OCR MEI C1 (Core Mathematics 1) 2006 June

1 The volume of a cone is given by the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\). Make \(r\) the subject of this formula.

2 One root of the equation \(x ^ { 3 } + a x ^ { 2 } + 7 = 0\) is \(x = - 2\). Find the value of \(a\).

3 A line has equation \(3 x + 2 y = 6\). Find the equation of the line parallel to this which passes through the point \(( 2,10 )\).

4 In each of the following cases choose one of the statements $$\mathrm { P } \Rightarrow \mathrm { Q } \quad \mathrm { P } \Leftrightarrow \mathrm { Q } \quad \mathrm { P } \Leftarrow \mathrm { Q }$$ to describe the complete relationship between P and Q .

1. P: \(x ^ { 2 } + x - 2 = 0\) Q: \(x = 1\)
2. \(\mathrm { P } : y ^ { 3 } > 1\) Q: \(y > 1\)

5 Find the coordinates of the point of intersection of the lines \(y = 3 x + 1\) and \(x + 3 y = 6\).

6 Solve the inequality \(x ^ { 2 } + 2 x < 3\).

7

1. Simplify \(6 \sqrt { 2 } \times 5 \sqrt { 3 } - \sqrt { 24 }\).
2. Express \(( 2 - 3 \sqrt { 5 } ) ^ { 2 }\) in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are integers.

8 Calculate \({ } ^ { 6 } \mathrm { C } _ { 3 }\).
Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 - 2 x ) ^ { 6 }\).

9 Simplify the following.

1. \(\frac { 16 ^ { \frac { 1 } { 2 } } } { 81 ^ { \frac { 3 } { 4 } } }\)
2. \(\frac { 12 \left( a ^ { 3 } b ^ { 2 } c \right) ^ { 4 } } { 4 a ^ { 2 } c ^ { 6 } }\)

10 Find the coordinates of the points of intersection of the circle \(x ^ { 2 } + y ^ { 2 } = 25\) and the line \(y = 3 x\). Give your answers in surd form.

12 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + 9 x ^ { 2 } + 20 x + 12\).

1. Show that \(x = - 2\) is a root of \(\mathrm { f } ( x ) = 0\).
2. Divide \(\mathrm { f } ( x )\) by \(x + 6\).
3. Express \(\mathrm { f } ( x )\) in fully factorised form.
4. Sketch the graph of \(y = \mathrm { f } ( x )\).
5. Solve the equation \(\mathrm { f } ( x ) = 12\).

13 Answer the whole of this question on the insert provided.
The insert shows the graph of \(y = \frac { 1 } { x } , x \neq 0\).

1. Use the graph to find approximate roots of the equation \(\frac { 1 } { x } = 2 x + 3\), showing your method clearly.
2. Rearrange the equation \(\frac { 1 } { x } = 2 x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac { p \pm \sqrt { q } } { r }\).
3. Draw the graph of \(y = \frac { 1 } { x } + 2 , x \neq 0\), on the grid used for part (i).
4. Write down the values of \(x\) which satisfy the equation \(\frac { 1 } { x } + 2 = 2 x + 3\).