If 1 is a multiple root of order 3 for the equation x⁴ - 2x³ + 2x - 1 = 0, then the other root is

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182.

The biquadratic equation, two of whose roots are $1 + \mathrm{i}, 1 - \sqrt{2}$, is

• x⁴ - 4x³ + 5x² - 2x - 2 = 0

• x⁴ + 4x³ - 5x² + 2x + 2 = 0

• x⁴ + 4x³ - 5x² + 2x - 2 = 0

• x⁴ + 4x³ + 5x² - 2x + 2 = 0

To remove the second term of the equation x⁴ - 8x³ + x² - x + 3 = 0, diminish the roots of the equation by

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If $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$ are the roots of the equation x³ + 4x + 1 = 0, then ${\left(\mathrm{\alpha } + \mathrm{\beta }\right)}^{-1} + {\left(\mathrm{\beta } + \mathrm{\gamma }\right)}^{-1} + {\left(\mathrm{\gamma } + \mathrm{\alpha }\right)}^{-1}$ is equal to

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Let a $\ne$ 0 and p(x) be a polynomial of degree greater than 2. If p(x) leaves remainders a and - a when divided respectively by x + a and x - a, then the remainder when p(x) is divided by x² - a² is:

• x

• - x

• - 2x

• 2x

If the amplitude of z - 2 - 3i is $\frac{\mathrm{\pi }}{4}$, then the locus of z = x + iy is :

• x + y - 1 = 0

• x - y - 1 = 0

• x + y + 1 = 0

• x - y + 1 = 0

If w is a complex cube root of unity, then 225 + (3w + 8 w²)² + (3w² + 8w)² is equal to

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If $\frac{\left(\mathrm{x} + 1\right)}{\left(2\mathrm{x} - 1\right)\left(3\mathrm{x} + 1\right)} = \frac{\mathrm{A}}{\left(2\mathrm{x} - 1\right)} + \frac{\mathrm{B}}{\left(3\mathrm{x} +\hspace{0.17em}1\right)}$, then 16A + 9B is equal to

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If (x - 2) is a common factor of the expressions x² + ax + b and x² + cx + d, then, $\frac{\mathrm{b} - \mathrm{d}}{\mathrm{c} - \mathrm{a}}$ is equal to

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If the roots of the equation 4x³ - 12x² + 11x + k = 0 are in arithmetic progression, then k is equal to

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