OCR MEI C1 (Core Mathematics 1) 2016 June

1 Find the value of each of the following.

1. \(3 ^ { 0 }\)
2. \(9 ^ { \frac { 3 } { 2 } }\)
3. \(\left( \frac { 4 } { 5 } \right) ^ { - 2 }\)

2 Find the coordinates of the point of intersection of the lines \(2 x + 3 y = 12\) and \(y = 7 - 3 x\).

3

1. Solve the inequality \(\frac { 1 - 2 x } { 4 } > 3\).
2. Simplify \(\left( 5 c ^ { 2 } d \right) ^ { 3 } \times \frac { 2 c ^ { 4 } } { d ^ { 5 } }\).

4 You are given that \(a = \frac { 3 c + 2 a } { 2 c - 5 }\). Express \(a\) in terms of \(c\).

5

1. Express \(\sqrt { 50 } + 3 \sqrt { 8 }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
2. Express \(\frac { 5 + 2 \sqrt { 3 } } { 4 - \sqrt { 3 } }\) in the form \(c + d \sqrt { 3 }\), where \(c\) and \(d\) are integers.

6 Find the binomial expansion of \(( 1 - 5 x ) ^ { 4 }\), expressing the terms as simply as possible.

7

1. Solve the equation \(( x - 2 ) ^ { 2 } = 9\).
2. Sketch the curve \(y = ( x - 2 ) ^ { 2 } - 9\), showing the coordinates of its intersections with the axes and its turning point.

8 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + a x + c\) and that \(\mathrm { f } ( 2 ) = 11\). The remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) is 8 . Find the values of \(a\) and \(c\).

9 Fig. 9 shows the curves \(y = \frac { 1 } { x + 2 }\) and \(y = x ^ { 2 } + 7 x + 7\). \begin{figure}[h] \end{figure}

1. Use Fig. 9 to estimate graphically the roots of the equation \(\frac { 1 } { x + 2 } = x ^ { 2 } + 7 x + 7\).
2. Show that the equation in part (i) may be simplified to \(x ^ { 3 } + 9 x ^ { 2 } + 21 x + 13 = 0\). Find algebraically the exact roots of this equation.
3. The curve \(y = x ^ { 2 } + 7 x + 7\) is translated by \(\binom { 3 } { 0 }\).
   (A) Show graphically that the translated curve intersects the curve \(y = \frac { 1 } { x + 2 }\) at only one point. Estimate the coordinates of this point.
   (B) Find the equation of the translated curve, simplifying your answer.

10 Fig. 10 shows a sketch of the points \(\mathrm { A } ( 2,7 ) , \mathrm { B } ( 0,3 )\) and \(\mathrm { C } ( 8 , - 1 )\). \begin{figure}[h] \end{figure}

1. Prove that angle ABC is \(90 ^ { \circ }\).
2. Find the equation of the circle which has AC as a diameter.
3. Find the equation of the tangent to this circle at A . Give your answer in the form \(a y = b x + c\), where \(a , b\) and \(c\) are integers.

11

1. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 5 x - 3\) with the axes.
2. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 5 x - 3\) and the line \(y = x + 3\).
3. Find the set of values of \(k\) for which the line \(y = x + k\) does not intersect the curve \(y = 2 x ^ { 2 } - 5 x - 3\). \section*{END OF QUESTION PAPER}