Sets A, B With 2 and 4 Elements — Subsets of A×B With 3+ Elements
A has 2 elements, B has 4 elements. Find the number of subsets of A×B having 3 or more elements, using the complement of small subset counts.
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Roots of x²+2x+2=0 — Find α¹⁵+β¹⁵
If α, β are the roots of x²+2x+2=0, find α¹⁵+β¹⁵ by converting the complex roots to polar form and using De Moivre’s theorem.
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AP: First Term 3, Sum of First 4 Terms = 1/5 of Next 4 — Sum of 20 Terms
An AP has first term 3, and the sum of its first 4 terms equals 1/5 of the sum of the next 4 terms. Find the sum of the first 20 terms.
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2×2 Matrix With Entries 0 or 1 — Variance of the Determinant
A 2×2 matrix has entries from {0,1}. Let X be its determinant. Find the variance of X by listing all 16 matrices and their determinant probabilities.
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16-Square Board: Probability Two Random Squares Share No Side
A 4×4 board has 16 squares. Two are chosen at random. Find the probability they share no side, by counting adjacent pairs directly.
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|z|≤1: Minimum Value of |z+1/2(3+4i)|
If z is a complex number with |z|≤1, find the minimum value of |z+1/2(3+4i)| by interpreting it as the distance from a fixed point to the unit disk.
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Vector d Perpendicular to Both a and b, With c·d=15
Given a=i+4j+2k, b=3i−2j+7k, c=2i−j+4k, find vector d perpendicular to both a and b with c·d=15, using the cross product a×b.
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The Value of 2sin(12°) − sin(72°)
Evaluate 2sin(12°)−sin(72°) by splitting 72° as 60°+12°, then recombining into a single sine of a difference angle using sin18°.
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Sum of Cubes of All Roots of x⁴−3x³−2x²+3x+1=0
Find α³+β³+γ³+δ³ for the roots of x⁴−3x³−2x²+3x+1=0 by factoring the quartic into two quadratics, avoiding solving for the roots directly.
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sec²(tan⁻¹α)+cosec²(cot⁻¹β)=36, α+β=8, α<β — Find α²+β
Given sec²(tan⁻¹α)+cosec²(cot⁻¹β)=36 and α+β=8 with α<β, find α²+β by converting to a system of equations in α+β and αβ.