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

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

• $\frac{121}{10}$

• $- \frac{121}{10}$

• $- \frac{72}{5}$

• $\frac{72}{5}$

363.

Let n > 2 be an integer. Suppose that there are n Metro stations in a city located around a circular path. Each pair of nearest stations is connected by a straight track only. Further, each pair of nearest station is connected by blue line, whereas all remaining pairs of stations are connected by red line. If number of red lines is 99 times the number of blue lines, then the value of n is

• 199

• 101

• 201

• 200

If the variance of the terms in an increasing A.P. b₁, b₂, b₃, ... , b₁₁ is 90, then the common difference of this A.P. is

$$\begin{gathered} \mathrm{If} {\mathrm{T}}_1, {\mathrm{T}}_2, {\mathrm{T}}_3 .... \mathrm{are} \mathrm{in} \mathrm{A}.\mathrm{P}. \mathrm{such} \mathrm{that} \\ {\mathrm{T}}_1 + {\mathrm{T}}_2 + ... + {\mathrm{T}}_{25} = {\mathrm{T}}_{28} +{\mathrm{T}}_{27} + ... {\mathrm{T}}_{40} \mathrm{and} \mathrm{first} \mathrm{term} \mathrm{is} 3 \\ \mathrm{then} \mathrm{value} \mathrm{of} \mathrm{common} \mathrm{difference} \mathrm{of} \mathrm{A}.\mathrm{P}. \mathrm{is} \end{gathered}$$

• $\frac{1}{2}$

• $\frac{1}{6}$

• 2

• 3

$\mathrm{S} = \left(2 {}^1\mathrm{P}_0 - 3{}^2\mathrm{P}_1 + 4{}^3\mathrm{P}_2 + ... 51 \mathrm{terms}\right) + \left(1! +\hspace{0.17em}2! + 3! - 4! + ...51\mathrm{terms}\right), \mathrm{find} \mathrm{S}$

• 1 + 51!

• 1 + 52!

• 1 + (50)51!

• 1 + (51)51!

If m arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that 4^th A.M. is equal to 2^nd G.M., then m is equal to :

$$\begin{gathered} \mathrm{If} 1 + \left(1 - 2^2 . 1\right) + \left(1 - 4^2 . 3\right) + \left(1 - 6^2 . 5\right) + ... + \left(1 - {20}^2 . 19\right) = \mathrm{\alpha } - 220\mathrm{\beta } \mathrm{then} \\ \mathrm{an} \mathrm{ordered} \mathrm{pair} \left(\mathrm{\alpha }, \mathrm{\beta }\right) = ? \end{gathered}$$

• (11, 97)

• (10, 103)

• (11, 103)

• (10, 97)

$$\begin{gathered} \mathrm{Let} \mathrm{\alpha } \mathrm{and} \mathrm{\beta } \mathrm{be} \mathrm{the} \mathrm{roots} \mathrm{of} {\mathrm{x}}^2 – 3\mathrm{x} + \mathrm{p} = 0 \mathrm{and} \mathrm{\gamma } \mathrm{and} \mathrm{\delta } \mathrm{be} \mathrm{the} \mathrm{roots} \mathrm{of} \\ {\mathrm{x}}^2 – 6\mathrm{x} + \mathrm{q} = 0. \mathrm{If} \mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }, \mathrm{\delta } \mathrm{from} \mathrm{a} \mathrm{geometric} \mathrm{progression}. \mathrm{Then} \mathrm{ratio} (2\mathrm{q} + \mathrm{p}) : (2\mathrm{q} – \mathrm{p}) \mathrm{is} \end{gathered}$$

• 3 : 1

• 5 : 3

• 9 : 7

• 33 : 31

$\mathrm{Let} {\left(2{\mathrm{x}}^2 + 3\mathrm{x} + 4\right)}^{10} = \sum _{\mathrm{r} = 0}^{20} {\mathrm{a}}_{\mathrm{r}}{\mathrm{x}}^{\mathrm{r}}. \mathrm{Then} \frac{{\mathrm{a}}_7}{{\mathrm{a}}_{13}} = ?$