Sigma n2

Q.

If $S$ is the sum of the first 10 terms of the series ${\tan }^{-1}\frac{1}{3}+{\tan }^{-1}\frac{1}{7}+{\tan }^{-1}\frac{1}{13}+{\tan }^{-1}\frac{1}{21}\dots$ then $\tan \left(S\right)$ is equal to:

1. $\frac{5}{11}$

2. $\frac{5}{6}$

3. $-\frac{6}{5}$

4. $\frac{10}{11}$

Q.

The value of $\cot \left[\sum _{n=1}^{23}{\cot }^{-1}\left(1+\sum _{k=1}^{n}2k\right)\right]$ is

1. $\frac{23}{25}$

2. $\frac{25}{23}$

3. $\frac{23}{24}$

4. $\frac{24}{23}$

Q.

If $a_1,a_2,...,a_n,$ are in AP with common difference $d\ne 0$, then $(\sin d)[sec{a}_{1}sec{a}_2+sec{a}_{2}sec{a}_3+...+sec{a}_{(n-1)}sec{a}_n]$ is equal to

1. $cot{a}_n+cot{a}_1$

2. $cot{a}_n-cot{a}_1$

3. $\tan a_n-\tan a_1$

4. $\tan a_n-\tan a_{(n-1)}$

Q.

If $x,y$ and $z$ are in AP and ${\tan }^{-1}x,{\tan }^{-1}y$and ${\tan }^{-1}z$ are also in AP, then

1. $x=y=z$

2. $x=y=-z$

3. $x=1,y=2,z=3$

4. $x=2,y=4,z=6$

5. $x=2,y=3z$

Q. If the sum of the first ten terms of the series ${\left(1\frac{3}{5}\right)}^2+{\left(2\frac{2}{5}\right)}^2+{\left(3\frac{1}{5}\right)}^2+4^2+{\left(4\frac{4}{5}\right)}^2+.....,$is $\frac{16}{5}m,$ then m is equal to:

1. 100
2. 99
3. 102
4. 101

Q. The sum of first $n$ terms of the series
$1^2+{2.2}^2+3^2+{2.4}^2+5^2+{2.6}^2+....$ is $\frac{n(n+1{)}^2}{2}$ when $n$ is even. When $n$ is odd the sum is

1. $\frac{n^2}{2}(n+1)$
2. $\frac{n}{2}(n-1{)}^2$
3. $\frac{n}{2}(n-1)$
4. ${\frac{n(n+1)}{2}}^2$