when a is max; cosa is minimum. a is max when it is right on the edge of the bowl; so height of ball is h; we take R-h to find the adjacent side of the triangle. hence R-h/R is equal to cosa at its minimum value
b)
cot\alpha = \frac{g}{w^2r} (dividing the 2 equations from part a)
=\frac{(R^2-r^2)^{0.5}}{r}=\frac{g}{w^2r} (from triangle formed by pendulum)
remove r at bottom and divide by R to get answer
c)
cos a is between a certain region; where it is equal or less than 1 or greater or equal to...
multiply weighting by mark(out of 100) for each btw; and add together, whoever has the highest score comes first:
96*15+89*25+88*20+91*40=9065
94*15+93*25+90*20+89*40=9095
that person came first im petty sure (if everything is correct); u guys are separated by a weighted average of 0.3%.
if two roots lie on unit circle and are reciprocal to each other:
(using fan96's and integral95's progress)
let x+iy denote one of these imaginary roots.
2x^2-2y^2+a^2+b^2=-3/4 if i didnt make any mistakes, from integral's working.
anyway if the root is on unit circle than x^2+y^2=1; x^2=1-y^2
a...
Re: HSC 2018 MX2 Integration Marathon
correct but the two tan signs are generally said to cancel each other out; although i understand it can vary from +-1. that being said if u look at example at the integral tanx/|tanx| it is a horizontal line being split up into positive and negative in pi/2...
ya;if u do solve it using calculations it will be nearly impossible; unless it is through induction. I still believe its a step up from the probability usually done in highschool.
a)
i) there are 5 items to be placed inside 3 boxes. how many ways can this happen. each box can contain any amount of items.
ii) if each box must contain at least one item; how many different ways are possible.
b)
i) There's 100 passengers and 100 seats in the airplane. The first...
making y the subject:
y = (-x^3+1)^1/3
factor out x^3
=(x^3)^1/3 * (-1+1/x^3)^1/3
=x (-1+1/x^3)^1/3
as x becomes larger; the part with 1/x^3 becomes smaller and the graph of y=x (-1 + 1/x^3 )^1/3 gets closer and closer to the graph y=x(-1+0)^1/3 or y=-x. of course there will be no solutions but...
Re: HSC 2018 MX2 Marathon
ya rip me. i misinterpreted what z as a variable point meant. i thought it meant z could be anything and that the line was dependent on what z was; although that doesnt make much sense once you think about it.
Re: HSC 2018 MX2 Marathon
i) the line created is bx+ay=0.since you said i) is not the locus of z; and z can be truly anything; then we can observe that bx+ay=0 only yields to purely real values; so it becomes the line y=0 on the imaginary number plane
ii) (x-p)^2+(y-q)^2= k^2
sub y=0 and...