OCR MEI C1 (Core Mathematics 1) 2006 January

\(n\) is a positive integer. Show that \(n^2 + n\) is always even. [2]

\includegraphics{figure_2} Fig. 2 shows graphs \(A\) and \(B\).

1. State the transformation which maps graph \(A\) onto graph \(B\). [2]
2. The equation of graph \(A\) is \(y = f(x)\). Which one of the following is the equation of graph \(B\)? \(y = f(x) + 2\) \quad \(y = f(x) - 2\) \quad \(y = f(x + 2)\) \quad \(y = f(x - 2)\) \(y = 2f(x)\) \quad \(y = f(x + 3)\) \quad \(y = f(x - 3)\) \quad \(y = 3f(x)\) [2]

Find the binomial expansion of \((2 + x)^4\), writing each term as simply as possible. [4]

Solve the inequality \(\frac{3(2x + 1)}{4} > -6\). [4]

Make \(C\) the subject of the formula \(P = \frac{C}{C + 4}\). [4]

When \(x^3 + 3x + k\) is divided by \(x - 1\), the remainder is 6. Find the value of \(k\). [3]

\includegraphics{figure_7} The line AB has equation \(y = 4x - 5\) and passes through the point B(2, 3), as shown in Fig. 7. The line BC is perpendicular to AB and cuts the \(x\)-axis at C. Find the equation of the line BC and the \(x\)-coordinate of C. [5]

1. Simplify \(5\sqrt{8} + 4\sqrt{50}\). Express your answer in the form \(a\sqrt{b}\), where \(a\) and \(b\) are integers and \(b\) is as small as possible. [2]
2. Express \(\frac{\sqrt{3}}{6 - \sqrt{3}}\) in the form \(p + q\sqrt{3}\), where \(p\) and \(q\) are rational. [3]

1. Find the range of values of \(k\) for which the equation \(x^2 + 5x + k = 0\) has one or more real roots. [3]
2. Solve the equation \(4x^2 + 20x + 25 = 0\). [2]

A circle has equation \(x^2 + y^2 = 45\).

1. State the centre and radius of this circle. [2]
2. The circle intersects the line with equation \(x + y = 3\) at two points, A and B. Find algebraically the coordinates of A and B. Show that the distance AB is \(\sqrt{162}\). [8]

1. Write \(x^2 - 7x + 6\) in the form \((x - a)^2 + b\). [3]
2. State the coordinates of the minimum point on the graph of \(y = x^2 - 7x + 6\). [2]
3. Find the coordinates of the points where the graph of \(y = x^2 - 7x + 6\) crosses the axes and sketch the graph. [5]
4. Show that the graphs of \(y = x^2 - 7x + 6\) and \(y = x^2 - 3x + 4\) intersect only once. Find the \(x\)-coordinate of the point of intersection. [3]

1. Sketch the graph of \(y = x(x - 3)^2\). [3]
2. Show that the equation \(x(x - 3)^2 = 2\) can be expressed as \(x^3 - 6x^2 + 9x - 2 = 0\). [2]
3. Show that \(x = 2\) is one root of this equation and find the other two roots, expressing your answers in surd form. Show the location of these roots on your sketch graph in part (i). [8]