11.

If $\frac{{\mathrm{x}}^2}{\left(2\mathrm{x} - 1\right)\left(\mathrm{x} + 2\right)\left(\mathrm{x} - 3\right)} = \mathrm{A} + \frac{\mathrm{B}}{2\mathrm{x} - 1} + \frac{\mathrm{C}}{\mathrm{x} + 2} + \frac{\mathrm{D}}{\mathrm{x} - 3}$, then A is equal to

• $\frac{1}{2}$

• $- \frac{1}{50}$

• $- \frac{8}{25}$

• $\frac{27}{25}$

$\sum _{\mathrm{n} = 1}^{\infty }\frac{2{\mathrm{n}}^2 + \mathrm{n} + 1}{\mathrm{n}!}$ is equal to

• 2e - 1

• 2e + 1

• 6e - 1

• 6e + 1

If $\left|\mathrm{a}\right| <\hspace{0.17em}1, \mathrm{b} = \sum _{\mathrm{k} = 1}^{\infty }\frac{{\mathrm{a}}^{\mathrm{k}}}{\mathrm{k}}$, then a is equal to

• $\sum _{\mathrm{k} = 1}^{\infty }\frac{{\left(- 1\right)}^{\mathrm{k}}{\mathrm{b}}^{\mathrm{k}}}{\mathrm{k}}$

• $\sum _{\mathrm{k} = 1}^{\infty }\frac{{\left(- 1\right)}^{\mathrm{k} - 1}{\mathrm{b}}^{\mathrm{k}}}{\mathrm{k}!}$

• $\sum _{\mathrm{k} = 1}^{\infty }\frac{{\left(- 1\right)}^{\mathrm{k}}{\mathrm{b}}^{\mathrm{k}}}{\left(\mathrm{k} - 1\right)!}$

• $\sum _{\mathrm{k} = 1}^{\infty }\frac{{\left(- 1\right)}^{\mathrm{k} - 1}{\mathrm{b}}^{\mathrm{k}}}{\left(\mathrm{k} + 1\right)!}$

E₁ : a + b + c = 0, if 1 is a root of ax² + bx + c = 0, E₂ : b² - a² = 2ac, if $\sin \left(\mathrm{\theta }\right), \cos \left(\mathrm{\theta }\right)$ are the roots of ax² + bx + c = 0 Which of the following is true ?

• E₁ is true, E₂ is true

• E₁ is true, E₂ is false

• E₁ is false, E₂ is true

• E₁ is false, E₂ is false

The roots of the equation x - 3x - 2 = 0 are

• - 1, - 1, 2

• - 1, 1, - 2

• - 1, 2, - 3

• - 1, - 1, - 2

$\mathrm{If} \mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma } \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} {\mathrm{x}}^3 + 2{\mathrm{x}}^2 - 3\mathrm{x} - 1 = 0 \mathrm{then} {\mathrm{\alpha }}^{- 2} + {\mathrm{\beta }}^{- 2} + {\mathrm{\gamma }}^{- 2} =$

• 12

• 13

• 14

• 15

If A + C = 2B, then $\frac{\cos \left(\mathrm{C}\right) - \cos \left(\mathrm{A}\right)}{\sin \left(\mathrm{A}\right) - \sin \left(\mathrm{C}\right)}$ is equal to

• $\cot \left(\mathrm{B}\right)$

• $\cot \left(2\mathrm{B}\right)$

• $\tan \left(2\mathrm{B}\right)$

• $\tan \left(\mathrm{B}\right)$

A + B = C $\Rightarrow$ ${\cos }^2\left(\mathrm{A}\right) + {\cos }^2\left(\mathrm{B}\right) + {\cos }^2\left(\mathrm{C}\right) - 2\cos \left(\mathrm{A}\right)\cos \left(\mathrm{B}\right)\cos \left(\mathrm{C}\right)$ is equal to

• 1

• 2

• 0

• 3

$\mathrm{If} \mathrm{m}\left[- 3, 4\right] + \mathrm{n}\left[4 - 3\right] = \left[10 - 11\right], \mathrm{then} 3\mathrm{m} + 7\mathrm{n} \mathrm{is} \mathrm{equal} \mathrm{to}$

• 3

• 5

• 10

• 1

If $\cos \left(2\mathrm{x}\right) = \left(\sqrt{2} + 1\right)\left(\cos \left(\mathrm{x}\right) - \frac{1}{\sqrt{2}}\right), \cos \left(\mathrm{x}\right) \ne \frac{1}{2} \mathrm{then} \mathrm{x} \in$

• $\left\{2\mathrm{n\pi } \pm \frac{\mathrm{\pi }}{3} : \mathrm{n} \in \mathrm{Z}\right\}$

• $\left\{2\mathrm{n\pi } \pm \frac{\mathrm{\pi }}{6} : \mathrm{n} \in \mathrm{Z}\right\}$

• $\left\{2\mathrm{n\pi } \pm \frac{\mathrm{\pi }}{2} : \mathrm{n} \in \mathrm{Z}\right\}$

• $\left\{2\mathrm{n\pi } \pm \frac{\mathrm{\pi }}{4} : \mathrm{n} \in \mathrm{Z}\right\}$