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.
Points A=2i+3j, B=pi+9j, C=i−j are collinear. Find p by setting the vectors AB and AC parallel using the 2D cross-product (determinant) condition.
Find the equation of the chord of the ellipse x²/25+y²/16=1 whose midpoint is (3,1), using the T=S₁ chord-of-midpoint formula.
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.
Two dice are rolled once. Find the probability of getting an even number on the first die or a total of 8, using the addition rule for probability.
Two arithmetic sequences share a first term and satisfy s2=2t2. Given the sum of 10 terms of one equals the sum of 15 terms of the other, find (s2−s1)/(t2−t1).
In triangle PQR, points A, B, C lie on QR, RP, PQ with QA/AR = RB/BP = PC/CQ = 1/2. Find the ratio of the area of triangle PQR to that of triangle ABC.
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.
Find the remainder when 19^200+23^200 is divided by 49, by writing 19 and 23 as 21∓2 and applying the binomial theorem twice.