2 Hints

As per usual, here’s where you’ll find the problem sheet hints!

  1. The techniques involved in this one are similar to those used in Tutorial Question 1 (and there were some questions on this last week too!)
    1. Firstly, note that this is an if and only if statement, so there are two things to prove! You can get most of the way between both statements by completing the square on the polynomial. For the “\(\Rightarrow\)” direction, remember square numbers are non-negative. For the “\(\Leftarrow\)” direction, make a specific choice of \(x\).
    2. Follow the hint here, and be careful when collecting terms together.
  3. Take cases on \(x\).
  4. Without loss of generality (WLOG), consider \(x \geq y\) (otherwise you can just swap them), and consider \(\lvert \sqrt{x} - \sqrt{y} \rvert^2\). On expanding, try and find a bound for the ‘middle’ term.
  5. Solve the modulus equation, and then use your solutions to formulate simultaneous equations for \(c\) and \(r\).
  6. You should only need the definitions given in lectures to solve this question. Make sure to write things logically!