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(a^3+1)(b^3+1)(c^3+1) \geq (\frac{ab+bc+ca+1}{2})^3 I found this question in the JRHS 2023 paper that was upload here recently. I skipped it when I did the practise test, but when I went over it again I still couldn't get it. I asked my teacher and she hasn't gotten it as of yet. Could anyone...

If the equation has a double root, show that I tried to solve this using calculus and a bunch of simultaneous equations, individually solving for p and q to get the LHS but cannot simplify for the RHS. I've worked on this for a couple hours and will probably get it eventually but if someone...

On a certain day, the depth of water in harbour at high tide at 5 am is 9 metres. At the following low tide at 11:20am the depth is 3 metres, Find the latest time before noon that a ship can enter the habour if a minimum depth of 7.5 metres is required. I keep messing this up idk why.

a) If u is purely real b) If u moves around a unit circle no idea for both of them, a hint would be appreciated

you have to start with this: where z, w are any complex numbers And prove: My method: let Z,W be conjugates of z,w - expand RHS then if the thing is true then: square both sides: and then I'm stuck. I saw another guy do the question on stack exchange and they basically got to the...

Find the equation of the tangent to the parabola y=2x^2 at (1,2). Calculate its point of intersection with the x-axis and the volume of the solid formed when the area between the parabola, the tangent line and the x axis is revolved about the axis. got a different answer, its apparently 2pi/3...

Prove that at least one of the numbers 4a-b^2, 4b-c^2, 4c-a^2 is less than or equal to 4. I don't really know how to approach this, any help would be appreciated.

i) x^3 + px + q = 0 has a root of multiplicity 2. Show 27q^2 + 4p^3=0 ii) Find the third root I know how to do i) but I'm not sure what the answer is for ii) I managed to get +-2sqrt(p/3) as an answer but I'm pretty sure that's not what they're looking for.