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If a, b, c are distinct then the value of x satisfying is

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Let be an identify in x, were a, b, c, d, e are independent of x. Then the value of e is

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Using the factor theorem it is found that a + b, b + c and c + a are three factors of the determinant . The other factor in the value of the determinant is

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If the determinant , then

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If ω is a non-real cube root of unity and n is not a multiple of 3, then is equal to