Download e-book for kindle: Algebraic Threefolds. Proc. conf. Varenna, 1981 by Alberto Conte

By Alberto Conte

ISBN-10: 3540115870

ISBN-13: 9783540115878

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Additional resources for Algebraic Threefolds. Proc. conf. Varenna, 1981

Sample text

The equation z 3 + kz 2 − 4 z − 12 = 0 has roots α, β and γ. (i) Write down the values of αβ + βγ + γα and αβγ, and express k in terms of α, β and γ. [3] [4] (ii) For the case where γ = –α, solve the equation and find the value of k. [4] (iii)For the case k = 5, find a cubic equation with roots 2 – α, 2 – β, 2 – γ. 4. The cubic equation 2 z 3 + pz 2 + qz + r = 0 has roots α k , α , kα . (i) Express p, q and r in terms of k and α. (ii) Show that 2q 3 = p 3 r . [3] [3] (iii)Solve the equation for the case where p = q = -3.

4] (ii) Show that (α + β)(γ + δ) = -81. [2] (iii)Find the quadratic equation which has roots α + β and γ + δ. [3] (iv) Find α + β and γ + δ. (v) Show that α 2 − 3(1 + 10)α + 4 = 0 , and find similar quadratic equations satisfied by β, γ and δ.

3] [3] (iii)Solve the equation for the case where p = q = -3. [4] 5. The equation x 4 − 6 x3 − 73x 2 + kx + m = 0 has two positive roots, α, β and two negative roots γ, δ. It is given that αβ = γδ = 4. [5] (i) Find the values of the constants k and m. [4] (ii) Show that (α + β)(γ + δ) = -81. [2] (iii)Find the quadratic equation which has roots α + β and γ + δ. [3] (iv) Find α + β and γ + δ. (v) Show that α 2 − 3(1 + 10)α + 4 = 0 , and find similar quadratic equations satisfied by β, γ and δ.

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Algebraic Threefolds. Proc. conf. Varenna, 1981 by Alberto Conte


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