Infinite GP

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. None of these

4. 3/2, 2

Q.

If sum of the series$\sum _{n=0}^{\infty }{r}^n=S \mathrm{for}\left|r\right|<1$, then the sum of the series $\sum _{n=0}^{\infty } r^{2n}$ is

1. $S^2$

2. $\frac{S^2}{2S+1}$

3. $\frac{2S}{S^2-1}$

4. $\frac{S^2}{2S-1}$

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. 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 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. $\{\frac{\pi }{6},\frac{5\pi }{6}\}$

Q. Divide
12$xy(9x^2-16y^2)÷4xy(3x+4y)$

Q. The least value of n ( a natural number), for which the sum S of the series $1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\cdots$ differs from $S_n$ by a quantity $<{10}^{-6}$ is

1. 20
2. 19
3. 21
4. None