Complex Numbers and Quadratic Equations

If $\left|\mathrm{z} + 4\right| \ge 3$ then the greatest and the least value of $\left|\mathrm{z} + 1\right|$ are

• - 1, 6

• 6, 0

• 6, 3

• None of these

If the area of the triangle on the complex plane formed by the points z, z + iz and iz is 200, then the value of 3lz I must be equal to

• 20

• 40

• 60

• 80

A cmplex number z is such that arg$\left(\frac{\mathrm{z} - 2}{\mathrm{z} + 2}\right) = \frac{\mathrm{z}}{3}$. The points representing this complex number will lie on

• an ellipse

• a parabola

• a circle

• a straight line

If $\left|{\mathrm{x}}^2 - \mathrm{x} - 6\right| = \mathrm{x} + 2$, then the values of x are

• - 2, 2, - 4

• - 2, 2, 4

• 3, 2, - 2

• 4, 4, 3

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are the roots of x² - ax + b = 0 and if ${\mathrm{\alpha }}^{\mathrm{n}} \mathrm{and} {\mathrm{\beta }}^{\mathrm{n}}$ = Vⁿ, then

• V_{n + 1} = aV_n + bV_{n - 1}

• V_{n + 1} = aV_n + aV_{n - 1}

• V_{n + 1} = aV_n - bV_{n - 1}

• V_{n + 1} = aV_{n - 1} + bV_n

The number of integral values of K, for which the equation $7\cos \left(\mathrm{x}\right) + 5\sin \left(\mathrm{x}\right) = 2\mathrm{K} + 1$ has a solution

• 4

• 8

• 10

• 12

If the complex numbers z₁, z₂ and z₃ are in AP, then they lie on a

• circle

• parabola

• line

• ellipse

The number of real solutions of the equation

$\left(\frac{9}{10}\right) = - 3 + \mathrm{x} - {\mathrm{x}}^2$ is

• 0

• 1

• 2

• None of these

If one root is square of the other root of the equation x² + px + q = 0 , then the relations between p and q is

• p³ - (3p - 1)q + q² = 0

• p³ - q(3p + 1) + q² = 0

• p³ + (3p - 1)q + q² = 0

• p³ + (3p + 1)q + q² = 0

The coefficient of x⁵³ in the following expansions

$\sum _{\mathrm{m} = 0}^{100}{}^{100}\mathrm{C}_{\mathrm{m}}{\left(\mathrm{x} - 3\right)}^{100 - \mathrm{m}} . 2^{\mathrm{m}}$ is

• ${}^{100}\mathrm{C}_{47}$

• ${}^{100}\mathrm{C}_{53}$

• - ${}^{100}\mathrm{C}_{53}$

• - ${}^{100}\mathrm{C}_{100}$