OCR MEI C1 (Core Mathematics 1) 2006 June

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. [3]

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

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

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

1. P: \(x^2 + x - 2 = 0\) Q: \(x = 1\) [1]
2. P: \(y^3 > 1\) Q: \(y > 1\) [1]

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

Solve the inequality \(x^2 + 2x < 3\). [4]

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

Calculate \(^6C_3\). Find the coefficient of \(x^3\) in the expansion of \((1 - 2x)^6\). [4]

Simplify the following.

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

Find the coordinates of the points of intersection of the circle \(x^2 + y^2 = 25\) and the line \(y = 3x\). Give your answers in surd form. [5]

A\((9, 8)\), B\((5, 0)\) and C\((3, 1)\) are three points.

1. Show that AB and BC are perpendicular. [3]
2. Find the equation of the circle with AC as diameter. You need not simplify your answer. Show that B lies on this circle. [6]
3. BD is a diameter of the circle. Find the coordinates of D. [3]

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

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

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 \(\frac{1}{x} = 2x + 3\), showing your method clearly. [3]
2. Rearrange the equation \(\frac{1}{x} = 2x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac{p \pm \sqrt{q}}{r}\). [5]
3. Draw the graph of \(y = \frac{1}{x} + 2\), \(x \neq 0\), on the grid used for part (i). [2]
4. Write down the values of \(x\) which satisfy the equation \(\frac{1}{x} + 2 = 2x + 3\). [2]