OCR MEI C1 (Core Mathematics 1) 2007 June

1 Solve the inequality \(1 - 2 x < 4 + 3 x\).

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

3 The converse of the statement ' \(P \Rightarrow Q\) ' is ' \(Q \Rightarrow P\) '.
Write down the converse of the following statement.
' \(n\) is an odd integer \(\Rightarrow 2 n\) is an even integer.'
Show that this converse is false.

4 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + k x + c\). The value of \(\mathrm { f } ( 0 )\) is 6, and \(x - 2\) is a factor of \(\mathrm { f } ( x )\).
Find the values of \(k\) and \(c\).

5

1. Find \(a\), given that \(a ^ { 3 } = 64 x ^ { 12 } y ^ { 3 }\).
2. Find the value of \(\left( \frac { 1 } { 2 } \right) ^ { - 5 }\).

6 Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 - 2 x ) ^ { 5 }\).

7 Solve the equation \(\frac { 4 x + 5 } { 2 x } = - 3\).

8

1. Simplify \(\sqrt { 98 } - \sqrt { 50 }\).
2. Express \(\frac { 6 \sqrt { 5 } } { 2 + \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are integers.

9

1. A curve has equation \(y = x ^ { 2 } - 4\). Find the \(x\)-coordinates of the points on the curve where \(y = 21\).
2. The curve \(y = x ^ { 2 } - 4\) is translated by \(\binom { 2 } { 0 }\). Write down an equation for the translated curve. You need not simplify your answer.

10 The triangle shown in Fig. 10 has height \(( x + 1 ) \mathrm { cm }\) and base \(( 2 x - 3 ) \mathrm { cm }\). Its area is \(9 \mathrm {~cm} ^ { 2 }\). \begin{figure}[h] \end{figure}

1. Show that \(2 x ^ { 2 } - x - 21 = 0\).
2. By factorising, solve the equation \(2 x ^ { 2 } - x - 21 = 0\). Hence find the height and base of the triangle. Section B (36 marks)

11 \begin{figure}[h] \end{figure} A circle has centre \(C ( 1,3 )\) and passes through the point \(A ( 3,7 )\) as shown in Fig. 11.

1. Show that the equation of the tangent at A is \(x + 2 y = 17\).
2. The line with equation \(y = 2 x - 9\) intersects this tangent at the point T . Find the coordinates of T .
3. The equation of the circle is \(( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 20\). Show that the line with equation \(y = 2 x - 9\) is a tangent to the circle. Give the coordinates of the point where this tangent touches the circle.

12

1. Write \(4 x ^ { 2 } - 24 x + 27\) in the form \(a ( x - b ) ^ { 2 } + c\).
2. State the coordinates of the minimum point on the curve \(y = 4 x ^ { 2 } - 24 x + 27\).
3. Solve the equation \(4 x ^ { 2 } - 24 x + 27 = 0\).
4. Sketch the graph of the curve \(y = 4 x ^ { 2 } - 24 x + 27\).

13 A cubic polynomial is given by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\).

1. Show that \(( x - 3 ) \left( 2 x ^ { 2 } + 5 x + 4 \right) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\). Hence show that \(\mathrm { f } ( x ) = 0\) has exactly one real root.
2. Show that \(x = 2\) is a root of the equation \(\mathrm { f } ( x ) = - 22\) and find the other roots of this equation.
3. Using the results from the previous parts, sketch the graph of \(y = \mathrm { f } ( x )\).