Let z₁ , z₂ and z₃ be non-zero complex numbers satisfying

z² = iz, where i = $\sqrt{ - 1}$.

Consider the following statements :

1. z₁z₂z₃ is purely imaginary
2. z₁z₂ + z₂z₃ + z₃z₁ is purely real

Which of the above statements is/are correct ?

• 1 only

• 2 only

• Both 1 and 2

• Neither 1 nor 2

Let, z be a complex number satisfying

$\left|\frac{\mathrm{z} - 4}{\mathrm{z} - 8}\right| = 1 \mathrm{and} \left|\frac{\mathrm{z}}{\mathrm{z} - 2}\right| = \frac{3}{2}$

What is $\left|\mathrm{z}\right| = ?$

• 6

• 12

• 18

• 36

Let, z be a complex number satisfying

$\left|\frac{\mathrm{z} - 4}{\mathrm{z} - 8}\right| = 1 \mathrm{and} \left|\frac{\mathrm{z}}{\mathrm{z} - 2}\right| = \frac{3}{2}$

What is $\left|\frac{\mathrm{z} - 6}{\mathrm{z} + 6}\right| = ?$

• 3

• 2

• 1

• 0

Let $\mathrm{\alpha } \mathrm{and} \mathrm{\beta } \left(\mathrm{\alpha } < \mathrm{\beta }\right)$ be the roots of the equation x² + bx + c = 0, where, b > 0 and c $<$ 0

Consider the following :

$$\begin{gathered} 1. \mathrm{\beta } < - \mathrm{\alpha } \\ 2. \mathrm{\beta } < \left|\mathrm{\alpha }\right| \end{gathered}$$

Which of the above is/are correct ?

• 1 only

• 2 only

• both 1 and 2

• neither 1 nor 2

45.

Let $\mathrm{\alpha } \mathrm{and} \mathrm{\beta } \left(\mathrm{\alpha } < \mathrm{\beta }\right)$ be the roots of the equation x² + bx + c = 0, where, b > 0 and c $<$ 0

Consider the following :

$$\begin{gathered} 1. \mathrm{\alpha } + \mathrm{\beta } + \mathrm{\alpha \beta } > 0 \\ 2. {\mathrm{\alpha }}^2\mathrm{\beta } + {\mathrm{\beta }}^2\mathrm{\alpha } > 0 \end{gathered}$$

Which of the above is/are correct ?

• 1 only

• 2 only

• both 1 and 2

• neither 1 nor 2

If one root of the equation (l - m)x² + lx+ 1 = 0 is double the other and l is real, then what is the greatest value of m ?

• $- \frac{9}{8}$

• $\frac{9}{8}$

• $- \frac{8}{9}$

• $\frac{8}{9}$

Suppose ${\mathrm{\omega }}_1 \mathrm{and} {\mathrm{\omega }}_2$, are two distinct cube roots of unity different from 1. Then what is ${\left({\mathrm{\omega }}_1 - {\mathrm{\omega }}_2\right)}^2$ equal to ?

• 3

• 1

•  - 1

•  - 3

Let  $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ be the roots of the equation :

${\mathrm{x}}^2 - \left(1 - 2{\mathrm{a}}^2\right)\mathrm{x} + \left(1 - 2{\mathrm{a}}^2\right) = 0$

Under what condition does the above equation have real roots ?

• ${\mathrm{a}}^2 < \frac{1}{2}$

• ${\mathrm{a}}^2 > \frac{1}{2}$

• ${\mathrm{a}}^2 \le \frac{1}{2}$

• ${\mathrm{a}}^2 \ge \frac{1}{2}$

Let  $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ be the roots of the equation :

${\mathrm{x}}^2 - \left(1 - 2{\mathrm{a}}^2\right)\mathrm{x} + \left(1 - 2{\mathrm{a}}^2\right) = 0$

Under what condition $\frac{1}{{\mathrm{\alpha }}^2} + \frac{1}{{\mathrm{\beta }}^2} < 1 ?$

• ${\mathrm{a}}^2 < \frac{1}{2}$

• ${\mathrm{a}}^2 > \frac{1}{2}$

• ${\mathrm{a}}^2 > 1$

• ${\mathrm{a}}^2 \in \left(\frac{1}{3}, \frac{1}{2}\right) \mathrm{only}$

What is $\sqrt{\frac{1 + {\mathrm{\omega }}^2}{1 + \mathrm{\omega }}}$ = ?, where $\mathrm{\omega }$ is cube root of unity

• 1

• $\mathrm{\omega }$

• $\mathrm{\omega }$²

• None of the above