Hard Questions (1 Viewer)

[Logix]

These are from the 2003 HSC.

4b (iii) After deriving the equation of the tangent to the standard hyperbola at P(asec @, btan @), show the tangent intersects the asymptotes at A and B. Prove the area OAB is ab. (I couldn't do the proof part, the rest is alright.)

4c) A hall has n doors. Suppose that n people each choose any door at random to enter the hall.
(i) In how many ways can this be done?
(ii) What is the probability that at least one door will not be chosen by any of the people?

6b) (iii) Prove by induction that Sn>=sqrt (n!) for all integers n>=1

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These are from the 2000 HSC.

5a) Consider the polynomial
P(x) = ax^4+bx^3+cx^2+dx+e
where a,b,c,d and e are integers. Suppose @ is an integer such that P(@) = 0
(i) Prove that @ divides e.
(ii) Prove that the polynomial
Q(x) = 4x^4-x^3+3x^2+2x-3 does not have an integer root.

5b) the whole question I couldnt do.

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Not too sure how to do this one either....

Show: 1/2<1/1+t^2<1

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Thats all for now

 

[ngai]

Show: 1/2<1/1+t^2<1

i assume thats 1/(1+t^2), not (1/1) + t^2
but still..
when t = 2,
the question says that 1/2 < 1/5 < 1
hence the question is wrong

 

[Logix]

oooops..hehe
i forgot to mention that 0< t <1
my bad

 

[Archman]

0 < t < 1
0 < t^2 < 1
1<1+t^2<2
1/1<1/(1+t^2)<1/2
edit: stupid html tags

6b) (iii) Prove by induction that Sn>=sqrt (n!) for all integers n>=1

yes you said its from the 2003 hsc, but you should really type out wat Sn is.


and for the area of the triangle OAB
say A is (p,q), B is (r,s)
now the length of perpendicular from A to OB is
|(ps - rq)/sqrt(r^2 + s^2)|
now area = 1/2 * the above crap * OB
= 1/2 * .................. * sqrt(r^2 + s^2)
stuff cancels out
= 1/2 * |ps - rq|
use this formula and the question will work out.

 

[Logix]

ok thx archman,

i thought for the 1/(1+t^2) question that u do it graphically, as in graph the function, and shade in from t=0 to t=1 or something.

is it possible to do it that way?

 

[Constip8edSkunk]

ok thx archman,

i thought for the 1/(1+t^2) question that u do it graphically, as in graph the function, and shade in from t=0 to t=1 or something.

is it possible to do it that way?

u could do it that way by differentiating and showing that it is a decreasinf function from 1 to 1/2 for t = 0 to 1 but proving it directly is alot simpler and easier to write