Worksheet 00: Maths revision

Worksheet 00: Maths revision#

These exercises form the prerequisites for this module. You should complete as many of the examples you need to feel confident with the material.

Please attempt the worksheet before your tutorial. Support is available in your tutorial or in the Class Team.

  1. Write the following numbers in decimal notation. Simplify as necessary.

    a. \(1^2\) b. \(1^0\) c. \((-1)^2\) d. \(2^2\) e. \(2^0\) f. \(2^{-2}\) g. \(10^2\) h. \(-10^0\) i. \(10^{-2}\) j. \(\frac{1}{2}\) k. \(\frac{1}{3}\) l. \(\frac{7}{9}\) m. \(\frac{1}{2} \times \frac{1}{20}\) n. \(3 \times \frac{7}{9}\) o. \(9 \times 0.\overline{1}\) p. \(0.9\overline{5} \times 10^3\) q. \(93 \times 10^{-3}\)

    Note that the bar (e.g. in \(0.9\overline{5}\)) denotes a recurring decimal.

  2. Argue (or, if you can, prove) that \(0.\overline{9} = 1\).

  3. a. What is the number of significant figures needed to express each of the items in Q1 in decimal?

    b. Write the answer to items k. to p. in Q1 above to 4, 5 and 6 significant figures respectively. Remember to round your answers as appropriate.

  4. Calculate

    a. \(\sqrt{4}\) b. \(\sqrt[3]{27}\).

    How many possible solutions exists for a. and b., respectively.

  5. A rectangle is 8cm wide and 6cm high.

    a. What is the area of the rectangle?

    b. What is the length of its diagonal?

    c. Suppose that the shape is tiled with small squares with dimensions 0.5cm by 0.5cm. How many tiles are needed to cover the rectangle?

    d. Suppose the shape is tiled with small squares with width \(w\) and height \(h\) (in cm). How many tiles are needed to cover the rectangle? Give your answer as a single mathematical expression in terms of \(w\) and \(h\).

  6. a. Simplify \(3 + 5 \times 2 - \frac{4}{8} + 3 (0.25)^{1/2}\).

    b. Rearrange the expression from a. to get \(2 \times 5 - 1/2 + \sqrt{2.25} + 3\).

  7. For each of the following linear equations (i.e., equations of lines)

    a. \(y = -1.5 x + 6\) b. \(y = 1.5 x - 6\) c. \(y = 1.5\) d. \(x = 1.5\),

    • state the slope;

    • state the \(y\)-axis intercept (if it exists);

    • plot the equation on paper;

    • state if this is a function. If not, why?

  8. What is the equation of the straight line that crosses the points \((1, -1)\) and \((7, 2)\)?

  9. For each of the following cases, consider the two equations and use an algebraic method to find where the lines intersect. Confirm your answer using your graph.

    a. \(y = -1.5x + 6\) and \(y = 1.5x - 6\) b. \(y = 1.5\) and \(x = 1.5\).

  10. For each of the following nonlinear equations:

    a. \(y = -x^2 + 4\) b. \(x = -y^4 + 4\) c. \(y = x^3\),

    • name the type of equation each describes;

    • plot the equation on paper;

    • state if this is a function. If not, why?

  11. In last year’s Numerical Computation class at Poppleton University, there were \(N\) registered students. Of these \(f\), \(s\) and \(t\) students achieved a first, second and third class mark on the exam, respectively, and everyone passed on their first attempt. If the number of second class marks was twice as large as the number of third class marks, and also three times as large as the number of firsts, what fraction of students received a second class mark on the exam?

Check your understanding#

Partial solutions are available to check your understanding.

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