Given the real part of (z−2i)/(z+2i) is zero, show z traces a circle of radius 2 about the origin, then find the maximum distance from that circle to (6,8).
Prove that sin2A+sin2B−sin2C=4cosA cosB sinC when A+B+C=180°, using the sum-to-product formula twice along with the supplementary-angle identities.
Find how many complex numbers z satisfy both |z|=1 and |z/z̄+z̄/z|=1 simultaneously, by reducing the condition to x²−y²=±1/2 on the unit circle.
Given x=−4 satisfies the determinant equation |x 2 3; 1 x 1; 3 2 x|=0, expand it into a cubic and find the sum of the remaining two roots.
Find the value of the product cos(π/4)cos(π/8)···cos(π/1024)sin(π/1024) using the repeated identity sinθcosθ=½sin2θ to telescope it down.
Bag 1 has 6 green, 3 red balls; Bag 2 has 5 green, 4 red. One ball moves from Bag 1 to Bag 2, then one is drawn from Bag 2. Find the probability it’s red.
Prove that the lines (x+1)/3=(y+3)/5=(z+5)/7 and (x−2)/1=(y−4)/3=(z−6)/5 intersect, by finding parameter values that satisfy all three coordinate equations.
Find the number of complex numbers z satisfying |z−i|=|z+i|=|z−1| simultaneously, by recognising z as the circumcenter of the triangle formed by i, −i, and 1.
In parallelogram ABCD with AB=q, AD=p, and acute ∠BAD, find the vector r for the altitude from vertex B to side AD, using vector projection.
Sets A, B have m, n elements, with A’s subset count 56 more than B’s. Find m and n via prime factorisation of 2^m−2^n=56, then the distance from P(m,n) to Q(−2,−3).