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 how many words can be formed from the letters of DAUGHTER such that all the vowels are not together, by subtracting the all-vowels-together count from the total.
Find the equation of a common tangent to the circle x²+y²−6x=0 and the parabola y²=4x, by writing the parabola’s tangent in slope form and applying the circle’s tangent-distance condition.
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.
In an arithmetic progression, if m times the mth term equals n times the nth term (m≠n), prove that the (m+n)th term must equal zero.
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.
Given x, y, z are unit vectors in 3D space, find the minimum value of |x+y|²+|y+z|²+|z+x|² by relating the sum to |x+y+z|² and using that a squared magnitude can’t be negative.
Find the sum of all possible values of θ ∈ [-π, 2π] for which (1+i cosθ)/(1-2i cosθ) is purely imaginary. Multiple-choice: 3π, 2π, 5π, or 4π.
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.