Progression

Q. $f\left(x\right)={\log }_{100x}\left(\frac{2{\log }_{10}x+1}{-x}\right)$ exists if $x\in$

1. $(0,{10}^{-2})\cup ({10}^{-2},{10}^{-1/2})$
2. $[{10}^{-2},{10}^{-1/2})$
3. $({10}^{-2},{10}^{-1/2})$
4. $(0,{10}^{-2})$

Q. If the greatest value of the term independent of $x$ in the expansion of ${(x\sin \alpha +a\frac{\cos \alpha }{x})}^{10}$ is $\frac{10!}{(5!{)}^2},$ then the value of $a$ is equal to

1. $-1$
2. $-2$
3. $2$
4. $1$

Q. If $\int \ln (x^2+x)dx=x\ln (x^2+x)+f\left(x\right)+c$, then $f\left(x\right)$ equals:

1. $2x+\ln (x+1)$
2. $2x-\ln (x+1)$
3. $-2x+\ln (x+1)$
4. $-2x-\ln (x+1)$

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. Which number would replace the blank in the following sequence

12, 25, 51, 103, , 415

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

Q. If all the letters of the word AGAIN are arranged in dictionary order, find the 43rd and 54th words.

Q. If the coefficients of $x^7$ and $x^8$ in the expansion of ${(2+\frac{x}{3})}^n$ are equal, then the value of $n$ is

1. $15$
2. $55$
3. $56$
4. $45$

Q. The coefficient of $x^n$ in the expansion of $(1-x)e^x$ is

1. $\frac{1}{n!}$
2. $\frac{1}{(n+1)!}$
3. $\frac{n-1}{n!}$
4. $\frac{1-n}{n!}$

Q. If $7^x=3^{{\log }_{9}7}\cdot 5^{{\log }_{25}49}$, then the value of $x$ is

1. $\frac{1}{2}$
2. $1$
3. $\frac{3}{2}$
4. $2$

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

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

Q. Match the following by appropriately matching the lists based on the information given in Column I and Column II​​​​​​
$\begin{array}{cccc}& ColumnI& & ColumnII\\ a.& \text{If}a,b,c\text{are in}G.P.,\text{then}& \text{p.}& A.P.\\ & {\log }_{a}10,{\log }_{b}10,{\log }_{c}10\text{are in}& & \\ b.& \text{If}\frac{a+b{e}^x}{a-b{e}^x}=\frac{b+c{e}^x}{b-c{e}^x}=\frac{c+d{e}^x}{c-d{e}^x},& \text{q.}& H.P.\\ & \text{then}a,b,c,d& & \\ c.& \text{If}a,b,c\text{are in}A.P.;& \text{r.}& G.P.\\ & a,x,b\text{are in}G.P.\text{and}& & \\ & b,y,c\text{are in}G.P.,& & \\ & \text{then}{x}^2,b^2,y^2\text{are in}& & \end{array}$

1. $a\to q,b\to p,c\to r$
2. $a\to p,b\to r,c\to q$
3. $a\to q,b\to r,c\to p$
4. $a\to r,b\to q,c\to p$

Q. if m and n are two natural numbers and mⁿ = 81. Then the value of n^mn is

Q. If $(1+x+x^2)(1-\frac{x}{1!}+\frac{x^2}{2!}-\frac{x^3}{3!}+\dots )=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+a_4{x}^4+...$ then,

1. $a_2=\frac{1}{2}$
2. $a_1=-1$
3. $a_3=\frac{1}{3}$
4. $a_4=-\frac{1}{2}$

Q. The $n^{\text{th}}$ term in the expansion of ${\log }_e\left(\frac{4}{3}\right)$ is

1. $\frac{(-1{)}^{n-1}}{n}$
2. $\frac{(-1{)}^{n-1}}{n{.3}^n}$
3. $\frac{(-1{)}^{n-1}}{n{.4}^n}$
4. $\frac{(-1{)}^{n-1}}{n{.2}^n}$

