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Given z^7 = 1 where z is not=1, deduce that z^3 + z^2 + z + 1 + 1/z + 1/z^2 + 1/z^3 = 0. How would you set out this question without assuming z is the complex root with smallest argument (cause it only says z is not=1)? Im a bit rusty on this topic so any help would be nice? thanks in advance

\int \cos^\frac{{5}}{2} (x) dx Can someone please help out with this one?

The diameter of ellipse x^2/a^2 + y^2/b^2 = 1 (a>b>0) through P(acos@, b sin@) meets circle x^2 + y^2 = a^2 at R(acos&,asin&). If the tangent to ellipse at P and tangent to the circle at R are concurrent with the right hand directrix of the ellipse, show that sec@ = 2/e. not sure where to...

A boy is standing on a hill inclined at angle @ to the horizontal. He throws the ball at angle of elevation 45 degrees and speed v m/s. If he can throw the ball 60m down the hill but only 30m up the hill, use the cartesian equation of the path y = x - (gx^2)/(v^2) to show that: tan @ = 1 -...

Glenn the bowler runs in to bowl and releases the ball 2.4m above the ground with speed 144km/h at an angle of 7 degrees below the horizontal. Take the origin to be the point where the ball is released and take g = 10m/s^2 Show that the coordinates of the ball t seconds after its initial...

A smooth piece of ice is projected up a smooth inclined surface. Its distrance up the surface at time t seconds is x = 6t - t^2. At what angle @ should the surface be inclined to the horizontal to produce these equations. the answer is 11 degrees and 47 minutes. how would you approach this...

Prove that a cyclic quadrilateral has the maximum area of all quadrilaterals with the same side lengths in hte same order. (by letting sides be a,b,c,d and letting P be the included angle of a,b and Q be included angle of c,d) I got up to the part where: dA/dP = absin(P+Q)/2sinQ = 0 P+Q =...

How do you prove Simpson's rule. I remember learning it in 3u but never actually proved it? Any ideas guys?

Find the volume of the torus with inner radius 3 and outer radius 6. This was all that was asked in the question. What the hell is a torus? Did they miss out some sort of equation or something?

need help with this question: int[1/(x^2(sqrt(x-1))] how are you meant to do these questions, lik wat to substitute? where do i look for clues? thanks

The ratio of the length of a chord of a circle to the diameter is x:1. The chord moves around the circle so that its length is unchanged (forming the locus of another circle). find the rato of hte areas of the 2 circles. my answer to the question is (1 - x^2): 1 the real answer is 0.25(4 -...

Why is int [1/sqrt(x^2 +a^2)]dx = ln |x+sqrt(x^2 +a^2)| a standard integral. Is there like a proper way to derive it other than differentiating the RHS.

A particle is projected with speed "V" m/sec from a height of "a" metres above a horizontal plane at an angle of elevation of @ degrees to the horizontal. If the range is "R", prove that: (sec^2@)(R^2) - 2(V^2/g)(tan@)R - (2aV^2)/g = 0

Find the gradient of the tangent to the curve at y=f<SUP>-1</SUP>(x) at the point (11,-1) where f(x)=x^3 - 12x. i keep gettin the wrong answer for some reason. its meant to be -1/9. can someone help out?

How would you set out the integration of: int [6/(49+25x^2)]dx Are we meant to use the standard forms or use substitutions?

Five families have 3 children each. Find the probability that at least one of these families has 3 boys. The answer's 0.487 but how do you get it?

Need help with 2 questions: (i) If n is a positive integer, use the bionmial theorem to prove (5+sqrt11)^n +(5-sqrt11)^n is an integer. (ii) Use bionomial expansions and the binomial theroem to find the value of (0.99)^13 to 5 siginificant figures. btw, is there a better way to do the last...

Prove that: (i) (n+1)P(r) = (n)P(r) + r[(n+1)P(r)] (ii) (n)P(r) = (n-2)P(r) + 2r[(n-2)P(r-1)] + r(r-1)[(n-2)P(r-2)] i have no idea on how to get around this question. plz help. thanks!

Find the locus of z such that Im[(z+1)/(z+i)]=Re[(z+1)/(z+i)] The answer's meant 2 be x^2 + y^2 = 1, but im getting (x+1)^2 + (y+1)^2 =1. Can someone confirm the right answer plz?

"show that the chord of contact of tangents from the point (a, -a) to the parabola x^2 = 4ay has length 5a" Can someone confirm whether theres a typo, cause i got 4a as the length. thanks