Practice questions

Various practice questions will be posted here as the semester progresses. These questions are from previous exams, assignments and tutorials. No solutions will be posted, so you should validate your answers using any appropriate software.

Tutorial 3, 2009

• Convert into decimal representation: (a) \((10011101)_2\); (b) \((0.001101)_2\)
• Convert into binary representation: (a) \((45.625)_{10}\); (b) \((0.1)_{10}\)
• Consider the following system of linear algebraic equations.
  • Use Gauss elimination (without pivoting) to solve these equations for \((x_1,x_2,x_3)\).
  • Validate your solution by comparing it to the one obtained with computer software.

\[ \begin{align*} \left\{\begin{array}{rcl} 2 x_1 -2x_2 +4 x_3 & = & 0 \\ x_1 -3 x_2 + 4x_3 & = & -1 \\ 3x_1 - x_2 +5x_3 &= & 0 \end{array}\right. \end{align*} \]

Tutorial 4, 2009

• We have already seen several elimination techniques for linear equation solving. In this question, you are to solve the following linear algebraic equations, using LU decomposition

\[\begin{align*} \left\{\begin{array}{rcl} 2 x_1 -2x_2 +4 x_3 & = & 0 \\ x_1 -3 x_2 + 4x_3 & = & -1 \\ 3x_1 - x_2 +5x_3 &= & 0 \end{array}\right. \end{align*} \]

• We have also seen that the LU decomposition technique can be used to calculate matrix inverses. Using the results obtained in the question above, compute the inverse of the following matrix,

\[ \begin{align*} A &= \left[ \begin{array}{rrr} 2 & -2 & 4 \\ 1 & -3 & 4 \\ 3 & -1 & 5 \end{array} \right] \end{align*} \]

Tutorial 5, 2009

In addition to elimination methods, we have seen that iterative methods can be used to solve systems of linear equations. In this question, you are to solve the following linear algebraic equations, using (a) the Jacobi method, (b) the Gauss-Seidel method and (c) the relaxed Gauss-Seidel method, with \(\omega=0.5\).

In each case, start from the initial guess \(x_1^{(0)}=x_2^{(0)}=x_3^{(0)}=0\) and perform 3 full iterations. Can anything be said regarding the convergence/divergence of these methods for that equation system? \[ \begin{align*} \left\{\begin{array}{rcl} 2 x_1 -2x_2 +4 x_3 & = & 0 \\ x_1 -3 x_2 + 4x_3 & = & -1 \\ 3x_1 - x_2 +5x_3 &= & 0 \end{array}\right. \end{align*} \]

Midterm, 2009

Grading: 5 points out of 22, 2 hour exam

Consider a mixing tank fed with two fluids A and B at volumetric flow rates \(F_{\sf A}\) and \(F_{\sf B}\) and temperatures \(T_{\sf A}\) and \(T_{\sf B}\), respectively. Both fluids have the same constant density \(\rho\) and specific heat capacity \(C_p\). The tank has one outlet stream and constant fluid volume \(V\). Moreover, the fluid in the tank looses heat to the environment at a rate \(q=k(T-T_w)\), where \(T\) denotes the current fluid temperature, \(T_w<T\) is the constant tank wall temperature, and \(k\) is a constant.

1. Derive a dynamical balance describing the evolution of the liquid temperature in the tank.
2. Can the liquid temperature in the tank at steady state exit be higher than both \(T_A\) and \(T_B\)? Explain.

Grading: 5 points out of 22, 2 hour exam

Consider the reaction \[{\rm P_2I_4} + n\,{\rm P_4} + p\,{\rm H_2O} \longrightarrow 4\,{\rm PH_4I} + q\,{\rm H_3PO_4} \] where \(n, p\) and \(q\) denote the stoichiometric coefficients for \(\rm P_4\), \(\rm H_2O\) and \(\rm H_3PO_4\) respectively.

• Derive the equations necessary to solve for \(n, p\) and \(q\) by equating atoms of \({\rm P}\), \({\rm H}\), and \({\rm O}\) on the reactant and product sides.
• Use Gauss elimination with partial pivoting to compute the solution. Show clearly all steps.