If the system of linear equations $$\begin{aligned} & 2x + 2ay + az = 0 \\ & 2x + 3by + bz = 0 \\ & 2x + 4cy + cz = 0 \end{aligned}$$ where $a, b, c \in R$ are non-zero and distinct; has a non-zero solution, then
(1) $\frac { 1 } { a } , \frac { 1 } { b } , \frac { 1 } { c }$ are in $A.P$.
(2) $a, b, c$ are in $G.P$.
(3) $a + b + c = 0$
(4) $a, b, c$ are in $A.P$.

Let $f : R \rightarrow R$ be a function which satisfies $f ( x + y ) = f ( x ) + f ( y ) , \forall x , y \in R$. If $f ( 1 ) = 2$ and $g ( n ) = \sum _ { k = 1 } ^ { ( n - 1 ) } f ( k ) , n \in N$ then the value of $n$, for which $g ( n ) = 20$, is
(1) 5
(2) 20
(3) 4
(4) 9

The sum of 10 terms of the series $\frac { 3 } { 1 ^ { 2 } \times 2 ^ { 2 } } + \frac { 5 } { 2 ^ { 2 } \times 3 ^ { 2 } } + \frac { 7 } { 3 ^ { 2 } \times 4 ^ { 2 } } + \ldots$ is :
(1) $\frac { 143 } { 144 }$
(2) $\frac { 99 } { 100 }$
(3) 1
(4) $\frac { 120 } { 121 }$

Let $S _ { 1 }$ be the sum of first $2 n$ terms of an arithmetic progression. Let $S _ { 2 }$ be the sum of first $4 n$ terms of the same arithmetic progression. If ( $S _ { 2 } - S _ { 1 }$ ) is 1000 , then the sum of the first $6 n$ terms of the arithmetic progression is equal to:
(1) 1000
(2) 7000
(3) 5000
(4) 3000

Let $S _ { n }$ denote the sum of first $n$-terms of an arithmetic progression. If $S _ { 10 } = 530 , S _ { 5 } = 140$, then $S _ { 20 } - S _ { 6 }$ is equal to:
(1) 1862
(2) 1842
(3) 1852
(4) 1872

Let $S _ { n } = 1 \cdot ( n - 1 ) + 2 \cdot ( n - 2 ) + 3 \cdot ( n - 3 ) + \ldots + ( n - 1 ) \cdot 1 , \quad n \geqslant 4$. The sum $\sum _ { n = 4 } ^ { \infty } \frac { 2 S _ { n } } { n ! } - \frac { 1 } { ( n - 2 ) ! }$ is equal to :
(1) $\frac { e - 2 } { 6 }$
(2) $\frac { e - 1 } { 3 }$
(3) $\frac { e } { 6 }$
(4) $\frac { \mathrm { e } } { 3 }$

If $\alpha , \beta$ are natural numbers such that $100 ^ { \alpha } - 199 \beta = ( 100 ) ( 100 ) + ( 99 ) ( 101 ) + ( 98 ) ( 102 ) + \ldots . + ( 1 ) ( 199 )$, then the slope of the line passing through $( \alpha , \beta )$ and origin is:
(1) 540
(2) 550
(3) 530
(4) 510

Let $a _ { 1 } , a _ { 2 } , \ldots , a _ { 21 }$ be an A.P. such that $\sum _ { n = 1 } ^ { 20 } \frac { 1 } { a _ { n } a _ { n + 1 } } = \frac { 4 } { 9 }$. If the sum of this A.P. is 189 , then $\mathrm { a } _ { 6 } \mathrm { a } _ { 16 }$ is equal to :
(1) 57
(2) 48
(3) 36
(4) 72

