ln $∆$ABC, if the sides a, b, c are in geometric progression and the largest angle exceeds the smallest angle by 60°, then cos(B) is equal to

• $\frac{\sqrt{13} + 1}{4}$

• $\frac{1 - \sqrt{13}}{4}$

• 1

• $\frac{\sqrt{13} - 1}{4}$

$$\begin{gathered} \mathrm{The} \mathrm{coefficient} \mathrm{of} {\mathrm{x}}^5 \mathrm{in} \mathrm{the} \mathrm{expansion} \mathrm{of} \\ (1 + \mathrm{x}{)}^{21} + (1 +\mathrm{x}{)}^{22} + ... + (1 + \mathrm{x}{)}^{30} \mathrm{is} \end{gathered}$$

• ${}_{31}\mathrm{C}_6 - {}_{21}\mathrm{C}_6$

• ${}_{51}\mathrm{C}_5$

• ${}_9\mathrm{C}_5$

• ${}_{30}\mathrm{C}_5 + {}_{20}\mathrm{C}_5$

If the roots of the equation x² -7x² + 14x - 8 = 0 are in geometric progression, then the difference between the largest and the smallest roots is

• 4

• 2

• $\frac{1}{2}$

• 3

214.

$\mathrm{p}, {\mathrm{x}}_1, {\mathrm{x}}_2 , ..., {\mathrm{x}}_{\mathrm{n}} \mathrm{and} \mathrm{q}, {\mathrm{y}}_1, {\mathrm{y}}_2, ..., {\mathrm{y}}_{\mathrm{n}}$ are two arithmetic progressions with common differences a and b respectively. If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are the arithmetic means of ${\mathrm{x}}_1, {\mathrm{x}}_2, ..., {\mathrm{x}}_{\mathrm{n}} \mathrm{and} {\mathrm{y}}_1, {\mathrm{y}}_2, ..., {\mathrm{y}}_{\mathrm{n}}$ respectively. Then the locus of P $\left(\mathrm{\alpha }, \mathrm{\beta }\right)$ is

• $\mathrm{a}\left(\mathrm{x} - \mathrm{p}\right) = \mathrm{b}\left(\mathrm{y} - \mathrm{q}\right)$

• $\mathrm{b}(\mathrm{x} - \mathrm{p}) = \mathrm{a}(\mathrm{y} - \mathrm{q})$

• $\mathrm{\alpha }\left(\mathrm{x} - \mathrm{p}\right) = \mathrm{\beta }\left(\mathrm{y} - \mathrm{q}\right)$

• $\mathrm{p}\left(\mathrm{x} - \mathrm{\alpha }\right) = \mathrm{q}\left(\mathrm{y} - \mathrm{\beta }\right)$

$\mathrm{The} \mathrm{sum} \mathrm{of} \mathrm{the} \mathrm{n} \mathrm{terms} \mathrm{of} \frac{1}{2 . 5} + \frac{1}{5 . 8} +\hspace{0.17em}\frac{1}{8 . 11} + ... \mathrm{is}$

• $\frac{3\mathrm{n}}{2\left(3\mathrm{n} + 2\right)}$

• $\frac{3\mathrm{n}}{3\mathrm{n} + 2}$

• $\frac{\mathrm{n}}{2\left(3\mathrm{n} + 2\right)}$

• $\frac{\mathrm{n}}{3\mathrm{n} + 2}$

$\mathrm{If} \mathrm{x} = \frac{1 . 3}{3 . 6} + \frac{1 . 3 . 5}{3 . 6. 9} + \frac{1 . 3 . 5 . 7}{3 . 6. 9 . 12} + ... \mathrm{to} \mathrm{infinite} \mathrm{terms}, \mathrm{then} 9{\mathrm{x}}^2 + 24\mathrm{x} = ?$

• 31

• 11

• 41

• 21

The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in

• $\left(- \infty , - 3\right) \cup \left(9, \infty \right)$

• $\left( - \infty , - 9\right) \cup \left(3, \infty \right)$

• $\left(- 3, \infty \right)$

• $\left( - \infty , 9\right)$

If |x| < 1, |y| < 1 and x $\ne$ y, then the sum to infinity of the following series (x + y) + (x² + xy + y²) + (x³ + x²y + xy² + y³) + ..... is :

• $\frac{\mathrm{x} +\hspace{0.17em}\mathrm{y} + \mathrm{xy}}{\left(1 - \mathrm{x}\right)\left(1 - \mathrm{y}\right)}$

• $\frac{\mathrm{x} + \mathrm{y} - \mathrm{xy}}{\left(1 - \mathrm{x}\right)\left(1 - \mathrm{y}\right)}$

• $\frac{\mathrm{x} + \mathrm{y} - \mathrm{xy}}{\left(1 - \mathrm{x}\right)\left(1 + \mathrm{y}\right)}$

• $\frac{\mathrm{x} + \mathrm{y} + \mathrm{xy}}{\left(1 + \mathrm{x}\right)\left(1 + \mathrm{y}\right)}$

Let S be the sum of the first 9 term of the series :(x + ka} + (x² + (k + 2)a} + {x³ + (k + 4)a} + {x⁴ + (k + 6)a} + ........ where a $\ne$ 0 and x $\ne$ 1. If $\mathrm{S} = \frac{{\mathrm{x}}^{10} - \mathrm{x} +\hspace{0.17em}45\mathrm{a}\left(\mathrm{x} - 1\right)}{\mathrm{x} - 1}$, then k is equal to

•  - 3

• 1

•  - 5

• 3

$$\begin{gathered} \mathrm{If} \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{first} 11 \mathrm{terms} \mathrm{of} \mathrm{an} \mathrm{A}.\mathrm{P}. , {\mathrm{a}}_1, {\mathrm{a}}_2, {\mathrm{a}}_3 ..... \mathrm{is} 0({\mathrm{a}}_1 \ne 0), \\ \mathrm{then} \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{the} \mathrm{A}.\mathrm{P}., {\mathrm{a}}_1, {\mathrm{a}}_3, {\mathrm{a}}_5, ..... {\mathrm{a}}_{23} \mathrm{is} {\mathrm{ka}}_1, \mathrm{where} \mathrm{k} \mathrm{is} \mathrm{equal} \mathrm{to} : \end{gathered}$$

• $\frac{121}{10}$

• $- \frac{121}{10}$

• $- \frac{72}{5}$

• $\frac{72}{5}$