Matrix A Satisfies A²⁰+αA¹⁹+βA=Given Matrix — Find β−α
Given a triangular matrix A satisfying A²⁰+αA¹⁹+βA equals a specific matrix, find β−α using the eigenvalues on A’s diagonal.
-
-
Circle Touches x-axis at (a,0), y-axis Intercept b — Find (2a, b²)
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).
-
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.
-
Three Circles of Radii a, b, c Touch Externally With x-axis as Common Tangent
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.
-
Sum of Prime Factors of |P⁻¹AP−2I| for Given 3×3 Matrices A, P
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.
-
sinx+sin²x=1: Find (cos¹²x+tan¹²x)+3(cos¹⁰x+tan¹⁰x+cos⁸x+tan⁸x)+(cos⁶x+tan⁶x)
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.
-
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.
-
Ellipse E and Hyperbola H Share Focal Distance 2√3, a−A=2 — Sum of Latus Rectums
Ellipse E and hyperbola H have the same distance between their foci (2√3), with a−A=2 and eccentricity ratio 1/3. Find the sum of their latus rectum lengths.
-
A and B Throw Dice Alternately, A Wins on Sum 5 Before B’s Sum 8
A and B alternately throw a pair of dice; A wins by throwing sum 5 before B throws sum 8. Full solution using independent events and an infinite GP.
-
Common Tangents to 4(x²+y²)=9 and y²=4x Meet at Q — Find l/e² of an Ellipse
The common tangents to 4(x²+y²)=9 and y²=4x meet at Q. An ellipse has semi-minor axis OQ and semi-major axis 6. Find l/e² for this ellipse.