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Test of Mathematics Solution Subjective 73 - Coefficients of a Quadratic

Test of Mathematics at the 10+2 Level This is a Test of Mathematics Solution Subjective 73 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.


Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta

Problem

Consider the equation {x^3 + Gx + H = 0}, where G and H are complex numbers. Suppose that this equation has a pair of complex conjugate roots. Show that both G and H are real.


Solution


Let three roots of the equation {x^3 + Gx + H = 0} are

{\alpha, \beta, \gamma}   [ Let {\alpha, \beta} are complex conjugates] .Now {\alpha \beta \gamma} = - H ... (i)

{\alpha + \beta +\gamma} = 0 ... (ii)

{\alpha \beta + \beta \gamma + \gamma \alpha} = G ... (iii)

From (ii) we get {\alpha + \beta + \gamma}

= 0 [ {\alpha, \beta} are complex conjugates so they are real]

{\Rightarrow} {\gamma} = real Now as {\gamma} = real

{\beta \gamma + \gamma \alpha} = {\gamma (\beta + \alpha)} ={real \times real} = {real} ... (iv) {\alpha, \beta} are complex conjugates so {\alpha \beta = real} ... (v) From (iv) & (v) we get

{\alpha \beta + \beta \gamma + \gamma \alpha} = {real + real = real} {\Rightarrow} G = real [from (iii)] Now {\alpha, \beta} is real and {\alpha, \beta} is real so {\alpha \beta \gamma} = real    {\Rightarrow} H = real.

Test of Mathematics at the 10+2 Level This is a Test of Mathematics Solution Subjective 73 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.


Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta

Problem

Consider the equation {x^3 + Gx + H = 0}, where G and H are complex numbers. Suppose that this equation has a pair of complex conjugate roots. Show that both G and H are real.


Solution


Let three roots of the equation {x^3 + Gx + H = 0} are

{\alpha, \beta, \gamma}   [ Let {\alpha, \beta} are complex conjugates] .Now {\alpha \beta \gamma} = - H ... (i)

{\alpha + \beta +\gamma} = 0 ... (ii)

{\alpha \beta + \beta \gamma + \gamma \alpha} = G ... (iii)

From (ii) we get {\alpha + \beta + \gamma}

= 0 [ {\alpha, \beta} are complex conjugates so they are real]

{\Rightarrow} {\gamma} = real Now as {\gamma} = real

{\beta \gamma + \gamma \alpha} = {\gamma (\beta + \alpha)} ={real \times real} = {real} ... (iv) {\alpha, \beta} are complex conjugates so {\alpha \beta = real} ... (v) From (iv) & (v) we get

{\alpha \beta + \beta \gamma + \gamma \alpha} = {real + real = real} {\Rightarrow} G = real [from (iii)] Now {\alpha, \beta} is real and {\alpha, \beta} is real so {\alpha \beta \gamma} = real    {\Rightarrow} H = real.

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