OCR MEI C1 (Core Mathematics 1) 2011 June

1 Solve the inequality \(6 ( x + 3 ) > 2 x + 5\).

3

1. Evaluate \(\left( \frac { 9 } { 16 } \right) ^ { - \frac { 1 } { 2 } }\).
2. Simplify \(\frac { \left( 2 a c ^ { 2 } \right) ^ { 3 } \times 9 a ^ { 2 } c } { 36 a ^ { 4 } c ^ { 12 } }\).

4 The point \(\mathrm { P } ( 5,4 )\) is on the curve \(y = \mathrm { f } ( x )\). State the coordinates of the image of P when the graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of

1. \(y = \mathrm { f } ( x - 5 )\),
2. \(y = \mathrm { f } ( x ) + 7\).

5 Find the coefficient of \(x ^ { 4 }\) in the binomial expansion of \(( 5 + 2 x ) ^ { 6 }\).

6 Expand \(( 2 x + 5 ) ( x - 1 ) ( x + 3 )\), simplifying your answer.

7 Find the discriminant of \(3 x ^ { 2 } + 5 x + 2\). Hence state the number of distinct real roots of the equation \(3 x ^ { 2 } + 5 x + 2 = 0\).

8 Make \(x\) the subject of the formula \(y = \frac { 1 - 2 x } { x + 3 }\).

9 A line \(L\) is parallel to the line \(x + 2 y = 6\) and passes through the point \(( 10,1 )\). Find the area of the region bounded by the line \(L\) and the axes.

10 Factorise \(n ^ { 3 } + 3 n ^ { 2 } + 2 n\). Hence prove that, when \(n\) is a positive integer, \(n ^ { 3 } + 3 n ^ { 2 } + 2 n\) is always divisible by 6 .

11

1. Find algebraically the coordinates of the points of intersection of the curve \(y = 4 x ^ { 2 } + 24 x + 31\) and the line \(x + y = 10\).
2. Express \(4 x ^ { 2 } + 24 x + 31\) in the form \(a ( x + b ) ^ { 2 } + c\).
3. For the curve \(y = 4 x ^ { 2 } + 24 x + 31\),
   (A) write down the equation of the line of symmetry,
   (B) write down the minimum \(y\)-value on the curve.

12 \begin{figure}[h] \end{figure} Fig. 12 shows the graph of \(y = \frac { 4 } { x ^ { 2 } }\).

1. On the copy of Fig. 12, draw accurately the line \(y = 2 x + 5\) and hence find graphically the three roots of the equation \(\frac { 4 } { x ^ { 2 } } = 2 x + 5\).
2. Show that the equation you have solved in part (i) may be written as \(2 x ^ { 3 } + 5 x ^ { 2 } - 4 = 0\). Verify that \(x = - 2\) is a root of this equation and hence find, in exact form, the other two roots.
3. By drawing a suitable line on the copy of Fig. 12, find the number of real roots of the equation \(x ^ { 3 } + 2 x ^ { 2 } - 4 = 0\).

13 \begin{figure}[h] \end{figure} Fig. 13 shows the circle with equation \(( x - 4 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 16\).

1. Write down the radius of the circle and the coordinates of its centre.
2. Find the \(x\)-coordinates of the points where the circle crosses the \(x\)-axis. Give your answers in surd form.
3. Show that the point \(\mathrm { A } ( 4 + 2 \sqrt { 2 } , 2 + 2 \sqrt { 2 } )\) lies on the circle and mark point A on the copy of Fig. 13. Sketch the tangent to the circle at A and the other tangent that is parallel to it.
   Find the equations of both these tangents.