Let α and β Be Two Roots of the Equation x² + 2x + 2 = 0. Then α¹⁵ + β¹⁵ Is Equal To
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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If the Matrix A = [[1,0,0],[0,2,0],[3,0,−1]] Satisfies A²⁰ + αA¹⁹ + βA = [[1,0,0],[0,4,0],[0,0,1]] for Real Numbers α and β, Then β − α Is Equal To
Given a triangular matrix A satisfying A²⁰+αA¹⁹+βA equals a specific matrix, find β−α using the eigenvalues on A’s diagonal.
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The Circle Which Touches the x-axis at (a, 0), a > 0, and Cuts Off an Intercept of Length b on the y-axis Is x² + y² − αx + βy + γ = 0. If It Lies Below the x-axis, the Ordered Pair (2a, b²) Is
A circle x²+y²−αx+βy+γ=0 touches the x-axis at (a,0) and cuts a y-axis intercept of length b, lying below the x-axis. Find the ordered pair (2a, b²).
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If the First Term of an Arithmetic Progression Is 3 and the Sum of the First 4 Terms Is Equal to 1/5th of the Sum of the Next 4 Terms, Then the Sum of the First 20 Terms Is
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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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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A Wheel Makes 360 Revolutions in 1 Minute. Find the Number of Radians It Turns Through in 1 Second
A wheel makes 360 revolutions in one minute. Find how many radians it turns through in 1 second, converting revolutions per minute into radians per second.
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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°.