Find the term independent of x in the expansion of [(x+1)/(x^(2/3)−x^(1/3)+1) − (x−1)/(x−x^(1/2))]^10, by simplifying both fractions using sum/difference-of-cubes and -squares factoring first.
Given the line ax+y=c is tangent to both the circle x²+y²=1 and the parabola y²=4√2x, find |c| by combining the circle’s tangent-distance condition with the parabola’s tangent-in-slope-form condition.
For an infinite GP a, ar, ar², … with Σar^n=57 and Σa³r^3n=9747, find a+18r by dividing the cubed-series sum by the cube of the original sum.
Find the greatest positive integer k such that 49^k+1 divides 1+49+49²+…+49¹²⁵, by writing the sum as a GP and factoring it as a difference of squares.
Given a(b−c)x²+b(c−a)x+c(a−b)=0 has equal roots, with a+c=5 and b=16/5, find a²+c² by recognising x=1 as a root and using the equal-roots condition to relate a, b, c.
Find the area of the region S={(x,y): x²+4x+2 ≤ y ≤ |x+2|} by substituting u=x+2 to reveal the region’s symmetry and simplify the intersection points.
Given z1,z2,z3 on |z|=1 with arg(z1)=−π/4, arg(z2)=0, arg(z3)=π/4, find α²+β² where |z1z̄2+z2z̄3+z3z̄1|²=α+β√2.
For real α, β let y_{α,β}(x) solve dy/dx+αy=xe^(βx), y(1)=1. S is the set of all such functions. Which given options belong to S? Multiple correct.
Find the sum of the squares of the modulus of complex numbers z=a+ib (a,b integers) satisfying |z−1|≤1 and |z−5|≤|z−5i|, by finding the disk and half-plane the condition defines.
Given α, β are the roots of x²−x+2=0 with Im(α)>Im(β), find α⁶+α⁴+β⁴−5α² by repeatedly reducing powers using α²=α−2.