the \(y\)-axis.
2. Show that 3 is a root of \(x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8 = - 4\). Hence solve this equation completely, giving the other roots in surd form.

13 A circle has equation \(( x - 5 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 20\).

1. State the coordinates of the centre and the radius of this circle.
2. State, with a reason, whether or not this circle intersects the \(y\)-axis.
3. Find the equation of the line parallel to the line \(y = 2 x\) that passes through the centre of the circle.
4. Show that the line \(y = 2 x + 2\) is a tangent to the circle. State the coordinates of the point of contact.