31.

If $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$ are the roots of the equation x³ + px² + qx + r = 0, then the coefficient of x in the cubic equation whose roots are $\mathrm{\alpha }\left(\mathrm{\beta } + \mathrm{\gamma }\right), \mathrm{\beta }\left(\mathrm{\gamma } + \mathrm{\alpha }\right) \mathrm{and} \mathrm{\gamma }\left(\mathrm{\alpha } + \mathrm{\beta }\right) \mathrm{is}$

• 2q

• q² + pr

• p² - qr

• r(pq - r)

Given that, ${\mathrm{a\alpha }}^2 + 2\mathrm{b\alpha } + \mathrm{c} \ne 0 \mathrm{and} \mathrm{that} \mathrm{the} \mathrm{system} \mathrm{of} \mathrm{equations}$

$$\begin{gathered} \left(\mathrm{a\alpha } + \mathrm{b}\right)\mathrm{x} + \mathrm{ay} + \mathrm{bz} = 0; \\ \left(\mathrm{b\alpha } + \mathrm{c}\right)\mathrm{x} +\hspace{0.17em}\mathrm{by} +\hspace{0.17em}\mathrm{cz} = 0; \\ \left(\mathrm{a\alpha } + \mathrm{b}\right)\mathrm{y} + \left(\mathrm{b\alpha } +\hspace{0.17em}\mathrm{c}\right)\mathrm{z} = 0 \\ \mathrm{has} \mathrm{a} \mathrm{non}-\mathrm{trival} \mathrm{solution}, \mathrm{then} \mathrm{a}, \mathrm{b} \mathrm{and} \mathrm{c} \mathrm{lie} \mathrm{in} \end{gathered}$$

• Arithmetic Progression

• Geometric Progression

   

• Harmonic Progression

• Arithmetico- geometric Progression

If a, b, c and d ∈ R such that a² + b² = 4 and ${\mathrm{c}}^2 + {\mathrm{d}}^2 = 4$and if (a + ib) = (c + id)² (x + iy), then x² + y² is equal to

• 4

• 3

• 2

• 1

If z is complex number such that $\left|\mathrm{z} - \frac{4}{\mathrm{z}}\right| = 2$, then the greatest value of $\left|\mathrm{z}\right|$ is

• $1 + \sqrt{2}$

• $\sqrt{2}$

• $\sqrt{3} + 1$

• $1 + \sqrt{5}$

$$\begin{gathered} \mathrm{If} \mathrm{\alpha } \mathrm{is} \mathrm{a} \mathrm{non}-\mathrm{real} \mathrm{root} \mathrm{of} \mathrm{the} \mathrm{equation} {\mathrm{x}}^6 - 1 = 0, \\ \mathrm{then} \frac{{\mathrm{\alpha }}^2 + {\mathrm{\alpha }}^3 + {\mathrm{\alpha }}^4 + {\mathrm{\alpha }}^5}{\mathrm{\alpha } + 1} = ? \end{gathered}$$

• $\mathrm{\alpha }$

• 1

• 0

• - 1

$\mathrm{The} \mathrm{minimum} \mathrm{value} \mathrm{of} 27{\tan }^2\left(\mathrm{\theta }\right) + 3{\cot }^2\left(\mathrm{\theta }\right) \mathrm{is}$

• 15

• 18

• 24

• 30

$\cos \left(36°\right) - \cos \left(72°\right) = ?$

• 1

• $\frac{1}{2}$

• $\frac{1}{4}$

• $\frac{1}{8}$

$$\begin{gathered} \mathrm{If} \tan \left(\mathrm{x}\right) + \tan \left(\mathrm{x} + \frac{\mathrm{\pi }}{3}\right) + \tan \left(\mathrm{x} + \frac{2\mathrm{\pi }}{3}\right) = 3, \\ \mathrm{then} \tan \left(3\mathrm{x}\right) = ? \end{gathered}$$

• 3

• 2

• 1

• 0

$\mathrm{If} 3\sin \left(\mathrm{x}\right) + 4\cos \left(\mathrm{x}\right) = 5, \mathrm{then} 6\tan \left(\frac{\mathrm{x}}{2}\right) - 9{\tan }^2\left(\frac{\mathrm{x}}{2}\right) = ?$

• 3

• 2

• 1

• 0

If a, b and c form a geometric progression with common ratio r, then the sum of the ordinates of the points of intersection of the line ax + by + c = 0 and the curve x + 2y² = 0 is

• $- \frac{{\mathrm{r}}^2}{2}$

• $- \frac{\mathrm{r}}{2}$

• $\frac{\mathrm{r}}{2}$

• r