The series of positive multiples of 3 is divided into sets: $\{ 3 \} , \{ 6,9,12 \} , \{ 15,18,21,24,27 \} , \ldots$ Then the sum of the elements in the $11 ^ { \text {th} }$ set is equal to $\_\_\_\_$.

Let $a_1, a_2, a_3, \ldots$ be an A.P. If $a_7 = 3$, the product $(a_1 a_4)$ is minimum and the sum of its first $n$ terms is zero then $n! - 4a_{n(n+2)}$ is equal to
(1) $\frac{381}{4}$
(2) 9
(3) $\frac{33}{4}$
(4) 24

The sum to 10 terms of the series $\frac{1}{1 + 1^2 + 1^4} + \frac{2}{1 + 2^2 + 2^4} + \frac{3}{1 + 3^2 + 3^4} + \ldots$ is:
(1) $\frac{59}{111}$
(2) $\frac{55}{111}$
(3) $\frac{56}{111}$
(4) $\frac{58}{111}$

Let $a, b, c > 1$, $a^{3}$, $b^{3}$ and $c^{3}$ be in A.P. and $\log_{a} b$, $\log_{c} a$ and $\log_{b} c$ be in G.P. If the sum of first 20 terms of an A.P., whose first term is $\frac{a + 4b + c}{3}$ and the common difference is $\frac{a - 8b + c}{10}$ is $-444$, then $abc$ is equal to
(1) 343
(2) 216
(3) $\frac{343}{8}$
(4) $\frac{125}{8}$

Let the digits $a , b , c$ be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?

Let $A _ { 1 } , A _ { 2 } , A _ { 3 }$ be the three A.P. with the same common difference $d$ and having their first terms as $A , A + 1 , A + 2$, respectively. Let $a , b , c$ be the $7 ^ { \text {th} } , 9 ^ { \text {th} } , 17 ^ { \text {th} }$ terms of $A _ { 1 } , A _ { 2 } , A _ { 3 }$, respectively such that $\left| \begin{array} { l l l } a & 7 & 1 \\ 2 b & 17 & 1 \\ c & 17 & 1 \end{array} \right| + 70 = 0$. If $a = 29$, then the sum of first 20 terms of an AP whose first term is $c - a - b$ and common difference is $\frac { d } { 12 }$, is equal to $\_\_\_\_$.

Let $f ( x ) = 2 x ^ { n } + \lambda , \lambda \in \mathbb { R } , \mathrm { n } \in \mathbb { N }$, and $f ( 4 ) = 133 , f ( 5 ) = 255$. Then the sum of all the positive integer divisors of $( f ( 3 ) - f ( 2 ) )$ is
(1) 61
(2) 60
(3) 58
(4) 59

The $8^{\text{th}}$ common term of the series $$\begin{aligned} & S_{1} = 3 + 7 + 11 + 15 + 19 + \ldots \\ & S_{2} = 1 + 6 + 11 + 16 + 21 + \ldots \end{aligned}$$ is

Let $S _ { K } = \frac { 1 + 2 + \ldots + K } { K }$ and $\sum _ { j = 1 } ^ { n } S ^ { 2 } { } _ { j } = \frac { n } { A } \left( B n ^ { 2 } + C n + D \right)$ where $A , B , C , D \in N$ and $A$ has least value then
(1) $A + C + D$ is not divisible by $D$
(2) $A + B = 5 ( D - C )$
(3) $A + B + C + D$ is divisible by 5
(4) $A + B$ is divisible by $D$

Let $a_{1} = 1, a_{2}, a_{3}, a_{4}, \ldots$ be consecutive natural numbers. Then $\tan^{-1}\left(\frac{1}{1 + a_{1}a_{2}}\right) + \tan^{-1}\left(\frac{1}{1 + a_{2}a_{3}}\right) + \ldots + \tan^{-1}\left(\frac{1}{1 + a_{2021}a_{2022}}\right)$ is equal to
(1) $\frac{\pi}{4} - \cot^{-1}(2022)$
(2) $\cot^{-1}(2022) - \frac{\pi}{4}$
(3) $\tan^{-1}(2022) - \frac{\pi}{4}$
(4) $\frac{\pi}{4} - \tan^{-1}(2022)$

Suppose f is a function satisfying $\mathrm { f } ( \mathrm { x } + \mathrm { y } ) = \mathrm { f } ( \mathrm { x } ) + \mathrm { f } ( \mathrm { y } )$ for all $\mathrm { x } , \mathrm { y } \in \mathbb { N }$ and $\mathrm { f } ( 1 ) = \frac { 1 } { 5 }$. If $\sum _ { n = 1 } ^ { m } \frac { f ( n ) } { n ( n + 1 ) ( n + 2 ) } = \frac { 1 } { 12 }$ then m is equal to $\_\_\_\_$ .

