let $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ be the roots of the quadratic equation ax² + bx + c = 0. Observe the lists given below

The correct match of List-I from List-II is

If $\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}, \overrightarrow{\mathrm{c}} = \hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{d}} =\hat{\mathrm{i}} - \hat{\mathrm{j}} - \hat{\mathrm{k}}$, then observe the following lists

Then, correct match of List_I to List-II is

The points in the set $\left\{\mathrm{z} \in \mathrm{C} : \arg \left(\frac{\mathrm{z} - 2}{\mathrm{z} - 6\mathrm{i}}\right) = \frac{\mathrm{\pi }}{2}\right\}$ (where C denotes the set of all complex numbers) lie on the curve which is a

• circle

• pair of lines

• parabola

• hyperbola

If w is a complex cube root of unity, then $\sin \left\{\left({\mathrm{w}}^{10} + {\mathrm{w}}^{23}\right)\mathrm{\pi } - \frac{\mathrm{\pi }}{4}\right\}$ is equal to

• $\frac{1}{\sqrt{2}}$

• $\frac{1}{2}$

• 1

• $\frac{\sqrt{3}}{2}$

If m₁, m₂, m₃ and m₄ respectively denote the moduli of the complex numbers 1 + 4i, 3 + i, 1 - i and 2 - 3i, then the correct one, among the following is

• m₁ < m₂ < m₃ < m₄

• m₄ < m₃ < m₂ < m₁

• m₃ < m₂ < m₄ < m₁

• m₃ < m₁ < m₂ < m₄

$\sqrt{3}\csc \left(20°\right) - \sec \left(20°\right)$ is equal to

• 2

• $2\sin \left(20°\right) - \csc \left(40°\right)$

• 4

• $4\sin \left(20°\right) . \csc \left(40°\right)$

If A = 35°, B = 15° and C = 40°, then tan(A) · tan(B) + tan(B) . tan(C) + tan(C) . tan(A) is equal to

• 0

• 1

• 2

• 3

If $\tan \left(\mathrm{\theta }\right) + \tan \left(\mathrm{\theta } + \frac{\mathrm{\pi }}{3}\right) + \tan \left(\mathrm{\theta } + \frac{2\mathrm{\pi }}{3}\right) = 3$, then which of the following is equal to 1 ?

• $\tan \left(2\mathrm{\theta }\right)$

• $\tan \left(3\mathrm{\theta }\right)$

• ${\tan }^2\left(\mathrm{\theta }\right)$

• ${\tan }^3\left(\mathrm{\theta }\right)$

$$\begin{gathered} \mathrm{If} : \mathrm{R}\to \mathrm{R} \mathrm{and} \mathrm{g} : \mathrm{R}\to \mathrm{R} \mathrm{are} \mathrm{defined} \mathrm{by} \mathrm{f}\left(\mathrm{x}\right) = \left|\mathrm{x}\right| \mathrm{and} \\ \mathrm{g}\left(\mathrm{x}\right) = \left[\mathrm{x} - 3\right] \mathrm{for} \mathrm{x} \in \mathrm{R}, \mathrm{then} \\ \left\{\mathrm{g}\left(\mathrm{f}\left(\mathrm{x}\right)\right) : - \frac{8}{5} < \mathrm{x} < \frac{8}{5}\right\} \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• {0, 1}

• {1, 2}

• {- 3, - 2}

• {2, 3}

$$\begin{gathered} \mathrm{f} : [-6, 6] \mathrm{R} \mathrm{is} \mathrm{defined} \mathrm{by} \mathrm{f}\left(\mathrm{x}\right) = {\mathrm{x}}^2 - 3 \\ \mathrm{for} \mathrm{x} \in \mathrm{R}, \mathrm{then} \\ \left(\mathrm{fofof}\right) (-1) + \left(\mathrm{fofof}\right) \left(0\right) + \left(\mathrm{fofof}\right)\left(1\right) \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• $\mathrm{f}\left(4\sqrt{2}\right)$

• $\mathrm{f}\left(3\sqrt{2}\right)$

• $\mathrm{f}\left(2\sqrt{2}\right)$

• $\mathrm{f}\left(\sqrt{2}\right)$