OCR MEI C1 (Core Mathematics 1)

1 Simplify \(( 3 x - 1 ) \left( 2 x ^ { 2 } - 5 x + 3 \right)\).

2 Make \(l\) the subject of the formula \(T = 2 \pi \sqrt { \frac { l } { g } }\).

3 Solve the inequality \(2 x ^ { 2 } - 7 x \geq 4\).

4 Simplify the following.

1. \(x ^ { \frac { 5 } { 2 } } \times \sqrt { x }\)
2. \(12 x ^ { - 5 } \div 3 x ^ { - 2 }\)

5 The vertices of a triangle have coordinates ( 1,5 ), ( \(- 3,7\) ) and ( \(- 2 , - 1\) ).
Show that the triangle is right-angled.

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

7 Find the coordinates of the points where the line \(y = 3 x - 2\) cuts the curve \(y = x ^ { 2 } + 4 x - 8\).

8 The lines \(y = 5 x - a\) and \(y = 2 x + 18\) meet at the point ( \(7 , b\) ).
Find the values of \(a\) and \(b\).

9 The graph shows the function \(y = x ^ { 2 } + b x + c\) where \(b\) and \(c\) are constants.
The point \(\mathrm { M } ( - 3 , - 16 )\) on the graph is the minimum point of the graph.
\includegraphics[max width=\textwidth, alt={}, center]{3b6291ef-bef9-49de-a20f-591e621bed65-2_478_948_1871_588}

1. Write down the function \(y = \mathrm { f } ( x )\) in completed square form.
2. Hence find the coordinates of the points where the curve cuts the axes.

1. Write down the equations of the circles A and B .
2. Find the \(x\) coordinates of the points where the two curves intersect.
3. Find the \(y\) coordinates of these points, giving your answers in surd form.

11 In this question \(\mathrm { f } ( x ) = x ^ { 3 } - 2 x ^ { 2 } - 4 x + k\).

1. You are asked to find the values of \(k\) which satisfy the following conditions.
   (A) The graph of \(y = \mathrm { f } ( x )\) goes through the origin.
   (B) The graph of \(y = \mathrm { f } ( x )\) intersects with the \(y\) axis at ( \(0 , - 2\) ).
   (C) ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\).
   (D) The remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) is 5 .
   (E) The graph of \(y = \mathrm { f } ( x )\) is as shown in the diagram below.
   \includegraphics[max width=\textwidth, alt={}, center]{3b6291ef-bef9-49de-a20f-591e621bed65-3_373_788_2131_584}
2. Find the solution of the equation \(\mathrm { f } ( x ) = 0\) when \(k = 8\). Sketch a graph of \(y = \mathrm { f } ( x )\) in this case.

12 ABCD is a parallelogram. The coordinates of \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D are (-2, 3), (2, 4), (8, -3) and ( \(4 , - 4\) ) respectively.
\includegraphics[max width=\textwidth, alt={}, center]{3b6291ef-bef9-49de-a20f-591e621bed65-4_592_725_387_492}

1. Prove that AB and BD are perpendicular.
2. Find the lengths of AB and BD and hence find the area of the parallelogram ABCD
3. Find the equation of the line CD and show that it meets the \(y\)-axis at \(\mathrm { X } ( 0 , - 5 )\).
4. Show that the lines BX and AD bisect each other.
5. Explain why the area of the parallelogram ABCD is equal to the area of the triangle BXC.
   Find the length of BX and hence calculate exactly the perpendicular distance of C from BX .