If z is a complex number such that z = r(cosθ +isinθ), where r is real, show that arg(z+r) = (1/2)θ .
Geometrically, on an Argand diagram, you have a position vector, from origin of length r and with angle
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If z is a complex number such that z = r(cosθ +isinθ), where r is real, show that arg(z+r) = (1/2)θ .
(1-α^2)(1-β^2)(1-γ^2)(1-δ^2)
Polynomial x^4 + qx^2 +rx + s = 0 has roots α, β, γ, δ. Find the value of the constant
term in the polynomial with roots 1-α^2 , 1-β^2 , 1-γ^2 , 1- δ^2.
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(1-α^2)(1-β^2)(1-γ^2)(1-δ^2)
=(1-α)(1-β)(1-γ)(1-δ)(1+α)(1+β)(1+γ)(1+δ)
=(1+q+r+s)(1+q-r+s)
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I got (q+s+1)²-r² which is equivalent but by first finding the equation which has roots (1-α²)(1-β²)(1-γ²)(1-δ²)(1-α^2)(1-β^2)(1-γ^2)(1-δ^2)
=(1-α)(1-β)(1-γ)(1-δ)(1+α)(1+β)(1+γ)(1+δ)
=(1+q+r+s)(1+q-r+s)
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just use the roots and to find the coefficientsExpand P(x) = (x-1)(x+2)(x-3)(x+1)
And for the sake of the exercise, please don't just expand.
Using t-result substitution and partial fractions I get down to:
You're correct.Using t-result substitution and partial fractions I get down to:
2 [int 1/(t²-t+1) dt - int 1/t² +1 dt]
2 [int dt/((t-1/2)² + 3/4)] - 2[int dt/(t²+1)]
2 [2/(√3)*tan-1 (t-1/2)/(√3/2) - tan-1 t] + C
which is both unsimplified and illegible because I can't latex
Pretty good question tbh.