Trace Algorithm or Flowchart

6 A student is tracing the following algorithm.
Line 10 Let \(A = 6\)
Line \(20 \quad\) Let \(B = 7\)
Line 30 Input \(C\)
Line 40 Let \(D = ( A + B ) / 2\)
Line \(50 \quad\) Let \(E = C - D ^ { 3 }\)
Line 60 If \(E ^ { 2 } < 1\) then go to Line 120
Line 70 If \(E > 0\) then go to Line 100
Line 80 Let \(B = D\)
Line 90 Go to Line 40
Line \(100 \quad\) Let \(A = D\)
Line 110 Go to Line 40
Line 120 Stop

1. Trace the algorithm in the case where the input value is \(C = 300\).
2. The algorithm is intended to find the approximate cube root of any input number. State two reasons why the algorithm is unsatisfactory in its present form.
   (3 marks)

3 An incomplete algorithm is specified in Fig. 3.
\(\mathrm { f } ( \mathrm { x } ) = \mathrm { x } ^ { 2 } - 2\)
Initial values: \(\mathrm { L } = 0 , \mathrm { R } = 2\).
Step 1 Compute \(\mathrm { M } = \frac { \mathrm { L } + \mathrm { R } } { 2 }\).
Step 2 Compute \(\mathrm { f } ( \mathrm { M } )\).
Step 3 If \(\mathrm { f } ( \mathrm { M } ) < 0\) change the value of L to that of M .
Otherwise change the value of \(R\) to that of \(M\).
Step 4 Go to Step 1. \section*{Fig. 3}

1. Apply two iterations of the algorithm.
2. After 10 iterations \(\mathrm { L } = 1.414063 , \mathrm { R } = 1.416016 , \mathrm { M } = 1.416016\) and \(\mathrm { f } ( \mathrm { M } ) = 0.005100\). Say what the algorithm achieves.
3. Say what is needed to complete the algorithm.

1 The flowchart below has positive inputs \(X , Y\) and \(M\).
\includegraphics[max width=\textwidth, alt={}, center]{74b6f747-7045-4902-8b21-0b59c007f7f6-2_1274_643_392_242}

1. Trace through the flowchart above using the inputs \(X = 1 , Y = 2\) and \(M = 2\). You only need to record values when they change.
2. Explain why the process in the flowchart is finite.

3. \begin{figure}[h] \end{figure} An algorithm for finding the positive real root of the equation \(8 x ^ { 4 } + 5 x - 12 = 0\) is described by the flow chart shown in Figure 2.

1. Use the flow chart, with \(a = 1\), to complete the table in the answer book, stating values to at least 6 decimal places. Give the final output correct to 5 decimal places. Given that the value of the input \(a\) is a non-negative real number,
2. determine the set of values for \(a\) that cannot be used to find the positive real root of \(8 x ^ { 4 } + 5 x - 12 = 0\) using this flow chart.