Let $a_1 = 8, a_2, a_3, \ldots, a_n$ be an A.P. If the sum of its first four terms is 50 and the sum of its last four terms is 170, then the product of its middle two terms is $\_\_\_\_$.

Let $a_1, a_2, \ldots, a_n$ be in A.P. If $a_5 = 2a_7$ and $a_{11} = 18$, then $12\left(\frac{1}{\sqrt{a_{10}} + \sqrt{a_{11}}} + \frac{1}{\sqrt{a_{11}} + \sqrt{a_{12}}} + \ldots + \frac{1}{\sqrt{a_{17}} + \sqrt{a_{18}}}\right)$ is equal to $\underline{\hspace{1cm}}$.

Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840, then the total number of persons who participated in the tournament is $\_\_\_\_$.

The sum of all those terms, of the arithmetic progression $3, 8, 13, \ldots, 373$, which are not divisible by 3, is equal to $\_\_\_\_$.

Let $S _ { a }$ denote the sum of first $n$ terms an arithmetic progression. If $S _ { 20 } = 790$ and $S _ { 10 } = 145$, then $S _ { 15 } - S _ { 5 }$ is
(1) 395
(2) 390
(3) 405
(4) 410

The number of common terms in the progressions $4,9,14,19 , \ldots$. up to $25 ^ { \text {th} }$ term and $3,6,9,12 , \ldots$. up to $37 ^ { \text {th} }$ term is:
(1) 9
(2) 5
(3) 7
(4) 8

In an A.P., the sixth term $\mathbf { a } _ { 6 } = 2$. If the $\mathbf { a } _ { 1 } \mathbf { a } _ { 4 } \mathbf { a } _ { 5 }$ is the greatest, then the common difference of the A.P., is equal to
(1) $\frac { 3 } { 2 }$
(2) $\frac { 8 } { 5 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 5 } { 8 }$

The value of $\frac { 1 \times 2 ^ { 2 } + 2 \times 3 ^ { 2 } + \ldots + 100 \times ( 101 ) ^ { 2 } } { 1 ^ { 2 } \times 2 + 2 ^ { 2 } \times 3 + \ldots + 100 ^ { 2 } \times 101 }$ is
(1) $\frac { 32 } { 31 }$
(2) $\frac { 31 } { 30 }$
(3) $\frac { 306 } { 305 }$
(4) $\frac { 305 } { 301 }$

Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10} = 390$ and the ratio of the tenth and the fifth terms is $15:7$, then $S_{15} - S_5$ is equal to:
(1) 800
(2) 890
(3) 790
(4) 690

If $\log _ { e } a , \log _ { e } b , \log _ { e } c$ are in an $A . P$. and $\log _ { e } a - \log _ { e } 2 b , \log _ { e } 2 b - \log _ { e } 3 c , \log _ { e } 3 c - \log _ { e } a$ are also in an $A . P$. , then $a : b : c$ is equal to
(1) $9 : 6 : 4$
(2) $16 : 4 : 1$
(3) $25 : 10 : 4$
(4) $6 : 3 : 2$

The $20 ^ { \text {th} }$ term from the end of the progression $20,19 \frac { 1 } { 4 } , 18 \frac { 1 } { 2 } , 17 \frac { 3 } { 4 } , \ldots , - 129 \frac { 1 } { 4 }$ is :-
(1) - 118
(2) - 110
(3) - 115
(4) - 100

Let three real numbers $a , b , c$ be in arithmetic progression and $a + 1 , b , c + 3$ be in geometric progression. If $a > 10$ and the arithmetic mean of $a , b$ and $c$ is 8 , then the cube of the geometric mean of $a , b$ and $c$ is
(1) 128
(2) 316
(3) 120
(4) 312

For $x \geqslant 0$, the least value of K , for which $4 ^ { 1 + x } + 4 ^ { 1 - x } , \frac { \mathrm {~K} } { 2 } , 16 ^ { x } + 16 ^ { - x }$ are three consecutive terms of an A.P., is equal to :
(1) 8
(2) 4
(3) 10
(4) 16

A software company sets up $m$ number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of $m$ is equal to:
(1) 150
(2) 180
(3) 160
(4) 125