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Degree of a Differential Equations

Let omega= co...

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Let omega= cos 3 degree+isin 3 degree then summation(r=1 to 10)(Re(omega^(2r-1))) equals

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ω

be the complex number

c o s \frac{2 π}{3} + i s i n \frac{2 π}{3}

. Then the number of distinct complex numbers z satisfying

∣ ∣ ∣ ∣ ∣ \begin{matrix} z + 1 & ω & ω^{2} ω & z + ω^{2} & 1 ω^{2} & 1 & z + ω \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0

P (x)

be a four-degree polynomial with leading coefficient equal to

1.

P (1) = - 1,

P (2) = 1,

P (3) = 3,

P (4) = 5,

lim x \to 1 \frac{P (x) + 3 - 2 x}{(x - 1)}

f (x)

be a polynomial of degree

4

having extreme values at

x = 1

x = 2

lim x \to 0 (\frac{f (x)}{x^{2}} + 1) = 3

f (- 1)

f (x)

be a polynomial of degree

4

having extreme values at

x = 1

x = 2

{lim}_{x \to 0} (\frac{f (x)}{x^{2}} + 1) = 3

f (- 1)

1, ω

ω^{2}

be the cube roots of unity. The least possible degree of a polynomial with real coefficients, having

2 ω^{2}, 3 + 4 ω, 3 + 4 ω^{2}

5 - ω - ω^{2}

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