OCR MEI C1 (Core Mathematics 1) 2010 June

Find the equation of the line which is parallel to \(y = 3x + 1\) and which passes through the point with coordinates \((4, 5)\). [3]

1. Simplify \((5a^2b)^3 \times 2b^4\). [2]
2. Evaluate \(\left(\frac{1}{16}\right)^{-1}\). [1]
3. Evaluate \((16)^{\frac{1}{2}}\). [2]

Make \(y\) the subject of the formula \(a = \frac{\sqrt{y} - 5}{c}\). [3]

Solve the following inequalities.

1. \(2(1 - x) > 6x + 5\) [3]
2. \((2x - 1)(x + 4) < 0\) [2]

1. Express \(\sqrt{48} + \sqrt{27}\) in the form \(a\sqrt{3}\). [2]
2. Simplify \(\frac{5\sqrt{7}}{3 - \sqrt{2}}\). Give your answer in the form \(\frac{b + c\sqrt{7}}{d}\). [3]

You are given that • the coefficient of \(x^3\) in the expansion of \((5 + 2x^2)(x^3 + kx + m)\) is 29, • when \(x^3 + kx + m\) is divided by \((x - 3)\), the remainder is 59. Find the values of \(k\) and \(m\). [5]

Expand \((1 + \frac{1}{2}x)^4\), simplifying the coefficients. [4]

Express \(5x^2 + 20x + 6\) in the form \(a(x + b)^2 + c\). [4]

Show that the following statement is false. $$x - 5 = 0 \Leftrightarrow x^2 = 25$$ [2]

1. Solve, by factorising, the equation \(2x^2 - x - 3 = 0\). [3]
2. Sketch the graph of \(y = 2x^2 - x - 3\). [3]
3. Show that the equation \(x^2 - 5x + 10 = 0\) has no real roots. [2]
4. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = 2x^2 - x - 3\) and \(y = x^2 - 5x + 10\). Give your answer in the form \(a \pm \sqrt{b}\). [4]

\includegraphics{figure_11} Fig. 11 shows the line through the points A \((-1, 3)\) and B \((5, 1)\).

1. Find the equation of the line through A and B. [3]
2. Show that the area of the triangle bounded by the axes and the line through A and B is \(\frac{32}{3}\) square units. [2]
3. Show that the equation of the perpendicular bisector of AB is \(y = 3x - 4\). [3]
4. A circle passing through A and B has its centre on the line \(x = 3\). Find the centre of the circle and hence find the radius and equation of the circle. [4]

You are given that \(f(x) = x^3 + 6x^2 - x - 30\).

1. Use the factor theorem to find a root of \(f(x) = 0\) and hence factorise \(f(x)\) completely. [6]
2. Sketch the graph of \(y = f(x)\). [3]
3. The graph of \(y = f(x)\) is translated by \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\). Show that the equation of the translated graph may be written as $$y = x^3 + 3x^2 - 10x - 24.$$ [3]