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

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!

205.

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

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

$$\begin{gathered} \mathrm{Let} {\mathrm{a}}_1, {\mathrm{a}}_2, .... {\mathrm{a}}_{\mathrm{n}} \mathrm{be} \mathrm{a} \mathrm{given} \mathrm{A}.\mathrm{P}. \mathrm{whose} \mathrm{common} \mathrm{difference} \mathrm{is} \mathrm{an} \mathrm{integer} \mathrm{and} {\mathrm{S}}_{\mathrm{n}}= {\mathrm{a}}_1 + {\mathrm{a}}_2 + ... + \mathrm{an}. \\ \mathrm{If} {\mathrm{a}}_1 = 1, {\mathrm{a}}_1 = 300 \mathrm{and} 15 \le \mathrm{n} \le 50, \mathrm{then} \mathrm{the} \mathrm{ordered} \mathrm{pair} ({\mathrm{S}}_{\mathrm{n} – 4}, {\mathrm{a}}_{\mathrm{n} – 4}) \mathrm{is} \mathrm{equal} \mathrm{to} : \end{gathered}$$

• (2490, 248)

• (2490, 249)

• (2480, 249)

• (2480, 248)

The minimum value of 2^sin(x)+ 2^cos(x) is :

• $2^{ - 1 + \frac{1}{\sqrt{2}}}$

• $2^{ - 1 + \sqrt{2}}$

• $2^{1 - \sqrt{2}}$

• $2^{1 - \frac{1}{\sqrt{2}}}$