Let $\mathrm{\alpha }, \mathrm{\beta }$ be the roots of the equation x² - ax + b = 0 and ${\mathrm{A}}_{\mathrm{n}} = {\mathrm{\alpha }}^{\mathrm{n}} + {\mathrm{\beta }}^{\mathrm{n}}$. Then, A_{n + 1} - aA_n + bA_{n - 1} is equal to

• - a

• b

• 0

• a - b

If b² $\ge$ 4ac for the equation ax⁴ + bx² + c = 0, then all the roots of the equation will be real if:

• b > 0, a < 0, c > 0

• b < 0, a > 0, c > 0

• b > 0, a > 0, c > 0

• b > 0, a > 0, c < 0

If x > 0 and log₃(x) + ${\log }_3\left(\sqrt{\mathrm{x}}\right) + {\log }_3\left(\sqrt[4]{\mathrm{x}}\right) + {\log }_3\left(\sqrt[8]{\mathrm{x}}\right) + {\log }_3\left(\sqrt[16]{\mathrm{x}}\right)$ = 4, then x equals:

• 9

• 81

• 1

• 27

The number of real roots of the equation ${\left(\mathrm{x} + \frac{1}{\mathrm{x}}\right)}^3 + \mathrm{x} + \frac{1}{\mathrm{x}} = 0$ is :

• 0

• 2

• 4

• 6

The imaginary part of $\frac{{\left(1 +\hspace{0.17em}\mathrm{i}\right)}^2}{\mathrm{i}\left(2\mathrm{i} - 1\right)}$ is :

• $\frac{4}{5}$

• 0

• $\frac{2}{5}$

• $- \frac{4}{5}$

If x = $\frac{1}{2}\left(\sqrt{3} + \mathrm{i}\right)$, then x³ is equal to

• 1

• - 1

• i

• - i

If the complex numbers sin(x) + icos(2x) and cos(x) - isin(2x) are complex conjugate to each other, then the value of x is

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{8}$

• $\frac{3\mathrm{\pi }}{4}$

• None of these

If the equation (a² + 4a + 3)x² + (a² - a - 2).x + a(a + 1) = 0 has more than two roots, then values of a is

• 0

• 1

• - 1

• None of the above

239.

If A(z₁), B(z₂), C(z₃) and P(z) represent complex numbers such that $\left|{\mathrm{z}}_1 - \mathrm{z}\right| = \left|{\mathrm{z}}_2 - \mathrm{z}\right| = \left|{\mathrm{z}}_3 - \mathrm{z}\right|$, then, A, B, C lies on

• a straight line

• a circle

• a parabola

• an ellipse

If the complex numbers z, z, and origin form vertices of an equilateral triangle, then the value of z₁² + z₂² will be

• z₁z₂

• z₁ + z₂

• 2z₁z₂

• z₁ - z₂