If Sm=Sn for an AP, Prove Sm+n=0
Prove that if the sum of the first m terms of an AP equals the sum of the first n terms (m≠n), then the sum of the first m+n terms must be zero.
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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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Determinant Dk Sum From k=1 to n Equals 96, Find n
A JEE Main 2023 problem: sum the determinant D_k of a given 3×3 matrix from k=1 to n, set the total to 96, and solve for n.
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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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Proof: P(A∪B) = 1 − P(A′)·P(B′) for Independent Events
A derivation of P(A∪B)=1−P(A’)P(B’) for independent events A and B, from NCERT Example 14, using the standard union and complement rules.
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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 A and B Are Symmetric Matrices of the Same Order, Show That AB Is Symmetric if and Only if AB = BA
Prove that for symmetric matrices A and B of the same order, the product AB is symmetric if and only if AB=BA, using the transpose-of-a-product rule.
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Shortest Distance Between Line y−x=1 and Curve x=y²
Find the shortest distance between the line y−x=1 and the parabola x=y² by parametrising a point on the curve and minimising the distance function.
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Sets A, B With 2 and 4 Elements — Subsets of A×B With 3+ Elements
A has 2 elements, B has 4 elements. Find the number of subsets of A×B having 3 or more elements, using the complement of small subset counts.