Sum of Infinite Terms of a GP

Q.

Sum of infinite terms $A,AR,A{R}^2,A{R}^3,\dots$ is $15$. If the sum of the squares of these terms is $150$, then find the sum of $A{R}^2,A{R}^4,A{R}^6,\dots$.

Q.

If $S$ is the sum to infinite of a G.P., whose first term is $a$, then the sum of the first $n$ terms is

1. $S{\left(1-\frac{a}{S}\right)}^n$

2. $S\left(1-{\left(1-\frac{a}{S}\right)}^n\right)$

3. $a\left(1-{\left(1-\frac{a}{S}\right)}^n\right)$

4. None of these

Q. If a = 3+2$\sqrt{2}$, find the value of $a^2+\frac{1}{a^2}$.

​​​​​​​[3 Marks]

Q. Evaluate : ${\left(\frac{3}{4}\right)}^0\times 2\frac{1}{4}-{\left(2\frac{1}{4}\right)}^0\times \frac{3}{4}$--

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

Q. If $x+\frac{1}{x}=2$, then $x^6+\frac{1}{x^6}=$ ______ .

1. 2

Q.

$x=1+a+a^2+...\infty (a<1)$
$y=1+b+b^2+...\infty (b<1)$
Then the value of $1+ab+a^2{b}^2+.....\infty$ is
[MNR 1980; MP PET 1985]

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

Q. Simplify: ${\left(\frac{81}{16}\right)}^{\frac{-3}{4}}\times \left\{{\left(\frac{25}{9}\right)}^{\frac{-3}{2}}÷{\left(\frac{5}{2}\right)}^{-3}\right\}$

Q.

In the question, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice:

Assertion: $(10+4{)}^3$ = 2744.
Reason: $(x+y{)}^3$ = $x^3+y^3+3x^{2}y+3x{y}^2$.

1. Both assertion (A) and reason (R) are true, but reason (R) is not the correct explanation of assertion (A).

2. Both assertion (A) and reason (R) are true, and reason (R) is the correct explanation of assertion (A).

3. Assertion (A) is false, but reason (R) is true.

4. Assertion (A) is true, but reason (R) is false.

Q.

8+88+888+........=

1. $\frac{80}{81}({10}^n-1)-\frac{8n}{9}$
   

2. $\frac{80}{81}({10}^n-1)+\frac{8n}{9}$
   

3. $\frac{10}{81}({10}^n-1)$
   

4. None of these

Q. If $S_1,S_2,S_3\cdots S_p$ are the sums of infinite geometric series whose first terms are 1, 2, 3, ... p and whose common
ratios are $\frac{1}{2},\frac{1}{3},\frac{1}{4},\cdots \frac{1}{p+1}respectively,$
Then $S_1+S_2+S_3+\cdots +S_p=$

1. p
2. None of these
3. $\frac{p(p+3)}{2}$
4. $\frac{p(p+1)}{2}$

Q. Mark the correct choice as:
Assertion(A): $343^{\frac{2}{3}}$ ÷ $7^{-1}$ = $7^3$
Reason(R): If 𝐚 be a real number, then $a^0$ = 𝟏.

1. Both A and R are true and R is the correct explanation of A.
2. A is false but R is true.
3. A is true but R is false.
4. Both A and R are true and R is not the correct explanation of A.

Q. An infinite G.P. has first term 'x' and sum '5', then x belongs to

1. $x<10-10$
2. $-10<x<0$
3. $0<x<10$
4. $x>10$

Q. If $x=\frac{1}{5-x}$ and $x\ne 5$, find : $x^3+\frac{1}{x^3}$.

1. $95$
2. $110$
3. $140$
4. $132$

Q. Find the missing term in the figure

1. $767701$
2. Can't be determined
3. None of these
4. $757701$

Q. Find the following product.
$2d{c}^2\times -6dc$

Q. If S is the sum to infinity of a G.P., whose first term is a, then the sum of the first n terms is [UPSEAT 2002]

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

Q. Range of $14,12,17,18,16$ and $x$ is $20$. Find $x$ $(x>0)$

1. $28$
2. $2$
3. $32$
4. Cannot be determined

Q. If $p\left(x\right)=x^3-5x^2+4x-3$ and $g\left(x\right)=x-2$, show that $p\left(x\right)$ is not a multiple of $g\left(x\right)$.

Q. The sides of a triangle $ABC$ are positive integers. The smallest side has length $l$. What of the following statements is true?

1. The area of ABC is always a rational number.
2. The perimeter of ABC is an even integer.
3. The information provided is not sufficient to conclude any of the statements A, B or C above.
4. The area of ABC is always an irrational number.

Q. If ${\left(\frac{5}{3}\right)}^2\times {\left(\frac{5}{3}\right)}^2=\frac{625}{m}$, then the value of $m$ is

Q. If the sum of the series $1+\frac{2}{x}+\frac{4}{x^2}+\frac{8}{x^3}+.....\infty$ is a finite number, then the complete solution of x will be -
[UPSEAT 2002]

1. x > 2
2. x < -2
3. (x < -2) U (x > 2)
4. None of these

Q. If $a>0$ and $\underset{x\to a}{lim}\frac{a^x-x^a}{x^x-a^a}=-1$ then $a$ =

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

Q. Find the value of ${\left(27\right)}^{-\frac{2}{3}}+{\left({\left(2^{-\frac{2}{3}}\right)}^{-\frac{5}{3}}\right)}^{-\frac{9}{10}}$

1. $\frac{2}{9}$
2. $\frac{11}{18}$
3. 1
4. $\frac{1}{9}$