Exercise Set 3

  1. Solve the following quadratic equations by factorisation.

    1. \(x^2+10x+25\)
    2. \(x^2-28x-60=0\)
    3. \(p^2=8p-15\)
    4. \(3x^2-14x+8=0\)

  2. Solve the following quadratic equations, giving results correct to 2 d.p.

    1. \(4x^2+x-3=0\)
    2. \(x^2+x=5\)
    3. \(x+\frac{1}{x}=5\)

  3. Solve the following sets of simultaneous equations.

    1. \(3x+4y=7\), \(5x+6y=11\)
    2. \(2x+y=7\), \(x^2-xy=6\)
    3. \(x + y=2\), \(x^2-xy+y^2 = 1\)

  4. Show that the following hyperbolic identity holds: \[\cosh^2(x)-\sinh^2(x)=1\] (you will first need to find the definitions of \(\cosh\) and \(\sinh\) in terms of \(e\) - these are in the lecture notes, or use Google).

  5. Use natural logarthims to make \(t\) the subject of the formula \[V=1-e^{\frac{-t}{RC}}.\]

  6. Find the solutions to the following equations without using a calculator.

    1. \(x=\log_4(64)\)
    2. \(x=\log_{101}(101)\)
    3. \(x=\ln(e)\)
    4. \(x=\log_{56}(1)\)
    5. \(10^x=100000\)
    6. \(2^x=128\)
    7. \(\log_3(x)=4\)
    8. \(\log_x(125)=3\)

  7. Calculate the following logarithms using a calculator.

    1. \(\ln(2)\)
    2. \(\log(20)\)
    3. \(\log_{16}(100)\) - do this using your \(\log_a(x)\) button and also using the change of base rule with your \(\log\) or \(\ln\) button.

  8. Calculate \(4321\times 9876\) using logs: convert both values to logs, add, then convert back.

  9. Simplify the following expression involving logarithms. \[\log_a(x^2)+3\log_a(x)-2\log_a(4x^2).\]

  10. Solve the following for \(x\). \[2\log_a(x)-\log_a(x-1)=\log_a(x+3).\]

  11. Solve the following for \(x\). \[\log_a(x^2-10)-\log_a(x)=2\log_a(3).\]

  12. Bacteria are undergoing cell division every 30 minutes. Approximating the number of bacteria as a continuous variable \(x\), if there are initially 5 bacteria, write an equation for the number of bacteria \(x\) at time \(t\) in the form \[x=A2^{kt}\] where \(t\) is measured in minutes. Also express this in the form \[x=Ae^{\lambda t}\] by finding the appropriate value of \(\lambda\).

  13. A quantity \(x\) is increasing exponentially with respect to time \(t\). We have the measurements \(x=21\) at \(t=0\) and \(x=156\) at \(t=10\). Find an equation for \(x\) in the form \(x=Ae^{kt}\).

  14. The number of specimens of an invasive plant species is observed to be increasing at a particular location. The number of plants observed last year was \(N=52\) and this year is \(N=76\). We shall approximate the number of plants as a continuous variable \(x\).

    1. Assuming continuous exponential growth \(x=Ae^{\lambda t}\), find the growth rate \(\lambda\).
    2. Approxiately how many plants are predicted by this model after 10 years from now?

  15. Solve the following inequalities.

    1. \(7-3x>x-5\)
    2. \(x^2\ge 4\)
    3. \(x^2-x-6<0\)
    4. \(x^2-3x-12\le 2x+2\)
    5. \(\frac{2}{2x-1} > \frac{3}{3x+1}\)
    6. \(\frac{2x}{x-5}\le 3\)
    7. \(x^3-9x^2<0\)