Sum of Infinite Terms of a GP

Q. The sum of first 9 terms of the series $\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\dots \dots$ is

1. 71
2. 96
3. 142
4. 192

Q.

The sum of the series $6+66+666+.......$upto $n$ terms is

1. $\frac{{10}^{n-1}-9n+10}{81}$

2. $\frac{2\left({10}^{n+1}-9n-10\right)}{27}$

3. $\frac{2\left({10}^n-9n-10\right)}{27}$

4. none of these

Q.

Find the sum of the sequence $0.4+0.44+0.444+...$ up to $n$ terms.

Q.

The sum of $1+\frac{2}{5}+\frac{3}{5^2}+\frac{4}{5^3}+..........$upto $n$ terms is

1. $\left(\frac{25}{16}\right)-\frac{\left(4n+5\right)}{\left(16·5^{n-1}\right)}$

2. $\left(\frac{3}{4}\right)-\frac{\left(2n+5\right)}{\left(16·5^{n-1}\right)}$

3. $\left(\frac{3}{7}\right)-\frac{\left(3n+5\right)}{\left(16·5^{n-1}\right)}$

4. $\left(\frac{1}{6}\right)-\frac{\left(5n+1\right)}{\left(3·5^{n+2}\right)}$

Q.

The first term of a G.P whose second term is$2$ and sum to infinity is$8$, will be

1. $6$

2. $3$

3. $4$

4. $1$

Q.

The sum of first $20$ terms of the sequence$0.7,0.77,0.777,...$, is

1. $\frac{7}{81}\left(179-{10}^{-20}\right)$

2. $\frac{7}{9}\left(99-{10}^{-20}\right)$

3. $\frac{7}{81}\left(179+{10}^{-20}\right)$

4. $\frac{7}{9}\left(99+{10}^{-20}\right)$

Q.

Find $\frac{dy}{dx}$ , if $x=2\cos \theta -\cos 2\theta$and $y=2\sin \theta -\sin 2\theta$.

1. $\tan \frac{3\theta }{2}$

2. $-\tan \frac{3\theta }{2}$

3. $cot\frac{3\theta }{2}$

4. $-cot\frac{3\theta }{2}$

Q.

The maximum value of the expression $\frac{1}{\left({\sin }^2\theta +3\sin \theta \cos \theta +5{\cos }^2\theta \right)}$ is

1. $2$

2. $1$

3. $3$

4. $4$

Q.

If the $4^{th},7^{th}$ and ${10}^{th}$ terms of a G.P. be $a,b,c$ respectively, then the relation between $a,b,c$ is

1. $b=\frac{\left(a+c\right)}{2}$

2. $a^2=bc$

3. $b^2=ac$

4. $c^2=ab$

Q.

If f is an even function defined on the interval (-5, 5), then the real values of x, satisfying the equation $f\left(x\right)=f\left(\frac{x+1}{x+2}\right)$ are ___.

1. $\frac{3\pm \sqrt{5}}{2}$

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

3. $\frac{-3\pm \sqrt{3}}{2}$

4. $\frac{-5\pm \sqrt{3}}{2}$

Q.

By what number should $(-15{)}^{-1}$ be divided so that the quotient may be equal to $(-5{)}^{-1}$ ?

Q.

Which term of the sequence 4, 9 , 14, 19, ...... is 124 ?

Q.

If $y=\cos \theta +i\sin \theta$, then the value of $y+1/y$ is equals to ?

1. $2\cos \theta$

2. $2\sin \theta$

3. $2\cos ec\theta$

4. $2\tan \theta$

Q.

If $\sin \left(\theta \right)+\sin \left(2\theta \right)+\sin \left(3\theta \right)=\sin \left(\alpha \right)$ and $\cos \left(\theta \right)+\cos \left(2\theta \right)+\cos \left(3\theta \right)=\cos \left(\alpha \right)$, then $\theta$ is equal to

1. $\frac{\alpha }{2}$

2. $\alpha$

3. $2\alpha$

4. $\frac{\alpha }{6}$

Q.

