Four Coplanar Points With Position Vectors a,b,c,d, Which Identity Holds
If four distinct points with position vectors a,b,c,d are coplanar, which scalar triple product identity must hold? A worked derivation from the coplanarity condition.
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Ellipse E1 With Minor Axis Equal to Next Ellipse’s Major Axis, Sum of Areas
Let E1: x²/9+y²/4=1. Ellipses Ei share E1’s centre and eccentricity, with minor axis of Ei equal to major axis of Ei+1. Find (5/π)ΣAi.
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Domain of log5(18x−x²−77) Is (α,β), Find α²+β²+γ²
If the domain of log5(18x−x²−77) is (α,β) and the domain of log(x−1)[(2x²+3x−2)/(x²−3x−4)] is (γ,δ), then α²+β²+γ² is: A)195 B)179 C)186 D)174.
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If Σr²(³⁰Cᵣ)²/(³⁰Cᵣ₋₁)=α·2²⁹ From r=1 to 30, Find α
A JEE Main 2025 binomial coefficient summation simplified by rewriting the ratio of consecutive coefficients, then splitting into two telescoping sums.
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If I = ∫96x²cos²x/(1+eˣ)dx = π(απ²+β), Find (α+β)²
A JEE Main 2025 definite integral solved using the King’s property trick for limits symmetric about zero, avoiding a direct, messier computation.
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Let A=[0 -2,2 0], M=ΣA²ᵏ, N=ΣA²ᵏ⁻¹ (k=1 to 10): Find MN²
Find MN² for a 2×2 matrix A where M and N are sums of its even and odd powers. Solved by spotting that A² is a scalar multiple of the identity.
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Let A, B=A−I, ω=(√3i−1)/2: Count n With Aⁿ+(ωB)ⁿ=A+B
Given A^n + (ωB)^n = A + B for ω a complex cube root of unity, how many n in 1..100 satisfy it? JEE Main 2022 matrix and complex numbers problem.
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If sin²10°sin20°sin40°sin50°sin70°=α−1/16sin10°, Find 16+1/α
A five-term trigonometric product from JEE Main 2022, reduced step by step using complementary angles and the sin a cos a identity.
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Position Vectors 4p+q-3r, -5p+q+2r, 2p-q+2r: Find α+2β+5γ
A JEE Main 2025 vector geometry problem solved using the Euler line property that the centroid divides the orthocenter-circumcenter segment 2:1.
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Two Ellipses, Eccentricity 1/√3: Area of Quadrilateral ABCD
Two ellipses with the same eccentricity, a given latus rectum product, and a known focal distance intersect at four points. A JEE Main 2025 area problem, solved.