Progression

Q. Which number would replace the blank in the following sequence

12, 25, 51, 103, , 415

1. 207
2. 205
3. 209
4. 208

Q. A line divides a plane into 2 regions. Two lines divide the plane into maximum 4 regions. If $L_n$ is the maximum number of regions divided by $n$ lines then the following is/are true?

1. $L_{20}=211$
2. $L_{10}=56$
3. $L_{15}=121$
4. $L_{25}=326$

Q. The expansion of $\frac{e^{7x}-e^x}{e^{4x}}$ is equal to

1. $2[1+\frac{(3x{)}^2}{2!}+\frac{(3x{)}^4}{4!}+...]$
2. $2[\frac{3x}{1!}+\frac{(3x{)}^3}{3!}+...]$
3. $2[1+\frac{(2x{)}^2}{2!}+\frac{(2x{)}^4}{4!}+...]$
4. $2[\frac{2x}{1!}+\frac{(2x{)}^3}{3!}+...]$

Q. The coefficient of $x^{10}$ in the expansion of $[1+x^2(1-x){]}^8$ is

1. $470$
2. $400$
3. $576$
4. $476$

Q. The expression $\frac{1}{x+1}+\frac{1}{2(x+1{)}^2}+\frac{1}{3(x+1{)}^3}+...$ is equal to

1. $\log \left[\frac{x}{x+1}\right]$
2. $\log \left[\frac{x+1}{x}\right]$
3. $\log \left[\frac{x+1}{x-1}\right]$
4. $\log \left[\frac{x-1}{x+1}\right]$

Q. The sixth term in the expansion of ${[\sqrt{\left\{2^{\log (10-3^x)}\right\}}+\sqrt[5]{5\left\{2^{(x-2)\log 3}\right\}}]}^m$ is equal to $21$. If it is known that the binomial coefficient of the $2^{\text{nd}},3^{\text{rd}}$ and $4^{\text{th}}$ terms in the expansion represents respectively the first, third and fifth terms of an A.P. (the symbol log stands for logarithm to the base $10$), then minimum value of expression is

1. $128$
2. $32$
3. None of these
4. $64$