Infinite GP

Q. For $k\in N,$ let $\frac{1}{\alpha (\alpha +1)(\alpha +2)\dots (\alpha +20)}=\sum _{k=0}^{20}\frac{A_k}{\alpha +k},$ where $\alpha >0.$ Then the value of $100{\left(\frac{A_{14}+A_{15}}{A_{13}}\right)}^2$ is equal to

Q. Let ${\Delta }_r=\left|\begin{array}{ccc}r-1& n& 6\\ (r-1{)}^2& 2n^2& 4n-2\\ (r-1{)}^3& 3n^3& 3n^2-3n\end{array}\right|$. Then the value of $\sum _{r=1}^n{\Delta }_r$ is:

1. $1$
2. $3$
3. $2$
4. $0$

Q.

The $5th$ term of the series $\frac{10}{9},\left(\frac{1}{3}\right)\left(\sqrt{\frac{20}{3}}\right),\frac{2}{3},......$.is

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

2. $1$

3. $\frac{2}{5}$

4. $\frac{\sqrt{2}}{5}$

Q. If $\frac{1}{1^4}+\frac{1}{2^4}+\frac{1}{3^4}+\cdots \text{upto}\infty =\frac{{\pi }^4}{90},$ then the value of $\frac{1}{1^4}+\frac{1}{3^4}+\frac{1}{5^4}+\cdots \text{upto}\infty$ is

1. $\frac{{\pi }^4}{45}$
2. $\frac{{\pi }^4}{96}$
3. $\frac{{\pi }^4}{124}$
4. $\frac{{\pi }^4}{196}$

Q. In an infinite G.P., if the first term is $x$ and sum of all terms is $5$, then

1. $x\in (0,10)$
2. $x\in (-\infty ,0)$
3. $x\in (10,\infty )$
4. $x\in [1,10]$

Q. If $x,y,z$ are in G.P. and $x+y,y+z,z+x$ are in A.P., where $x\ne y\ne z$, then common ratio of the G.P. is

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

Q. Find the sum of the following geometric series:
$(x+y)+(x^2+xy+y^2)+(x^3+x^{2}y+x{y}^2+y^3)+\cdots$ to n terms;

Q. If $11+103+1005+\cdots n\text{terms}=\frac{{10}^{n+1}+x{n}^2+y}{9}$, then the value of $x-y$ is

1. $19$
2. $1$
3. $-1$
4. $-19$

Q. The sum of infinite terms of a G.P. is x and on squaring the each term of it, the sum will be y, then the common ratio of this series is

1. $\frac{x^2-y^2}{x^2+y^2}$
2. $\frac{x^2+y^2}{x^2-y^2}$
3. $\frac{x^2-y}{x^2+y}$
4. $\frac{x^2+y}{x^2-y}$

Q. The value of $\underset{0}{\overset{1}{\int }}\frac{x^a-1}{\ln x}dx,$ (where $a$ is parameter) is equal to

1. $\ln |a-1|$
2. $\ln |a+1|$
3. $a\ln |a+1|$
4. $a\ln |a-1|$

Q. Let $a+a{r}_1+a{r_1}^2+...$ and $a+a{r}_2+a{r_2}^2+...$ be two infinite series of positive numbers with the same first term, where $r_1,r_2<1$. The sum of the first series is $r_1$ and the sum of the second series is $r_2$ then the value of $r_1+r_2$ is

Q. The sum of the series $\sum _{n=1}^{\infty }\frac{n^2+6n+10}{(2n+1)!}$ is equal to :

1. $\frac{41}{8}e+\frac{19}{8}{e}^{-1}-10$
2. $-\frac{41}{8}e+\frac{19}{8}{e}^{-1}-10$
3. $\frac{41}{8}e-\frac{19}{8}{e}^{-1}-10$
4. $\frac{41}{8}e+\frac{19}{8}{e}^{-1}+10$

Q. If $\sum _{k=1}^{\infty }\frac{1}{(k+2)\sqrt{k}+k\sqrt{k+2}}=\frac{\sqrt{a}+\sqrt{b}}{\sqrt{c}},$ where $a,b,c\in N$ and $a,b,c\in [1,15],$ then $a+b+c$ is equal to

Q. If $x,y,z$ are in G.P. and $x+y,y+z,z+x$ are in A.P., where $x\ne y\ne z$, then common ratio of the G.P. is

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

Q. $\sum _{i=1}^{\infty }\sum _{j=1}^{\infty }\sum _{k=1}^{\infty }\frac{1}{a^{i+j+k}}$ is equal to $(\text{where}|a|>1)$

1. $(a-1{)}^{-3}$
2. $\frac{3}{a-1}$
3. $\frac{3}{a^3-1}$
4. None of these

Q.

