OCR MEI C1 (Core Mathematics 1) 2015 June

1 Make \(r\) the subject of the formula \(A = \pi r ^ { 2 } ( x + y )\), where \(r > 0\).

2 A line \(L\) is parallel to \(y = 4 x + 5\) and passes through the point \(( - 1,6 )\). Find the equation of the line \(L\) in the form \(y = a x + b\). Find also the coordinates of its intersections with the axes.

3 Evaluate the following.

1. \(200 ^ { \circ }\)
2. \(\left( \frac { 25 } { 9 } \right) ^ { - \frac { 1 } { 2 } }\)

4 Solve the inequality \(\frac { 4 x - 5 } { 7 } > 2 x + 1\).

5 Find the coordinates of the point of intersection of the lines \(y = 5 x - 2\) and \(x + 3 y = 8\).

6

1. Expand and simplify \(( 3 + 4 \sqrt { 5 } ) ( 3 - 2 \sqrt { 5 } )\).
2. Express \(\sqrt { 72 } + \frac { 32 } { \sqrt { 2 } }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.

7 Find and simplify the binomial expansion of \(( 3 x - 2 ) ^ { 4 }\).

8 Fig. 8 shows a right-angled triangle with base \(2 x + 1\), height \(h\) and hypotenuse \(3 x\). \begin{figure}[h] \end{figure} Fig. 8

1. Show that \(h ^ { 2 } = 5 x ^ { 2 } - 4 x - 1\).
2. Given that \(h = \sqrt { 7 }\), find the value of \(x\), giving your answer in surd form.

9 Explain why each of the following statements is false. State in each case which of the symbols ⇒, ⟸ or ⇔ would make the statement true.

1. ABCD is a square \(\Leftrightarrow\) the diagonals of quadrilateral ABCD intersect at \(90 ^ { \circ }\)
2. \(x ^ { 2 }\) is an integer \(\Rightarrow x\) is an integer

10 You are given that \(\mathrm { f } ( x ) = ( x + 3 ) ( x - 2 ) ( x - 5 )\).

1. Sketch the curve \(y = \mathrm { f } ( x )\).
2. Show that \(\mathrm { f } ( x )\) may be written as \(x ^ { 3 } - 4 x ^ { 2 } - 11 x + 30\).
3. Describe fully the transformation that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 11 x - 6\).
4. Show that \(\mathrm { g } ( - 1 ) = 0\). Hence factorise \(\mathrm { g } ( x )\) completely.

11 \begin{figure}[h] \end{figure} Fig. 11 shows a sketch of the circle with equation \(( x - 10 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 125\) and centre C . The points \(\mathrm { A } , \mathrm { B }\), D and E are the intersections of the circle with the axes.

1. Write down the radius of the circle and the coordinates of C .
2. Verify that B is the point \(( 21,0 )\) and find the coordinates of \(\mathrm { A } , \mathrm { D }\) and E .
3. Find the equation of the perpendicular bisector of BE and verify that this line passes through C .

12

1. Find the set of values of \(k\) for which the line \(y = 2 x + k\) intersects the curve \(y = 3 x ^ { 2 } + 12 x + 13\) at two distinct points.
2. Express \(3 x ^ { 2 } + 12 x + 13\) in the form \(a ( x + b ) ^ { 2 } + c\). Hence show that the curve \(y = 3 x ^ { 2 } + 12 x + 13\) lies completely above the \(x\)-axis.
3. Find the value of \(k\) for which the line \(y = 2 x + k\) passes through the minimum point of the curve \(y = 3 x ^ { 2 } + 12 x + 13\).