Q. The coefficient of the term independent of $x$ in the expansion of$(1+x+2x^3){(\frac{3}{2}{x}^2-\frac{1}{3x})}^9$, is

1. $\frac{1}{3}$
2. $\frac{19}{54}$
3. $\frac{17}{54}$
4. $\frac{1}{4}$

Q. If $a^2+4b^2=12ab,$ then $\log \left(a+2b\right)$ is

1. $\frac{\text{1}}{\text{2}}\left[\text{log}\text{a}\text{+}\text{log}\text{b}\text{-}\text{log2}\right]$
2. $\frac{\text{1}}{\text{2}}\left|\text{log}\text{a}\text{+}\text{log}\text{b + 4}\text{log}\text{2}\right|$
3. $\text{log}\frac{\text{a}}{\text{2}}\text{+}\text{log}\frac{\text{b}}{\text{2}}\text{+}\text{log2}$
4. $\frac{\text{1}}{\text{2}}\left|\text{log}\text{a}-\text{log}\text{b + 4}\text{log}\text{2}\right|$

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

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

Q. The differential coefficient of $a^{{\log }_{10}\left({\text{cosec}}^{-1}x\right)}$ is

1. $\frac{a^{{\log }_{10}\left({\text{cosec}}^{-1}x\right)}}{[\text{cosec}{]}^{-1}x}\frac{1}{x\sqrt{x^2-1}}{\log }_{10}$ a
2. $-\frac{a^{{\log }_{10}\left({\text{cosec}}^{-1}x\right)}}{[\text{cosec}{]}^{-1}x}\frac{1}{|x|\sqrt{x^2-1}}{\log }_{10}$ a
3. $\frac{a^{{\log }_{10}\left({\text{cosec}}^{-1}x\right)}}{[\text{cosec}{]}^{-1}x}\frac{1}{|x|\sqrt{x^2-1}}{\log }_{a}10$
4. $\frac{a^{{\log }_{10}\left({\text{cosec}}^{-1}x\right)}}{[\text{cosec}{]}^{-1}x}\frac{1}{x\sqrt{x^2-1}}{\log }_{a}10$

Q. Prove that coefficient of $x^4$ in (1 + x - $2x^2{)}^6$ is -45 and if complete expansion of the expression is $1+a_{1}x+a_2{x}^2+....+a_{1}2x^{12}$
Prove that $a_2+a_4+a_6+...+a_{12}=31$

Q. Supposse $a,b,c$ are in A.P. and $a^2,b^2,c^2$ are in G.P. If $a<b<c$ and $a+b+c=\frac{3}{2}$, then the value of $a$ is

1. $\frac{1}{2\sqrt{2}}$
2. $\frac{1}{2\sqrt{3}}$
3. $\frac{1}{2}-\frac{1}{\sqrt{3}}$
4. $\frac{1}{2}-\frac{1}{\sqrt{2}}$

Q. $\underset{0}{\overset{2}{\int }}x\left[x\right]dx.$

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. Area of the region bounded by $y=|x|$ and $y=1-|x|$ is

1. $\frac{1}{3}$ sq. units
2. $\frac{1}{2}$ sq. unit
3. 1 sq. units
4. 2 sq. units

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

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

Q. If $I_n=\int (logx{)}^{n}dx$, then $I_n+n{I}_{n-1}=$

1. $x(logx{)}^n$
   
2. $(xlogx{)}^n$
   
3. $(logx{)}^{n-1}$
   
4. $n(logx{)}^n$

Q.

Find the sum of the products of the corresponding terms of the sequences 2, 4, 8, 16, 32 and 128, 32, 8, 2, .

Q. Supposse $a,b,c$ are in A.P. and $a^2,b^2,c^2$ are in G.P. If $a<b<c$ and $a+b+c=\frac{3}{2}$, then the value of $a$ is

1. $\frac{1}{2\sqrt{2}}$
2. $\frac{1}{2\sqrt{3}}$
3. $\frac{1}{2}-\frac{1}{\sqrt{3}}$
4. $\frac{1}{2}-\frac{1}{\sqrt{2}}$

Q.

Write the first five terms of the following sequence and obtain the corresponding series:

Q.

Find the sum of the products of the corresponding terms of the sequences 2, 4, 8, 16, 32 and 128, 32, 8, 2, .