OCR MEI C1 (Core Mathematics 1)

1

1. Statement P is \(a + b = 4\).
   Statement Q is \(\quad a = 1\) and \(b = 3\).
   Which one of the following is correct? $$\mathrm { P } \Rightarrow \mathrm { Q } , \quad \mathrm { P } \Leftrightarrow \mathrm { Q } , \quad \mathrm { P } \Leftarrow \mathrm { Q }$$
2. Statement R is \(\quad x = 2\). Statement S is \(\quad x ^ { 2 } = 4\). Which one of the following is correct? $$R \Rightarrow S , \quad R \Leftrightarrow S , \quad R \Leftarrow S$$

2 Find the equation of the straight line which is parallel to the line \(y = 3 x + 5\) and which goes through the point \(( 2,12 )\).

3 Find the term which has the highest coefficient in the expansion of \(( 1 + x ) ^ { 8 }\).

4 The surface area of the surface of a cylinder is given by the formula $$A = 2 \pi r ( r + h )$$ Rearrange this formula so that \(h\) is the subject.

5 Solve the following equations.

1. \(\quad 2 ^ { x } = \frac { 1 } { 8 }\).
2. \(\quad x ^ { - \frac { 1 } { 2 } } = \frac { 1 } { 4 }\)

6 Find the positive integer values of \(x\) for which $$\frac { 1 } { 2 } ( 26 - 2 x ) \geq 2 ( 3 + x )$$

7 The remainder when \(x ^ { 3 } - 2 x + 4\) is divided by ( \(x - 2\) ) is twice the remainder when \(x ^ { 2 } + x + k\) is divided by ( \(x + 1\) ).
Find the value of \(k\).

8 Find the values of \(a\) and \(b\) for which \(\frac { 4 } { ( 2 \sqrt { 3 } - 1 ) } = a + b \sqrt { 3 }\).

9 Find the coordinates of the points where the curve \(y = x ^ { 2 } - 2 x - 8\) meets the line \(y = x + 2\).

10 The diagram shows the graph of \(y = \mathrm { f } ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{4c556b8e-1a19-4480-bf2a-0ef9e67f98b4-3_507_1085_933_383} A is the minimum point of the curve at \(( 3 , - 4 )\) and B is the point \(( 5,0 )\).
On separate diagrams on graph paper, draw the graphs of the following. In each case give the coordinates of the images of the points A and B .

1. \(\quad y = \mathrm { f } ( x ) + 2\),
2. \(y = \mathrm { f } ( x + 2 )\).

11 Fig. 11 shows the graph of \(y = a x ^ { 2 } + b x + c\). \begin{figure}[h] \end{figure}

1. Explain why a must be negative.
2. State two factors of \(y = a x ^ { 2 } + b x + c\).
3. Hence, or otherwise, find the values of \(a , b\) and \(c\). Another function is given by \(y = x ^ { 2 } - 4 x + 10\).
4. Write this in completed square form.
5. Explain why the graphs of these two functions never meet.

12 The function \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = x ^ { 3 } + 6 x ^ { 2 } + 5 x - 12\).

1. Show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
2. Find the other factors of \(\mathrm { f } ( x )\).
3. State the coordinates where the graph of \(y = \mathrm { f } ( x )\) cuts the \(x\) axis. Hence sketch the graph of \(y = \mathrm { f } ( x )\).
4. On the same graph sketch also \(y = x ( x - 1 ) ( x - 2 )\) Label the two points of intersection of the two curves A and B .
5. By equating the two curves, show that the \(x\) coordinates of A and B satisfy the equation \(3 x ^ { 2 } + x - 4 = 0\).
   Solve this equation to find the \(x\)-coordinates of A and B .

13 In Fig.13, XP and XQ are the perpendicular bisectors of AC and BC respectively. \begin{figure}[h] \end{figure}

1. Find the coordinates of X .
2. Hence show that \(\mathrm { AX } = \mathrm { BX } = \mathrm { CX }\).
3. The circumcircle of a triangle is the circle which passes through the vertices of the triangle.
   Write down the equation of the circumcircle of the triangle ABC .
4. Find the coordinates of the points where the circle cuts the \(x\) axis.