Vn Method

Q.

${11}^2+{12}^2+{13}^2+...{20}^2=$

1. $2481$

2. $2483$

3. $2485$

4. $2487$

Q.

If $f\left(x\right)={\cos }^{2}x+se{c}^{2}x$, then

1. $f\left(x\right)<1$

2. $1<f\left(x\right)<2$

3. $f\left(x\right)$greater than or equal to $2$

4. $f\left(x\right)=1$

Q. The sum of the following series
$1+6+\frac{9(1^2+2^2+3^2)}{7}+\frac{12(1^2+2^2+3^2+4^2)}{9}+\frac{15(1^2+2^2+...+5^2)}{11}+...\text{up to}15\text{terms}$, is :

1. $7820$
2. $7830$
3. $7520$
4. $7510$

Q.

The sum of the series $\sum _{\mathrm{n}=1}^{\infty }\frac{\left[{\mathrm{n}}^2+6\mathrm{n}+10\right]}{(2\mathrm{n}+1)!}$ is equal to:

1. $\left(\frac{41}{8}\right)\mathrm{e}+\left(\frac{19}{8}\right){\mathrm{e}}^{-1}-10$

2. $\left(-\frac{41}{8}\right)\mathrm{e}+\left(\frac{19}{8}\right){\mathrm{e}}^{-1}-10$

3. $\left(\frac{41}{8}\right)\mathrm{e}-\left(\frac{19}{8}\right){\mathrm{e}}^{-1}-10$

4. $\left(\frac{41}{8}\right)\mathrm{e}+\left(\frac{19}{8}\right){\mathrm{e}}^{-1}+10$

Q. Sum of n terms of the series $\sqrt{2}+\sqrt{8}+\sqrt{18}+\sqrt{32}+........$ is

1. 
2. 
3. 
4. 

Q. If $S=\sum _{n=2}^{\infty }\frac{3n^2+1}{(n^2-1{)}^3}$ then $16S$ is

Q.

Find $\frac{4}{5}$ of $20$.

Q.

The value of $I={\int }_0^1\sqrt{\frac{1-x}{1+x}}dx$ is

1. $\frac{\mathrm{\pi }}{2}+1$

2. $\frac{\mathrm{\pi }}{2}-1$

3. $-1$

4. $1$

Q. If $a+b+c+d=63,$ where $a,b,c,d\in I^{+},$ then the maximum value of $ab+bc+cd$ is

1. $990$
2. $991$
3. $992$
4. $994$

Q. If $\frac{\sqrt{5+x}+\sqrt{5-x}}{\sqrt{5+x}-\sqrt{5-x}}=4$, then value of $x$ is

1. $\frac{20}{17}$
2. $\frac{18}{7}$
3. $\frac{43}{7}$
4. $\frac{40}{17}$

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:

1. $x>2$

2. $x>-2$

3. $x>\frac{1}{2}$

4. None of these

Q. Let $S=\frac{1}{6}+\frac{1}{24}+\frac{1}{60}+\frac{1}{120}+\cdots \text{upto}\infty$. Then the value of $2S$ is

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

Q.

${lim}_{n\to \infty }[\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}\cdots \frac{1}{(2n+1)(2n+3)}]$ is equa to

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

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

3. 0

4. 2

Q. The sum to infinity of the series
$\frac{1}{3}+\frac{3}{3\cdot 7}+\frac{5}{3\cdot 7\cdot 11}+\frac{7}{3\cdot 7\cdot 11\cdot 15}+\cdots$ is

Q.

Statement I. The sum of the series $1+(1+2+4)+(4+6+9)+(9+12+16)+\dots +(361+380+400)$ is $8000$.

Statement II$\sum _{k=1}^n\left[k^3\right.\left.-(k-1{)}^3\right]=n^3$ , for any natural number $n.$

1. Statement I is false, Statement II is true

2. Statement I is true, Statement II is true; Statement II is a correct explanation of Statement I

3. Statement I is true, Statement II is true; Statement II is not a correct explanation of Statement I

4. Statement I is true, Statement II is false

Q. Let $r^{\text{th}}$ term of a series be given by $T_r=\frac{r}{1-3r^2+r^4}.$ Then $-2\sum _{r=1}^{\infty }{T}_r$ is

Q. The sum to $50$ terms of the series $\frac{3}{1^2}+\frac{5}{1^2+2^2}+\frac{7}{1^2+2^2+3^2}+\dots$ is

1. $\frac{100}{17}$
2. $\frac{200}{51}$
3. $\frac{150}{17}$
4. $\frac{50}{17}$

Q. The $n^{\text{th}}$ term of a sequence of numbers is $a_n$ and given by the formula $a_n=a_{n-1}+2n$ for $n\ge 2$ and $a_1=1.$
The sum of first $20$ terms is

1. $3060$
2. $2679$
3. $3059$
4. $2680$

Q. If $a_k=\frac{1}{k(k+1)},fork=1,2,\mathrm{3.....}n,then{\left(\sum _{k=1}^n{a}_k\right)}^2=$

1. 
2. 
3. 
4. 

Q. Show that $\text{f(x)}=2\text{x}+{\text{cot}}^{-1}\text{x}+\text{log}(\sqrt{1+{\text{x}}^2}-\text{x})$ is increasing in $R$.

Q. $\sum _{r=1}^{99}r!(r^2+r+1)=$

1. $102!-100!$
2. $100(100!)-1$
3. $99(100!)-1$
4. $100(99!)-1$

Q. If $S_r$ denotes the sum of the infinite geometric series whose first term is $r$ and common ratio is $\frac{1}{1+r}$, where $r\in N$, then the value of $\sum _{r=1}^{10}{S}_r^2$ is

1. $385$
2. $659$
3. $384$
4. $505$

Q. The sum of $10$ terms of the series $2\cdot 5+5\cdot 8+8\cdot 11+\dots$ is

1. $3630$
2. $3610$
3. $3620$
4. $3600$

Q. The sum of $10$ terms of the series $\mathrm{1.3.5}+\mathrm{3.5.7}+\mathrm{5.7.9}+\dots \dots$ is

1. $28720$
2. $28780$
3. $28700$
4. $28680$

Q.

If $A,B,C$ are acute angles such that $\tan A+\tan B+\tan C=\tan A\tan B\tan C$. Then $cotAcotBcotC$ is

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

2. $\le \frac{1}{2\sqrt{3}}$

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

4. None of these

Q. The sum of first 10 terms of the series $1\cdot 2\cdot 3+2\cdot 3\cdot 4+3\cdot 4\cdot 5+.........$ is

1. 6006
2. 2970
3. 3990
4. 4290

Q. Suppose $\alpha ,\beta$ and $\theta$ are angles satisfying $0<\alpha <\theta <\beta <\frac{\pi }{2}$, then $\frac{\sin \alpha -\sin \beta }{\cos \beta -\cos \alpha }=$

1. $\tan \theta$
2. $-\tan \theta$
3. $\cot \theta$
4. $-\cot \theta$

Q.

Find$(-15)-(-18)$

Q. If the sum to $n$ terms of the series $\frac{3}{1^2\times 2^2}+\frac{5}{2^2\times 3^2}+\frac{7}{3^2\times 4^2}+\dots \dots$ is $0.99$, then the value of $n$ is

Q. If $\alpha =1+\frac{1}{2}+\frac{1}{3}+\cdots \cdots +\frac{1}{101}$ and $\beta =\sum _{r=1}^{99}\frac{r}{(102-r)(101-r)}$, then the value of $\alpha +\beta$ is