If $\mathrm{\alpha }, \mathrm{\beta }$ are the roots of the equation x² + bx + c = 0 and a$\mathrm{\alpha } + \mathrm{h}, \mathrm{\beta } + \mathrm{h}$ are the roots of the equation x² + qx + r = 0, then h is equal to

• b + q

• b - q

• $\frac{1}{2}\left(\mathrm{b} + \mathrm{q}\right)$

• $\frac{1}{2}\left(\mathrm{b} - \mathrm{q}\right)$

${20}^{2 - 3{\mathrm{x}}^2} = {\left(40\sqrt{5}\right)}^{3{\mathrm{x}}^2 - 2}$, then x is equal to

• $\pm \sqrt{\frac{3}{2}}$

• $\pm \sqrt{\frac{2}{3}}$

• $\pm \sqrt{\frac{4}{3}}$

• $\pm \sqrt{\frac{5}{4}}$

Each of the roots of the equation x³ - 6x² + 6x - 5 = 0 are increased by h. So that the new transformed equation does not contain x term, then h is equal to

• 1

• 2

• $\frac{1}{2}$

• $\frac{1}{3}$

44.

The roots of the equation x³ - 14x² + 56x - 64 = 9 are in

• AGP

• HP

• AP

• GP

If 1 is a multiple root of order 3 for the equation x⁴ - 2x³ + 2x - 1 = 0, then the other root is

• 0

• - 1

• 1

• 2

$\underset{\mathrm{x}\to 0}{\lim }\frac{\sin \left(\mathrm{x}\right){\sin }^{-1}\left(\mathrm{x}\right)}{{\mathrm{x}}^2}$ is equal to

• 0

• 1

• - 1

• $\infty$

The biquadratic equation, two of whose roots are $1 + \mathrm{i}, 1 - \sqrt{2}$, is

• x⁴ - 4x³ + 5x² - 2x - 2 = 0

• x⁴ + 4x³ - 5x² + 2x + 2 = 0

• x⁴ + 4x³ - 5x² + 2x - 2 = 0

• x⁴ + 4x³ + 5x² - 2x + 2 = 0

iF a is a complex number and b is a real number, then the equation $\overline{\mathrm{a}} + \mathrm{a} + \mathrm{b} = 0$ represents a

• straight line

• parabola

• circle

• hyperbola

$$\begin{gathered} \mathrm{If} \mathrm{\theta } = \frac{\mathrm{\pi }}{6}, \mathrm{then} \mathrm{the} 10\mathrm{th} \mathrm{term} \mathrm{of} 1 + \left(\cos \left(\mathrm{\theta }\right) + \mathrm{isin}\left(\mathrm{\theta }\right)\right) + {\left(\cos \left(\mathrm{\theta }\right) + \mathrm{isin}\left(\mathrm{\theta }\right)\right)}^2 + \\ {\left(\cos \left(\mathrm{\theta }\right) + \mathrm{isin}\left(\mathrm{\theta }\right)\right)}^3 + ...\mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• $\mathrm{i}$

• - 1

• 1

• $- \mathrm{i}$

$\frac{\sin \left(5\mathrm{\theta }\right)}{\sin \left(\mathrm{\theta }\right)} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $16co{s}^4\left(\theta \right) - 12co{s}^2\left(\theta \right)+ 1$

• $16{\cos }^4\left(\mathrm{\theta }\right)+ 12{\cos }^2\left(\mathrm{\theta }\right) + 1$

• $16{\cos }^4\left(\mathrm{\theta }\right) - 12{\cos }^2\left(\mathrm{\theta }\right) - 1$

• $16{\cos }^4\left(\mathrm{\theta }\right) + 12{\cos }^2\left(\mathrm{\theta }\right) - 1$