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 range [a,b] of f(x)=1/(2+sin3x+cos3x), then find α/β where α, β are the AM and GM of a and b, using the fact that sin3x+cos3x ranges over [−√2,√2].
Given a non-empty set X, define R on P(X), the power set of X, by ARB iff A⊆B. Determine, with justification, whether R is an equivalence relation on P(X).
A box has 4 red and 5 black marbles. Find the probability distribution of the number of red marbles in a random draw of 3, then its mean and standard deviation.
Given |z+2|=1 and Im((z+1)/(z+2))=1/5, find |Re(z+2)| by extracting y directly from the imaginary-part condition and using the circle equation for x.
Given |z|≥2, find the minimum value of |z+1/2| by writing z in polar form and minimising the distance from z to the point −1/2.
Find dy/dx where y=a^(t+1/t) and x=(t+1/t)^a, by treating u=t+1/t as an intermediate variable so the parameter t’s own derivative cancels out.
Given positive reals a,b,c in AP with abc=4, find the minimum value of b using the AM-GM inequality on a, b, c.
Find where the tangents to y=(x−2)²−1 at its points of intersection with the line x−y=3 meet, by finding the two intersection points, their slopes, and solving the tangent-line system.
Find the sum of the alternating series 1/(1·2)−1/(2·3)+1/(3·4)−… to infinity, by splitting each term via partial fractions and matching against the log(1+x) expansion.