OCR MEI C3 (Core Mathematics 3)

1 Solve the equation \(| 3 - 2 x | = 4 | x |\).

2 Express \(1 < x < 3\) im th \(\quad | x - a | < b\), where \(a\) and \(b\) are to be determined.

3 Fig. 1 shows the graphs of \(y = | x |\) and \(y = a | x + b |\), where \(a\) and \(b\) are constants. The intercepts of \(y = a | x + b |\) with the \(x\)-and \(y\)-axes are \(( - 1,0 )\) and \(\left( 0 , \frac { 1 } { 2 } \right)\) respectively. \begin{figure}[h] \end{figure}

1. Find \(a\) and \(b\).
2. Find the coordinates of the two points of intersection of the graphs.

4 Solve the inequality \(| 2 x + 1 | \geqslant 4\).

5 Solve the equation \(| 2 x - 1 | = | x |\).
[0pt] [4]

6 Given that \(\mathrm { f } ( x ) = | x |\) and \(\mathrm { g } ( x ) = x + 1\), sketch the graphs of the composite functions \(y = \mathrm { fg } ( x )\) and \(y = \operatorname { gf } ( x )\), indicating clearly which is which.

7 Solve the inequality \(| x - 1 | < 3\).

8 Fig. 4 shows a sketch of the graph of \(y = 2 | x - 1 |\). It meets the \(x\) - and \(y\)-axes at ( \(a , 0\) ) and ( \(0 , b\) ) respectively. \begin{figure}[h] \end{figure} Find the values of \(a\) and \(b\).

9 Solve the inequality \(| 2 x - 1 | \leqslant 3\).

10 Fig. 1 shows the graphs of \(y = | x |\) and \(y = | x - 2 | + 1\). The point P is the minimum point of \(y = | x - 2 | + 1\), and Q is the point of intersection of the two graphs. \begin{figure}[h] \end{figure}

1. Write down the coordinates of P .
2. Verify that the \(y\)-coordinate of Q is \(1 \frac { 1 } { 2 }\).

11 Solve the equation \(| 3 x - 2 | = x\).

12 Solve the equation \(| 3 x + 2 | = 1\).