OCR MEI C1 (Core Mathematics 1) 2012 June

Find the equation of the line with gradient \(-2\) which passes through the point \((3, 1)\). Give your answer in the form \(y = ax + b\). Find also the points of intersection of this line with the axes. [3]

Make \(b\) the subject of the following formula. $$a = \frac{3}{5}b^2c$$ [3]

1. Evaluate \(\left(\frac{1}{5}\right)^{-2}\). [2]
2. Evaluate \(\left(\frac{8}{27}\right)^{\frac{2}{3}}\). [2]

Factorise and hence simplify the following expression. $$\frac{x^2 - 9}{x^2 + 5x + 6}$$ [3]

1. Simplify \(\frac{10\sqrt{6}}{3}{\sqrt{24}}\). [3]
2. Simplify \(\frac{1}{4 - \sqrt{5}} + \frac{1}{4 + \sqrt{5}}\). [2]

1. Evaluate \(^5C_3\). [1]
2. Find the coefficient of \(x^3\) in the expansion of \((3 - 2x)^5\). [4]

Find the set of values of \(k\) for which the graph of \(y = x^2 + 2kx + 5\) does not intersect the \(x\)-axis. [4]

The function \(f(x) = x^4 + bx + c\) is such that \(f(2) = 0\). Also, when \(f(x)\) is divided by \(x + 3\), the remainder is \(85\). Find the values of \(b\) and \(c\). [5]

Simplify \((n + 3)^2 - n^2\). Hence explain why, when \(n\) is an integer, \((n + 3)^2 - n^2\) is never an even number. Given also that \((n + 3)^2 - n^2\) is divisible by \(9\), what can you say about \(n\)? [4]

\includegraphics{figure_10} Fig. 10 is a sketch of quadrilateral ABCD with vertices A \((1, 5)\), B \((-1, 1)\), C \((3, -1)\) and D \((11, 5)\).

1. Show that \(AB = BC\). [3]
2. Show that the diagonals AC and BD are perpendicular. [3]
3. Find the midpoint of AC. Show that BD bisects AC but AC does not bisect BD. [5]

A cubic curve has equation \(y = f(x)\). The curve crosses the \(x\)-axis where \(x = -\frac{1}{2}\), \(-2\) and \(5\).

1. Write down three linear factors of \(f(x)\). Hence find the equation of the curve in the form \(y = 2x^3 + ax^2 + bx + c\). [4]
2. Sketch the graph of \(y = f(x)\). [3]
3. The curve \(y = f(x)\) is translated by \(\begin{pmatrix} 0 \\ -8 \end{pmatrix}\). State the coordinates of the point where the translated curve intersects the \(y\)-axis. [1]
4. The curve \(y = f(x)\) is translated by \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\) to give the curve \(y = g(x)\). Find an expression in factorised form for \(g(x)\) and state the coordinates of the point where the curve \(y = g(x)\) intersects the \(y\)-axis. [4]

\includegraphics{figure_12} Fig. 12 shows the graph of \(y = \frac{-1}{x - 3}\).

1. Draw accurately, on the copy of Fig. 12, the graph of \(y = x^2 - 4x + 1\) for \(-1 < x < 5\). Use your graph to estimate the coordinates of the intersections of \(y = \frac{-1}{x - 3}\) and \(y = x^2 - 4x + 1\). [5]
2. Show algebraically that, where the curves intersect, \(x^3 - 7x^2 + 13x - 4 = 0\). [3]
3. Use the fact that \(x = 4\) is a root of \(x^3 - 7x^2 + 13x - 4 = 0\) to find a quadratic factor of \(x^3 - 7x^2 + 13x - 4\). Hence find the exact values of the other two roots of this equation. [5]