OCR MEI C1 (Core Mathematics 1) 2011 January

1 Find the equation of the line which is parallel to \(y = 5 x - 4\) and which passes through the point (2, 13). Give your answer in the form \(y = a x + b\).

2

1. Write down the value of each of the following.
   (A) \(4 ^ { - 2 }\)
   (B) \(9 ^ { 0 }\)
2. Find the value of \(\left( \frac { 64 } { 125 } \right) ^ { \frac { 4 } { 3 } }\).

3 Simplify \(\frac { \left( 3 x y ^ { 4 } \right) ^ { 3 } } { 6 x ^ { 5 } y ^ { 2 } }\).

4 Solve the inequality \(5 - 2 x < 0\).

5 The volume \(V\) of a cone with base radius \(r\) and slant height \(l\) is given by the formula $$V = \frac { 1 } { 3 } \pi r ^ { 2 } \sqrt { l ^ { 2 } - r ^ { 2 } }$$ Rearrange this formula to make \(l\) the subject.

6 Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \(( 2 - 3 x ) ^ { 5 }\), simplifying each term.

7

1. Express \(\frac { 81 } { \sqrt { 3 } }\) in the form \(3 ^ { k }\).
2. Express \(\frac { 5 + \sqrt { 3 } } { 5 - \sqrt { 3 } }\) in the form \(\frac { a + b \sqrt { 3 } } { c }\), where \(a , b\) and \(c\) are integers.

8 Find the coordinates of the point of intersection of the lines \(x + 2 y = 5\) and \(y = 5 x - 1\).

9 Fig. 9 shows a trapezium ABCD , with the lengths in centimetres of three of its sides. \begin{figure}[h] \end{figure} This trapezium has area \(140 \mathrm {~cm} ^ { 2 }\).

1. Show that \(x ^ { 2 } + 2 x - 35 = 0\).
2. Hence find the length of side AB of the trapezium.

10 Select the best statement from $$\begin{aligned} & \mathbf { P } \Rightarrow \mathbf { Q }
& \mathbf { P } \Leftarrow \mathbf { Q }
& \mathbf { P } \Leftrightarrow \mathbf { Q } \end{aligned}$$ none of the above
to describe the relationship between P and Q in each of the following cases.

1. P : WXYZ is a quadrilateral with 4 equal sides
   \(\mathrm { Q } : \mathrm { WXYZ }\) is a square
2. P: \(n\) is an odd integer Q : \(\quad ( n + 1 ) ^ { 2 }\) is an odd integer
3. P: \(n\) is greater than 1 and \(n\) is a prime number Q : \(\sqrt { n }\) is not an integer

11 The points \(A ( - 1,6 ) , B ( 1,0 )\) and \(C ( 13,4 )\) are joined by straight lines.

1. Prove that the lines AB and BC are perpendicular.
2. Find the area of triangle ABC .
3. A circle passes through the points A , B and C . Justify the statement that AC is a diameter of this circle. Find the equation of this circle.
4. Find the coordinates of the point on this circle that is furthest from B .

12

1. You are given that \(\mathrm { f } ( x ) = ( 2 x - 5 ) ( x - 1 ) ( x - 4 )\).
   (A) Sketch the graph of \(y = \mathrm { f } ( x )\).
   (B) Show that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 20\).
2. You are given that \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 40\).
   (A) Show that \(\mathrm { g } ( 5 ) = 0\).
   (B) Express \(\mathrm { g } ( x )\) as the product of a linear and quadratic factor.
   (C) Hence show that the equation \(\mathrm { g } ( x ) = 0\) has only one real root.
3. Describe fully the transformation that maps \(y = \mathrm { f } ( x )\) onto \(y = \mathrm { g } ( x )\).

13 \begin{figure}[h] \end{figure} Fig. 13 shows the curve \(y = x ^ { 4 } - 2\).

1. Find the exact coordinates of the points of intersection of this curve with the axes.
2. Find the exact coordinates of the points of intersection of the curve \(y = x ^ { 4 } - 2\) with the curve \(y = x ^ { 2 }\).
3. Show that the curves \(y = x ^ { 4 } - 2\) and \(y = k x ^ { 2 }\) intersect for all values of \(k\).