The Equation of the Chord of the Ellipse x²/25 + y²/16 = 1 Whose Midpoint Is (3, 1) Is
Find the equation of the chord of the ellipse x²/25+y²/16=1 whose midpoint is (3,1), using the T=S₁ chord-of-midpoint formula.
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Two APs: s1=t1≠0, s2=2t2 — Find (s2−s1)/(t2−t1)
Two arithmetic sequences share a first term and satisfy s2=2t2. Given the sum of 10 terms of one equals the sum of 15 terms of the other, find (s2−s1)/(t2−t1).
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Triangle PQR, QA/AR=RB/BP=PC/CQ=1/2 — Area(PQR)/Area(ABC)
In triangle PQR, points A, B, C lie on QR, RP, PQ with QA/AR = RB/BP = PC/CQ = 1/2. Find the ratio of the area of triangle PQR to that of triangle ABC.
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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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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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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.
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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 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.