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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Let A = {z ∈ C : |(z+1)/(z−1)| < 1} and B = {z ∈ C : arg((z−1)/(z+1)) = 2π/3}. Then A ∩ B Is
Let A: |(z+1)/(z−1)|<1 and B: arg((z−1)/(z+1))=2π/3 in the complex plane. Find A∩B by converting both conditions into geometric loci.
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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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Let S Be a Set of 10 Distinct Primes and A the Set of Products of Two or More Elements of S. If P = {(x,y): x ∈ S and y ∈ A and y Is Divisible by x}, Then n(P) Is Equal To
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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If the Midpoint of a Chord of the Ellipse x²/9 + y²/4 = 1 Is (√2, 4/3) and the Length of the Chord Is 2√α/3, Then α Is
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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Evaluate the Integral of cos x/[(4 + sin²x)(5 − 4cos²x)] dx
Step-by-step solution to the integral of cos(x)/[(4+sin²x)(5−4cos²x)] dx, using a sine substitution and partial fractions to reduce it to two arctangent terms.
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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 Monthly Incomes of Aryan and Babban Are in the Ratio 3:4 and Their Monthly Expenditures Are in the Ratio 5:7. If Each Saves ₹15,000 per Month, Find Their Monthly Incomes Using the Matrix Method
The monthly incomes of Aryan and Babban are in the ratio 3:4 and expenditures in the ratio 5:7. If each saves ₹15,000, find their incomes using the matrix method.
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f(x)=2x/√(1+9x²) Composed 10 Times, Find √(3α+1)
Consider f:R→R defined by f(x)=2x/√(1+9x²). If (f∘f∘⋯∘f, 10 times)(x)=2^10x/√(1+9αx²), find the value of √(3α+1).