Triangle ABC: y=x Bisects ∠B, AC is 2x−y=2, 2AB=BC, A=(4,6) — Find α+2β
In triangle ABC, y=x bisects angle B, AC is 2x−y=2, and 2AB=BC. Given A=(4,6) and B=(α,β), find α+2β using the angle bisector’s side-ratio property.
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Three Circles of Radii a, b, c (a < b < c) Touch Each Other Externally. If They Have the x-axis as a Common Tangent, Then
Three circles of radii a, b, c touch each other externally with the x-axis as a common tangent. Derive the relation between a, b, c using centre coordinates and distance.
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Let A = [[2,1,2],[6,2,11],[3,3,2]] and P = [[1,2,0],[5,0,2],[7,1,5]]. The Sum of the Prime Factors of |P⁻¹AP − 2I| Is Equal To
Given 3×3 matrices A and P, find the sum of the prime factors of |P⁻¹AP−2I| using the property that similar matrices share the same determinant after shifting by 2I.
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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 sin x + sin²x = 1, x ∈ (0, π/2), Then (cos¹²x + tan¹²x) + 3(cos¹⁰x + tan¹⁰x + cos⁸x + tan⁸x) + (cos⁶x + tan⁶x) Is Equal To
Given sinx+sin²x=1 for x in (0,π/2), evaluate a large trigonometric expression by first proving tanx=cosx, then spotting a binomial cube pattern.
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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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Find the Positive Square Root of 3+√5
Find the positive square root of 3+√5 by expressing it as √x+√y, splitting the equation into a rational and irrational part, and solving for x and 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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Limit of Σ(k³+6k²+11k+5)/(k+3)! From k=1 to n as n→∞
Evaluate the infinite series limit of (k³+6k²+11k+5)/(k+3)! by rewriting the numerator in terms of j=k+3 to reveal a telescoping factorial pattern.