Let $a _ { 1 } , \quad a _ { 2 } , \quad a _ { 3 } , \quad \ldots$ be an A.P. If $\frac { a _ { 1 } + a _ { 2 } + \ldots + a _ { 10 } } { a _ { 1 } + a _ { 2 } + \ldots + a _ { p } } = \frac { 100 } { p ^ { 2 } }$, then $\frac{a_{11}}{a_{10}}$ is equal to:

$\frac { 1 } { 3 ^ { 2 } - 1 } + \frac { 1 } { 5 ^ { 2 } - 1 } + \frac { 1 } { 7 ^ { 2 } - 1 } + \ldots + \frac { 1 } { ( 201 ) ^ { 2 } - 1 }$ is equal to
(1) $\frac { 101 } { 404 }$
(2) $\frac { 25 } { 101 }$
(3) $\frac { 101 } { 408 }$
(4) $\frac { 99 } { 400 }$

A function $f ( x )$ is given by $f ( x ) = \frac { 5 ^ { x } } { 5 ^ { x } + 5 }$, then the sum of the series $f \left( \frac { 1 } { 20 } \right) + f \left( \frac { 2 } { 20 } \right) + f \left( \frac { 3 } { 20 } \right) + \ldots + f \left( \frac { 39 } { 20 } \right)$ is equal to:
(1) $\frac { 19 } { 2 }$
(2) $\frac { 49 } { 2 }$
(3) $\frac { 39 } { 2 }$
(4) $\frac { 29 } { 2 }$

Let $A = \{ n \in N : n$ is a 3-digit number $\}$, $B = \{ 9 k + 2 : k \in N \}$ and $C = \{ 9 k + l : k \in N \}$ for some $l ( 0 < l < 9 )$. If the sum of all the elements of the set $A \cap ( B \cup C )$ is $274 \times 400$, then $l$ is equal to

Let $A _ { 1 } , A _ { 2 } , A _ { 3 } , \ldots$ be squares such that for each $n \geq 1$, the length of the side of $A _ { n }$ equals the length of diagonal of $A _ { n + 1 }$. If the length of $A _ { 1 }$ is 12 cm, then the smallest value of $n$ for which area of $A _ { n }$ is less than one is

A set of 20 tuning forks is arranged in a series of increasing frequencies. If each fork gives 4 beats with respect to the preceding fork and the frequency of the last fork is twice the frequency of the first, then the frequency of last fork is $\_\_\_\_$ Hz.

If $\frac { 1 } { 2 \cdot 3 ^ { 10 } } + \frac { 1 } { 2 ^ { 2 } \cdot 3 ^ { 9 } } + \ldots + \frac { 1 } { 2 ^ { 10 } \cdot 3 } = \frac { K } { 2 ^ { 10 } \cdot 3 ^ { 10 } }$, then the remainder when $K$ is divided by 6 is
(1) 2
(2) 3
(3) 4
(4) 5

Suppose $a _ { 1 } , a _ { 2 } , \ldots , a _ { \mathrm { n } } , \ldots$ be an arithmetic progression of natural numbers. If the ratio of the sum of the first five terms to the sum of first nine terms of the progression is 5 : 17 and $110 < a _ { 15 } < 120$, then the sum of the first ten terms of the progression is equal to
(1) 290
(2) 380
(3) 460
(4) 510

If $\left\{ a _ { i } \right\} _ { i = 1 } ^ { \mathrm { n } }$, where $n$ is an even integer, is an arithmetic progression with common difference 1 , and $\sum _ { i = 1 } ^ { n } a _ { i } = 192 , \sum _ { i = 1 } ^ { \frac { n } { 2 } } a _ { 2 i } = 120$, then $n$ is equal to
(1) 18
(2) 36
(3) 96
(4) 48

The sum $1 + 2 \cdot 3 + 3 \cdot 3^2 + \ldots + 10 \cdot 3^9$ is equal to
(1) $\frac{2 \cdot 3^{12} + 10}{4}$
(2) $\frac{19 \cdot 3^{10} + 1}{4}$
(3) $5 \cdot 3^{10} - 2$
(4) $\frac{9 \cdot 3^{10} + 1}{2}$

