Product of Rational Roots of (x²−9x+11)²−(x−4)(x−5)=3
Find the product of all rational roots of (x²−9x+11)²−(x−4)(x−5)=3 by substituting y=x²−9x+11 to reduce it to a simple quadratic in y.
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Line Through A(4,3), Slope>1, Meets x−y−2=0 at B — Find B’s Line
A line through A(4,3) with slope greater than 1 meets x−y−2=0 at B, with AB=√29/3. Find which of four given lines B lies on.
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Integers Between 100 and 1000 With Digit Sum Equal to 14
Count the 3-digit integers between 100 and 1000 whose digits sum to 14, using stars-and-bars with inclusion-exclusion for the digit-range constraints.
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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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Committee of 4 From 8 Boys, 4 Girls — P(Exactly 2 Girls | At Least 1 Girl)
A committee of 4 is chosen from 8 boys and 4 girls. Given at least one girl is selected, find the probability of exactly 2 girls, using the conditional probability formula.
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α,β Roots of x²−x+1=0 — Find α¹⁰¹+β¹⁰⁷
If α,β are the distinct complex roots of x²−x+1=0, find α¹⁰¹+β¹⁰⁷ by recognising the roots as negatives of cube roots of unity.
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If 3a+2b=5c and 8a−7b=4c, Are a,b,c Collinear and Is |a|>|b|?
Given 3a+2b=5c and 8a−7b=4c, determine whether |a|>|b| and whether a, b, c are collinear vectors, by eliminating c between the two equations.
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Sum of nCr/(r+1) From r=0 to n
A step-by-step derivation of the summation of nCr/(r+1) from r=0 to n, using the identity that turns each term into a binomial coefficient one order up.
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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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The Remainder When 2021^2023 Is Divided by 7
Find the remainder when (2021)^2023 is divided by 7 — a modular arithmetic problem solved by reducing the base first, then finding a repeating power cycle.