Given Pn=α^n−β^n for roots of x²−ax−b=0, and values of P3 through P6, find a and b using a linear recurrence, then compute |α⁴+β⁴|.
Find point P on the parabola y²=8x at minimum distance from the centre C of x²+(y+6)²=1, then find the equation of the circle through C centred at P.
An ellipse centred at (1,0) has latus rectum 1/2 and its minor axis subtends 60° at the foci. Find the square of the sum of its major and minor axis lengths.
Two circles in the first quadrant with radii r₁, r₂ touch both coordinate axes, each cutting a 2-unit intercept on x+y=2. Find r₁²+r₂²−r₁r₂.
Evaluate the sum 1²·15C1+2²·15C2+…+15²·15C15 using the standard identity for Σr²·nCr, then find m+n+k from its prime factorisation.
Roots α, β of 2x²+cosθ·x−1=0 depend on θ∈(0,2π). Find 16(M+m) where m, M are the minimum and maximum values of α⁴+β⁴ over all θ.
For A=[1,−2,α;α,2,−1] and B=[2,α;−1,2;4,−5], find the absolute value of the sum of all complex α for which det(AB)=0, by computing AB directly.
A committee of 4 is chosen from 8 boys and 6 girls, with at least one girl. Find the probability the committee has more girls than boys, using conditional probability.
If z1, z2 are distinct complex numbers with |z1−2z2|/|1/2−z1z̄2|=2, determine which circle condition z1 and z2 must satisfy, by expanding both sides.
If α, β are roots of 2z²−3z−2i=0, find 16·Re(E)·Im(E) where E=(α^19+β^19+α^11+β^11)/(α^15+β^15), using a Newton’s-identity shortcut.