Remainder When 19^200+23^200 Is Divided by 49
Find the remainder when 19^200+23^200 is divided by 49, by writing 19 and 23 as 21∓2 and applying the binomial theorem twice.
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Distance of Origin From Centroid — Two Sides and Orthocenter Given
Given two sides of a triangle and its orthocenter, find the distance from the origin to the centroid by locating the altitudes and the third vertex.
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cot x = −5/√11, x∈(π/2,π) — Evaluate a Multi-Angle Trig Expression
Given cot x = −5/√11 with x in the second quadrant, evaluate (sin(11x/2))(sin6x−cos6x)+(cos(11x/2))(sin6x+cos6x) by collapsing it to sin(x/2)+cos(x/2).
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2×2 Matrix A With A’=αA+I and det(A²−A)=4, Sum of All Possible α
Let A be a 2×2 matrix with real entries such that A’=αA+I, α∈R−{−1,1}. If det(A²−A)=4, find the sum of all possible values of α.
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y=(x−y dx/dy)sin(x/y), x(1)=π/2 — Find cos(x(2))
Solve the differential equation y=(x−y dx/dy)sin(x/y) with x(1)=π/2, using the substitution v=x/y to reduce it to a separable equation.
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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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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.