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Announcement For most students in this forum, the interesting and lively on-going discussion of problems and solutions in these threads is sufficient extra-school mathematical study material. However, for some students who may need private, confidential one-on-one help, I am available online...

Not really difficult (like a HSC Q7 polynomial question, all bark no bite), but adds depth to your polynomial knowledge. Let P(x) be the quadratic ax^2+bx+c. Suppose that P(x)=x has unequal roots. Show that the roots are also roots of P(P(x)=x. Find a quadratic equation for the other roots of...

A particle moves in a straight line subject only to a resistive force proportional to its speed. Its speed falls from 1200 m/s to 800 m/s over 1400 m. Find the time taken to the nearest 0.01 sec. Note :if writing integrals w/ limits, use notation I{a--->b}f(x)dx integral upper limit b...

Give Spice Girl's integral notation a run, btw where's Spice? I{0--->pi/2}f(x)dx integral upper limit pi/2, lower limit 0 wrt x, and f(x)=1/(1+tan^.25 (x))

If 4 distinct points of the curve y=4x^4+14x^3+6x-10 are collinear, then their mean x-coordinates is a constant k. Find k. Trivial if you find the key!

A frictionless* frog jumps from the ground with speed V at an unknown angle to the horizontal. It swallows a fly at a height h. Show that the frog should position itself within a radius of V/g SQRT(V^2-2gh) of the point below gulp point. This problem should come with a warning and a...

a,b both positive and a+b < ab. Prove a+b > 4.

Find the sum :nC1 (1^2) + nC2 (2^2)+...+nCn (n^2)

Sorry for the long absence. :confused: offers a lame "Sydney" excuse: got sidetracked by real estate! Two months till the big exam, I'd like to help. I'll pose an elegant problem now and then. What's an elegant problem? A problem charactererized by the brevity of its statement, giving...

Yes, harder 2 unit. Through the magic of compounding, capital C becomes C(1+r)^n after n years. How much do we need to invest to be able to withdraw $1 at the end of year 1, 4 at the end of year 2, 9 at the end of year 3, 16 at the end of year 4, and so on in perpetuity? Actuaries...

P(x) is a polynomial with integral coefficients. The leading coefficient, the constant term, and P(1) are all odd. Show that P(x) has no rational roots.

I{a-->b}f(x)dx integral, upper bound b, lower bound a integrate f(x) w.r.t. x. Must give Spice Girl's Integration notation above a road test. Using sin2x=2sinxcosx or otherwise, find I{pi/2-->0} Ln(sinx)dx

Following Q has some complex number, some algebra, some trigonometry, some calculus - a good practise question. Maximize |z^3-z+2| when |z|=1.

A question worthy of q7 3U, or a definite Harder 3Unit. The chord AB is normal to the parabola x^2=4ay. Find the point A which minimizes the length of this chord.

Here's the hyperbola morph of the ellipse question 8, 1995. a)Consider the line y=mx+c and the hyperbola H, x^2/a^2-y^2/b^2=1. Show that the conditions for cutting, touching and avoiding are c^2>(am)^2-b^2, c^2=(am)^2-b^2, and c^2<(am)^2-b^2 respectively. b)The point M(X_0,Y_0) lies...

P is an arbitrary pt. on the ellipse and line L is the tangent to the ellipse at P. The pts. S' and S are the foci of the ellipse. Let S" be the reflection of S across the L. i) Prove that the focal chords through P are equally inclined. ii) Fully describe the path of S" as P moves on the...

The tangent at P(asec@,btan@) on a hyperbola meets the asymptotes at QR. Show that QR is twice the distance of the chord joining point P with the intersection of the asymptotes. Note: this question is a morph of Geha's question for the special rectangular hyperbola case ie. P(cp,c/p).

Prove that the area of the triangle formed by the tangent to the hyperbola and the asymptotes is a constant.

For those who were requesting a conic problem. An interesting but fairly easy question. Find the smallest area of the triangle formed by the tangent line to the ellipse (s-major a, s-minor b) with the coordinate axes in the first quadrant.

Anyone for harder 3Unit? Montana duck hunters area all perfect shots. Ten Montana hunters are in a duck blind when 10 ducks fly over. All 10 hunters pick a duck at random to shoot at, and all 10 hunters fire at the same time. How many ducks could be expected to escape, on average, if...