The product of infinite terms in $x^{\frac{1}{2}}.x^{\frac{1}{4}}.x^{\frac{1}{8}}...........\infty$ is

1. 0

2. infinity

3. 1

4. x

Q. If $f:[1,\infty )\to B$ defined by the function $f\left(x\right)=x^2-2x+6$ is a surjection, then $B$ is equals to

1. $[1,\infty )$
2. $[6,\infty )$
3. $[5,\infty )$
4. $[2,\infty )$

Q.

If sum of infinite terms of a G.P. is 3 and sum of squares of its terms is 3, then its first term and common ratio are
[RPET 1999]

1. 3/2, 1/2

2. 1, 1/2

3. 3/2, 2

4. None of these

Q.

Match the statements of Column I with values of Column II
$\begin{array}{cc}ColumnI& ColumnII\\ \left(A\right)\int \frac{e^{2x}-2e^x}{e^{2x}+1}dx=Aln(e^{2x}+1)+Bta{n}^{-1}\left(e^x\right)+c& \left(p\right)A=-\frac{1}{2},B=-\frac{1}{4}\\ \left(B\right)\int \sqrt{x+\sqrt{x^2+2}}dx=A\{x+\sqrt{x^2+2}{\}}^{\frac{3}{2}}+\frac{B}{\sqrt{x+\sqrt{x^2+2}}}+c& \left(q\right)A=\frac{1}{2},B=-2\\ \left(C\right)\int \frac{cos8x-cos7x}{1+2cos5x}dx=Asin3x+Bsin2x+c& \left(r\right)A=\frac{1}{3},B=-2\\ \left(D\right)\int \frac{lnx}{x^3}dx=A\frac{lnx}{x^2}+\frac{B}{x^2}+c& \left(s\right)A=\frac{1}{3},B=-\frac{1}{2}\end{array}$

1. A-s , B-r , C-q , D-p

2. A-p , B-s , C-r , D-q

3. A-p , B-q , C-r , D-s

4. A-q , B-r , C-s , D-p

Q. $\frac{(625{)}^{6.25}\times (25{)}^{2.6}}{(625{)}^{6.75}\times (5{)}^{1.2}}=?$

1. $5$
2. $10$
3. $15$
4. $25$

Q. Let ${\Delta }_r=\left|\begin{array}{ccc}r-1& n& 12\\ (r-1{)}^2& 2n^2& 8n-4\\ (r-1{)}^3& 3n^3& 6n^2-6n\end{array}\right|$. Then the value of $\sum _{r=1}^n{\Delta }_r$ is:

1. $1$
2. $0$
3. $2$
4. $3$

Q. If $\underset{0}{\overset{1}{\int }}{e}^{x^2}(x-\alpha )dx=0,$ then which of the following is true for $\alpha$?

1. $1<\alpha <2$
2. $\alpha <0$
3. $0<\alpha <1$
4. $\alpha =0$

Q. Minimise and maximise $z=5x+2y$ subject to the following constraints :

$x-2y\le 2$

$3x+2y\le 12$

$-3x+2y\le 3$

$x\ge 0,y\ge 0$

Q. If $(1-y)(1+2x+4x^2+8x^3+16x^4+32x^5)=1-y^6,$ where $y\ne 1,x\ne \frac{1}{2}$, then the value of $\frac{y}{x}$ is

1. $\frac{1}{2}$
2. $2$
3. $\frac{25}{24}$
4. $\frac{24}{25}$

Q. ${\sum }_{i=1}^n{\sum }_{j=1}^i{\sum }_{k=1}^{j}1$ is equal to

1. $\frac{n(n+1)}{2}$
2. ${\left[\frac{n(n+1)}{2}\right]}^2$
3. $\frac{n(n+1)(n+2)}{6}$
4. $\frac{n(n+1)(2n+1)}{6}$

Q. Sum of infinite number of terms in G.P. is 20 and sum of their square is 100. The common ratio of G.P. is
[AIEEE 2002]

1. 5

2. 3/5

3. 8/5
4. 1/5

Q. Evaluate: $\int \frac{x^2}{x^4+x^2-2}dx$.

Q. Evaluate the definite integral ${\int }_0^2\frac{6x+3}{x^2+4}dx$

Q. If $(1-y)(1+2x+4x^2+8x^3+16x^4+32x^5)=1-y^6,$ where $y\ne 1,x\ne \frac{1}{2}$, then the value of $\frac{y}{x}$ is

1. $\frac{1}{2}$
2. $2$
3. $\frac{25}{24}$
4. $\frac{24}{25}$

Q. The value of x in $(0,\pi )$ which satisfy the equation
$8^{-+|cosx|+co{s}^{2}x+|co{s}^{3}x|+\cdots to\infty }=4^3$ is

1. $\{\frac{\pi }{2},\frac{3\pi }{4}\}$
2. $\{\frac{\pi }{4},\frac{3\pi }{4}\}$
3. $\{\frac{\pi }{3},\frac{2\pi }{3}\}$
4. $\{\frac{\pi }{6},\frac{5\pi }{6}\}$