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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Out of 16 Squares on a Board, Two Squares Are Chosen at Random. The Probability That They Have No Side in Common Is
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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Let a = i + 4j + 2k, b = 3i − 2j + 7k and c = 2i − j + 4k. Find a Vector d Which Is Perpendicular to Both a and b, and Such That 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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If sec²(tan⁻¹α) + cosec²(cot⁻¹β) = 36 and α + β = 8 With α < β, Then α² + β Is
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.
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The Product of All the Rational Roots of the Equation (x² − 9x + 11)² − (x − 4)(x − 5) = 3 Is Equal To
Find the product of all rational roots of (x²−9x+11)²−(x−4)(x−5)=3 by substituting y=x²−9x+11 to reduce it to a simple quadratic in y.
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A Line With Slope Greater Than One Passes Through the Point A(4, 3) and Intersects the Line x − y − 2 = 0 at the Point B. If the Length of AB Is √29/3, Then B Also Lies on the Line
A line through A(4,3) with slope greater than 1 meets x−y−2=0 at B, with AB=√29/3. Find which of four given lines B lies on.
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Integers Between 100 and 1000 With Digit Sum Equal to 14
Count the 3-digit integers between 100 and 1000 whose digits sum to 14, using stars-and-bars with inclusion-exclusion for the digit-range constraints.