OCR MEI C1 (Core Mathematics 1) 2009 June

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

Make \(a\) the subject of the formula \(s = ut + \frac{1}{2}at^2\). [3]

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

Solve the inequality \(x(x - 6) > 0\). [2]

1. Calculate \(^5C_3\). [2]
2. Find the coefficient of \(x^3\) in the expansion of \((1 + 2x)^5\). [2]

Prove that, when \(n\) is an integer, \(n^3 - n\) is always even. [3]

Find the value of each of the following.

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

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

1. Express \(x^2 + 6x + 5\) in the form \((x + a)^2 + b\). [3]
2. Write down the coordinates of the minimum point on the graph of \(y = x^2 + 6x + 5\). [2]

Find the real roots of the equation \(x^4 - 5x^2 - 36 = 0\) by considering it as a quadratic equation in \(x^2\). [4]

\includegraphics{figure_11} Fig. 11 shows the line joining the points A \((0, 3)\) and B \((6, 1)\).

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

1. You are given that \(\text{f}(x) = (x + 1)(x - 2)(x - 4)\).
   1. Show that \(\text{f}(x) = x^3 - 5x^2 + 2x + 8\). [2]
   2. Sketch the graph of \(y = \text{f}(x)\). [3]
   3. The graph of \(y = \text{f}(x)\) is translated by \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\). 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. [3]
2. Show that 3 is a root of \(x^3 - 5x^2 + 2x + 8 = -4\). Hence solve this equation completely, giving the other roots in surd form. [5]

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]
2. State, with a reason, whether or not this circle intersects the \(y\)-axis. [2]
3. Find the equation of the line parallel to the line \(y = 2x\) that passes through the centre of the circle. [2]
4. Show that the line \(y = 2x + 2\) is a tangent to the circle. State the coordinates of the point of contact. [5]