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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205.

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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$\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$ are the roots of the equation x³ - 10x² + 7x + 8=0. Match the following and choose the correct answer.

$$\begin{gathered} A. \mathrm{\alpha } + \mathrm{\beta } + \mathrm{\gamma } \left(1\right) - \frac{43}{7} \\ \mathrm{B}. {\mathrm{\alpha }}^2 + {\mathrm{\beta }}^2 + {\mathrm{\gamma }}^2 \left(2\right) - \frac{7}{8} \\ \mathrm{C}. \frac{1}{\mathrm{\alpha }} + \frac{1}{\mathrm{\beta }} + \frac{1}{\mathrm{\gamma }} \left(3\right) 86 \\ \mathrm{D}. \frac{\mathrm{\alpha }}{\mathrm{\beta \gamma }} + \frac{\mathrm{\beta }}{\mathrm{\gamma \alpha }} + \frac{\mathrm{\gamma }}{\mathrm{\alpha \beta }} \left(4\right) 0 \\ \left(10\right) \end{gathered}$$

If f(x) is apolynomial ofdegreen with rational coefficients and 1 + 2i , $2 - \sqrt{3}$ and 5 are three roots of f(x) = 0 , then the least value of n is

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Match the following elements of $\left[\begin{array}{ccc}1& - 1& 0\\ 0& 4& 2\\ 3& - 4& 6\end{array}\right]$ with their co-factors and choose the correct answer

$$\begin{gathered} \mathrm{Element} \mathrm{Co}-\mathrm{factor} \\ \mathrm{A} - 1 \left(1\right) - 2 \\ \mathrm{B} 1 \left(2\right) 32 \\ \mathrm{C} 3 \left(3\right) 4 \\ \mathrm{D} 6 \left(4\right) 6 \\ \left(5\right) - 6 \end{gathered}$$