The number of arrangements that can be made out of the letter of the word "SUCCESS" so that the all Ss do not come together is?

1. $60$

2. $120$

3. $360$

4. $420$

Q.

The sum of the first five terms of the series $3+4\frac{1}{2}+6\frac{3}{4}+...$ will be

1. $39\frac{9}{16}$

2. $18\frac{3}{16}$

3. $39\frac{7}{16}$

4. $13\frac{9}{16}$

Q. If each term of an infinite G.P. is thrice the sum of the terms following it, then the common ratio of the G.P. is .

1. $\frac{1}{4}$
2. $\frac{1}{3}$
3. $\frac{1}{6}$
4. $\frac{1}{2}$

Q.

The sum of the series $3·6+4·7+5·8+.......$ upto $\left(n-2\right)$ terms

1. $n^3+n^2+n+2$

2. $\frac{1}{6}\left(2n^3+12n^2+10n-84\right)$

3. $n^3+n^2+n$

4. none of these

Q.

If the sum of first $n$ natural numbers is $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.$ times the sum of their squares, then the value of $n$ is:

1. $5$

2. $6$

3. $7$

4. $8$

Q.

The value of ${\int }_3^5\frac{x^2}{x^2-4}dx=?$

1. $2-\log \left(\frac{15}{7}\right)$

2. $2+\log \left(\frac{15}{7}\right)$

3. $2+4\log \left(3\right)-4\log \left(7\right)+4\log \left(5\right)$

4. $2-{\tan }^{-1}\left(\frac{15}{7}\right)$

Q.

The approximate value of $\sqrt[3]{28}$ is

1. $3.0037$

2. $3.037$

3. $3.0086$

4. $3.37$

Q. The sum of the series $1+2.2+{3.2}^2+{4.2}^3+{5.2}^4+\cdots +{100.2}^{99}$ is

1. ${100.2}^{100}$
2. ${99.2}^{100}$
3. ${99.2}^{100}-1$
4. ${99.2}^{100}+1$

Q. Which term of the GP 1, 2, 4, 8, .... is 512 ?

1. 10
2. 9
3. 8
4. 11

Q.

If $\text{A}$ and$\text{B}$ are two events such that $\text{P(A)≠ 0,}$$\text{P(B)≠ 1}$then $\text{P}\left(\frac{\text{A}}{\text{B}}\right)\text{=?}$

1. $1-\text{P}\left(\frac{\text{A}}{\text{B}}\right)$

2. $\text{1-P}\left(\frac{\text{A}}{\text{B}}\right)$

3. $\frac{\text{[1-P(AUB)]}}{\text{P(B)}}$

4. $\frac{\text{P(A)}}{\text{P(B)}}$

Q.

Write the Noun Form of Perfect

Q.

If $x_i>0$, $i=1,2,3,\dots 50$and $(x_1+x_2+x_3+\dots +x_{50})=50$, then the minimum value of $\left[\frac{1}{x_1}+\frac{1}{x_2}+\dots .+\frac{1}{x_{50}}\right]$ equals

1. $50$

2. ${50}^2$

3. ${50}^3$

4. none of these

Q.

If $s$ is the sum of an infinite G.P, the first term $a$ then the common ratio $r$ given by

1. $\frac{a-s}{s}$

2. $\frac{s-a}{s}$

3. $\frac{a}{1-s}$

4. $\frac{s-a}{a}$

Q.

Simplify it using brackets:-

$105\times 104$

Q.

If $x>0$ and ${\log }_{3}x+{\log }_3\sqrt{x}+{\log }_3\sqrt[4]{x}+.......=4$, then $x$ equals

1. $9$

2. $81$

3. $1$

4. $27$

Q.

Express $0.\overline{2}$ as a fraction in its lowest form.

1. $\frac{2}{9}$

2. $\frac{4}{9}$

3. $\frac{1}{9}$

4. $\frac{5}{9}$