Hey there Sign in to join this conversationNew here? Join for free

Step II 1999, q2

Announcements Posted on
Study Help needs new mods! 14-04-2014
Post on TSR and win a prize! Find out more... 10-04-2014
    • Thread Starter
    • 1 follower
    Offline

    ReputationRep:
    I've done the question, but I have no idea how many marks I would get because of formality/rigour/correct use of signs etc.

    If someone could give me some general guidance & marks out of 20 I would appreciate it hugely. It's just so I don't walk naively into the exam in about 2 months and lose huge marks on silly things, cheers.






    Here is a given solution:

    nx^2+2x\sqrt{pn^2+q}+rn+s=0 p\not=r, p>0\ and\ n\in \Bbb{Z}^{+}

    i) Where p=3 q=50 r=2 and s=15 find the set of values for n where the equation above has no real roots.

    For non-real roots we need 4(pn^2+q)-4n(rn+s)<0 \Leftrightarrow pn^2+q<n^2r+ns \Leftrightarrow n^2(p-r)-ns+q<0
    Inserting the given values of p, q, r and s:
    n^2-15n+50<0 \Leftrightarrow (n-\frac{15}{2})^2-\frac{25}{4}<0\Rightarrow \pm (n-\frac{15}{2})<\frac{5}{2}
    Solving for the positive case we have n<10 and the negative case -n<\frac{1}{2}(5-15) \Leftrightarrow n>5
    I.e. the equation lacks real roots when n>5 and n<10

    ii) Prove that if p<r and 4q(p-r)&gt;s^2 then the above equation has no real roots fo any n.

    Recall from i) that for non-real roots n^2(p-r)-ns+q&lt;0
    Solving for n gives n= \frac{s^2\pm\sqrt{s^2-4q(p-r)}}{2(p-r)} and \sqrt{s^2-4q(p-r)} is obviously complex when 4q(p-r)&gt;s^2 and thus there are no real solutions for any n.


    iii) Find when the above equation has real roots when n=1, (p-r)=1 and q=\frac{s^2}{8}
    For real roots (p-r)n^2-ns+q\geq0, and with the above values this transfers to finding where 1-s+\frac{s^2}{8}\geq0 \Leftrightarrow (\frac{s}{2\sqrt{2}}-\sqrt{2})^2\geq1 \Rightarrow \pm(\frac{s}{2\sqrt{2}}-\sqrt{2})\geq1
    Solving the positive case gives s\geq2\sqrt{2}(1+\sqrt{2})=4+2\s  qrt{2} as desired and the negative case gives -s\geq2\sqrt{2}(1-\sqrt{2})=2\sqrt{2}-4\Leftrightarrow s\geq4-2\sqrt{2} as desired.

Reply

Submit reply

Register

Thanks for posting! You just need to create an account in order to submit the post
  1. this can't be left blank
    that username has been taken, please choose another Forgotten your password?

    this is what you'll be called on TSR

  2. this can't be left blank
    this email is already registered. Forgotten your password?

    never shared and never spammed

  3. this can't be left blank

    6 characters or longer with both numbers and letters is safer

  4. this can't be left empty
    your full birthday is required
  1. By completing the slider below you agree to The Student Room's terms & conditions and site rules

  2. Slide the button to the right to create your account

    Slide to join now Processing…

    You don't slide that way? No problem.

Updated: May 9, 2012
Article updates
Reputation gems:
You get these gems as you gain rep from other members for making good contributions and giving helpful advice.