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 Committee of 4 Students Is Selected at Random From a Group Consisting of 8 Boys and 4 Girls. Given That There Is at Least One Girl in the Committee, Find the Probability That There Are Exactly 2 Girls
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 Is
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.
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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?
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Find Vector d Perpendicular to Both b and c, With d·a = 21
Given vectors a=4i+5j−k, b=i−4j+5k, c=3i+j−k, find vector d perpendicular to both b and c such that d·a=21, by solving a system of three linear equations.
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G.P. of Increasing Positive Terms, a1a5=28 and a2+a4=29, Find a6
Let a1,a2,a3,… be a G.P. of increasing positive terms. If a1a5=28 and a2+a4=29, then a6 is equal to: (A) 628 (B) 812 (C) 526 (D) 784.