If $a _ { 1 } , a _ { 2 } , a _ { 3 } \ldots$ and $b _ { 1 } , b _ { 2 } , b _ { 3 } \ldots$ are A.P. and $a _ { 1 } = 2 , a _ { 10 } = 3 , a _ { 1 } b _ { 1 } = 1 = a _ { 10 } b _ { 10 }$ then $a _ { 4 } b _ { 4 }$ is equal to
(1) $\frac { 28 } { 27 }$
(2) $\frac { 28 } { 24 }$
(3) $\frac { 23 } { 26 }$
(4) $\frac { 22 } { 23 }$

Let $f : N \rightarrow R$ be a function such that $f( x + y ) = 2 f(x) f(y)$ for natural numbers $x$ and $y$. If $f(1) = 2$, then the value of $\alpha$ for which $\sum _ { k = 1 } ^ { 10 } f ( \alpha + k ) = \frac { 512 } { 3 } ( 2 ^ { 20 } - 1 )$ holds, is
(1) 3
(2) 4
(3) 5
(4) 6

If $b_n = \int_0^{\frac{\pi}{2}} \frac{\cos^2 nx}{\sin x} dx$, $n \in \mathbb{N}$, then
(1) $b_3 - b_2, b_4 - b_3, b_5 - b_4$ are in an A.P. with common difference $-2$
(2) $\frac{1}{b_3 - b_2}, \frac{1}{b_4 - b_3}, \frac{1}{b_5 - b_4}$ are in an A.P. with common difference $2$
(3) $b_3 - b_2, b_4 - b_3, b_5 - b_4$ are in a G.P.
(4) $\frac{1}{b_3 - b_2}, \frac{1}{b_4 - b_3}, \frac{1}{b_5 - b_4}$ are in an A.P. with common difference $-2$

Let $a, b$ be two non-zero real numbers. If $p$ and $r$ are the roots of the equation $x ^ { 2 } - 8ax + 2a = 0$ and $q$ and $s$ are the roots of the equation $x ^ { 2 } + 12bx + 6b = 0$, such that $\frac { 1 } { p }, \frac { 1 } { q }, \frac { 1 } { r }, \frac { 1 } { s }$ are in A.P., then $a ^ { - 1 } - b ^ { - 1 }$ is equal to $\_\_\_\_$.

$\frac { 2 ^ { 3 } - 1 ^ { 3 } } { 1 \times 7 } + \frac { 4 ^ { 3 } - 3 ^ { 3 } + 2 ^ { 3 } - 1 ^ { 3 } } { 2 \times 11 } + \frac { 6 ^ { 3 } - 5 ^ { 3 } + 4 ^ { 3 } - 3 ^ { 3 } + 2 ^ { 3 } - 1 ^ { 3 } } { 3 \times 15 } + \ldots \ldots + \frac { 30 ^ { 3 } - 29 ^ { 3 } + 28 ^ { 3 } - 27 ^ { 3 } + \ldots + 2 ^ { 3 } - 1 ^ { 3 } } { 15 \times 63 }$ is equal to $\_\_\_\_$ .

Let $A = \sum _ { i = 1 } ^ { 10 } \sum _ { j = 1 } ^ { 10 } \min \{ i , j \}$ and $B = \sum _ { i = 1 } ^ { 10 } \sum _ { j = 1 } ^ { 10 } \max \{ i , j \}$. Then $A + B$ is equal to $\_\_\_\_$.

Let $a _ { 1 } = b _ { 1 } = 1$, $a _ { n } = a _ { n - 1 } + 2$ and $b _ { n } = a _ { n } + b _ { n - 1 }$ for every natural number $n \geq 2$. Then $\sum _ { n = 1 } ^ { 15 } a _ { n } \cdot b _ { n }$ is equal to $\_\_\_\_$.