αx+βy=109 Is Chord of x²/9+y²/4=1 With Midpoint (5/2,1/2), Find α+β
Given αx+βy=109 is the equation of a chord of x²/9+y²/4=1 with midpoint (5/2,1/2), find α+β using the T=S₁ chord-of-midpoint formula.
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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.
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Circle x²+y²=8x Meets Hyperbola x²/9−y²/4=1 at A,B — Centroid of PAB
A and B are intersection points of a circle and hyperbola; P moves on 2x−3y+4=0. Find the line on which the centroid of triangle PAB always lies.
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Binomial (1+x)^(n+4): 5th, 6th, 7th Coefficients in AP, Find Largest Coefficient
For some n≠10, the 5th, 6th and 7th coefficients of (1+x)^(n+4) are in AP. Find n, then the largest coefficient of the expansion.
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AP: a1+(a5+a10+…+a2020)+a2024=2233, Find Sum to 2024 Terms
For an AP a1,…,a2024 with a1+(a5+a10+…+a2020)+a2024=2233, find a1+a2+…+a2024 by summing the selected terms as their own smaller AP.
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S Is 10 Distinct Primes, A = Products of Two or More Elements, Find n(P)
S is a set of 10 distinct primes and A is the set of products of two or more elements of S. Find n(P) where P={(x,y): x∈S, y∈A, x divides y}.
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Midpoint of a Chord of the Ellipse x²/9+y²/4=1 Is (√2, 4/3), Find α
If the midpoint of a chord of the ellipse x²/9+y²/4=1 is (√2, 4/3), and the chord’s length is 2√α/3, then α is: (1) 18 (2) 22 (3) 26 (4) 20.
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3 Girls and 4 Boys in a Queue, B1 and B2 Not Adjacent to Each Other
In a group of 3 girls and 4 boys, there are two boys B1 and B2. The number of ways they can stand in a queue with girls together, boys together, but B1, B2 not adjacent.
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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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Is f(x)=(2^x−2^−x)/(2^x+2^−x) One-One, Onto, Both or Neither?
The function f:(−∞,∞)→(−∞,1), defined by f(x)=(2^x−2^−x)/(2^x+2^−x), is: neither one-one nor onto, onto but not one-one, both, or one-one but not onto?