sequences and series

The lines 2x + 3y = 6 , 2x + 3y = 8 cut the X-axis at A and B, respectively. A line L drawn through the point (2, 2) meets the X-axis as C in such away that abscissae of A, B and C are in arithmetic progression. Then, the equation of the line L is

• 2x + 3y = 10

• 8x + 2y = 10

• 2x - 3y = 10

• 8x - 2y = 10

If the polar of a point on the circle x² + y² = p² with respect to the circle x² + y² = q² touches the circle x² + y² = r, then p, q, r are in

• AP

• GP

• HP

• AGP

$1 + \frac{1}{4} + \frac{1}{4} . \frac{3}{8} + \frac{1}{4} . \frac{3}{8} . \frac{5}{12 } + . . . \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\sqrt{2}$

• $\frac{1}{\sqrt{2}}$

• $\sqrt{3}$

• $\frac{1}{\sqrt{3}}$

$\frac{2}{2!} + \frac{2 + 4}{3!} + \frac{2 + 4 + 6}{4!} + ...\mathrm{is} \mathrm{equal} \mathrm{to}$

• e

• ${\mathrm{e}}^{ - 1}$

• ${\mathrm{e}}^{ - 2}$

• ${\mathrm{e}}^{ - 3}$

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

• AGP

• HP

• AP

• GP

$1 + \frac{1 + 2}{2!} + \frac{1 + 2 + 2^2}{3!} + ...$ is equal to

• e² + e

• e²

• e² - 1

• e² - e

If the altitude of a triangle are in arithmatic progression, then the sides of the triangles are in

• AP

• HP

• GP

• AGP

If t_n = $\frac{1}{4}\left(\mathrm{n} + 2\right)\left(\mathrm{n} + 3\right)$ for n = 1, 2, 3 ..., then $\frac{1}{{\mathrm{t}}_1} + \frac{1}{{\mathrm{t}}_2} + ... + \frac{1}{{\mathrm{t}}_{2003}}$ is equal to

• $\frac{4006}{3006}$

• $\frac{4003}{3007}$

• $\frac{4006}{3008}$

• $\frac{4006}{3009}$

$\frac{1}{2!} + \frac{1 + 2}{3!} + \frac{{\displaystyle 1 + 2 +3}}{4!} + ... \mathrm{is} \mathrm{equal} \mathrm{to} :$

• $\frac{\mathrm{e}}{2}$

• $\frac{\mathrm{e}}{3}$

• $\frac{\mathrm{e}}{4}$

• $\frac{\mathrm{e}}{5}$

The value of the series ${\mathrm{xlog}}_{\mathrm{e}}\left(\mathrm{a}\right) + \frac{{\mathrm{x}}^3}{3!}{\left({\log }_{\mathrm{e}}\left(\mathrm{a}\right)\right)}^3$ + $\frac{{\mathrm{x}}^5}{5!}{\left({\log }_{\mathrm{e}}\left(\mathrm{a}\right)\right)}^5$ + ... is

• $\cosh \left({\mathrm{xlog}}_{\mathrm{e}}\left(\mathrm{a}\right)\right)$

• $\coth \left({\mathrm{xlog}}_{\mathrm{e}}\left(\mathrm{a}\right)\right)$

• $\sinh \left({\mathrm{xlog}}_{\mathrm{e}}\left(\mathrm{a}\right)\right)$

• $\tanh \left({\mathrm{xlog}}_{\mathrm{e}}\left(\mathrm{a}\right)\right)$