OCR MEI C1 (Core Mathematics 1)

1 Solve the inequality \(2 ( x - 3 ) < 6 x + 15\).

2 Make \(r\) the subject of \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\).

3 In each case, choose one of the statements $$\mathbf { P } \Rightarrow \mathbf { Q } \quad \mathbf { P } \Leftarrow \mathbf { Q } \quad \mathbf { P } \Leftrightarrow \mathbf { Q }$$ to describe the complete relationship between P and Q .

1. For \(n\) an integer: P: \(n\) is an even number
   Q: \(n\) is a multiple of 4
2. For triangle ABC : P: \(\quad \mathrm { B }\) is a right-angle
   Q: \(\quad \mathrm { AB } ^ { 2 } + \mathrm { BC } ^ { 2 } = \mathrm { AC } ^ { 2 }\)

4 Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 2 + 3 x ) ^ { 5 }\).

5 Find the value of the following.

1. \(\left( \frac { 1 } { 3 } \right) ^ { - 2 }\)
2. \(16 ^ { \frac { 3 } { 4 } }\)

6 The line \(L\) is parallel to \(y = - 2 x + 1\) and passes through the point \(( 5,2 )\).
Find the coordinates of the points of intersection of \(L\) with the axes.

7 Express \(x ^ { 2 } - 6 x\) in the form \(( x - a ) ^ { 2 } - b\).
Sketch the graph of \(y = x ^ { 2 } - 6 x\), giving the coordinates of its minimum point and the intersections with the axes.

8 Find, in the form \(y = m x + c\), the equation of the line passing through \(\mathrm { A } ( 3,7 )\) and \(\mathrm { B } ( 5 , - 1 )\).
Show that the midpoint of AB lies on the line \(x + 2 y = 10\).

11 A cubic polynomial is given by \(\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 10 x + 8\).

1. Show that \(( x - 1 )\) is a factor of \(\mathrm { f } ( x )\). Factorise \(\mathrm { f } ( x )\) fully.
   Sketch the graph of \(y = f ( x )\).
2. The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { - 3 } { 0 }\). Write down an equation for the resulting graph. You need not simplify your answer.
   Find also the intercept on the \(y\)-axis of the resulting graph.