4. Suppose $a , b , c$ are I A.P. and $a ^ { 2 } , b ^ { 2 } , c ^ { 2 }$ are in G.P. If $\mathrm { a } < \mathrm { b } < \mathrm { c }$ and $a + b + c = 3 / 2$, then the value of $a$ is
(A) $\quad 1 / 2 \sqrt { } 2$
(B) $1 / 2 \sqrt { } 3$
(C) $\quad 1 / 2 - 1 / \sqrt { } 3$
(D) $\quad 1 / 2 - 1 / \sqrt { } 3$

10. Let $f ( x ) = a x ^ { 2 } + b x + c$, $a ^ { 1 } 0$ and $D = b ^ { 2 } - 4 a c$. If $a + b , a ^ { 2 } + b ^ { 2 }$ and $a ^ { 3 } + b ^ { 3 }$ are in G.P., then :
(a) $\quad \mathrm { D } ^ { 1 } 0$
(b) $\quad \mathrm { bD } ^ { 1 } 0$
(c) $\quad \mathrm { cD } ^ { 1 } 0$
(d) $\quad b c ^ { 1 } 0$

Suppose four distinct positive numbers $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 }$ are in G.P. Let $b _ { 1 } = a _ { 1 }$, $b _ { 2 } = b _ { 1 } + a _ { 2 } , b _ { 3 } = b _ { 2 } + a _ { 3 }$ and $b _ { 4 } = b _ { 3 } + a _ { 4 }$. STATEMENT-1 : The numbers $b _ { 1 } , b _ { 2 } , b _ { 3 } , b _ { 4 }$ are neither in A.P. nor in G.P. and
STATEMENT-2 : The numbers $b _ { 1 } , b _ { 2 } , b _ { 3 } , b _ { 4 }$ are in H.P.
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True

Let $\mathrm { S } _ { \mathrm { k } } , \mathrm { k } = 1,2 , \ldots , 100$, denote the sum of the infinite geometric series whose first term is $\frac { \mathrm { k } - 1 } { \mathrm { k } ! }$ and the common ratio is $\frac { 1 } { \mathrm { k } }$. Then the value of $\frac { 100 ^ { 2 } } { 100 ! } + \sum _ { \mathrm { k } = 1 } ^ { 100 } \left| \left( \mathrm { k } ^ { 2 } - 3 \mathrm { k } + 1 \right) \mathrm { S } _ { \mathrm { k } } \right|$ is

Let $$M = \left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : x ^ { 2 } + y ^ { 2 } \leq r ^ { 2 } \right\} ,$$ where $r > 0$. Consider the geometric progression $a _ { n } = \frac { 1 } { 2 ^ { n - 1 } } , n = 1,2,3 , \ldots$. Let $S _ { 0 } = 0$ and, for $n \geq 1$, let $S _ { n }$ denote the sum of the first $n$ terms of this progression. For $n \geq 1$, let $C _ { n }$ denote the circle with center ( $S _ { n - 1 } , 0$ ) and radius $a _ { n }$, and $D _ { n }$ denote the circle with center ( $S _ { n - 1 } , S _ { n - 1 }$ ) and radius $a _ { n }$. Consider $M$ with $r = \frac { 1025 } { 513 }$. Let $k$ be the number of all those circles $C _ { n }$ that are inside $M$. Let $l$ be the maximum possible number of circles among these $k$ circles such that no two circles intersect. Then
(A) $k + 2 l = 22$
(B) $2 k + l = 26$
(C) $2 k + 3 l = 34$
(D) $3 k + 2 l = 40$

Let $$M = \left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : x ^ { 2 } + y ^ { 2 } \leq r ^ { 2 } \right\} ,$$ where $r > 0$. Consider the geometric progression $a _ { n } = \frac { 1 } { 2 ^ { n - 1 } } , n = 1,2,3 , \ldots$. Let $S _ { 0 } = 0$ and, for $n \geq 1$, let $S _ { n }$ denote the sum of the first $n$ terms of this progression. For $n \geq 1$, let $C _ { n }$ denote the circle with center ( $S _ { n - 1 } , 0$ ) and radius $a _ { n }$, and $D _ { n }$ denote the circle with center ( $S _ { n - 1 } , S _ { n - 1 }$ ) and radius $a _ { n }$. Consider $M$ with $r = \frac { \left( 2 ^ { 199 } - 1 \right) \sqrt { 2 } } { 2 ^ { 198 } }$. The number of all those circles $D _ { n }$ that are inside $M$ is
(A) 198
(B) 199
(C) 200
(D) 201

In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equals
(1) $\frac { 1 } { 2 } ( 1 - \sqrt { 5 } )$
(2) $\frac { 1 } { 2 } \sqrt { 5 }$
(3) $\sqrt { 5 }$
(4) $\frac { 1 } { 2 } ( \sqrt { 5 } - 1 )$

The difference between the fourth term and the first term of a Geometrical Progression is 52. If the sum of its first three terms is 26, then the sum of the first six terms of the progression is
(1) 63
(2) 189
(3) 728
(4) 364

If $( 10 ) ^ { 9 } + 2 ( 11 ) ^ { 1 } ( 10 ) ^ { 8 } + 3 ( 11 ) ^ { 2 } ( 10 ) ^ { 7 } + \ldots\ldots + 10 ( 11 ) ^ { 9 } = k ( 10 ) ^ { 9 }$, then $k$ is equal to:
(1) 100
(2) 110
(3) $\frac { 121 } { 10 }$
(4) $\frac { 441 } { 100 }$

If $m$ is the A.M. of two distinct real numbers $l$ and $n$ $(l, n > 1)$ and $G_1, G_2$ and $G_3$ are three geometric means between $l$ and $n$, then $G_1^4 + 2G_2^4 + G_3^4$ equals:
(1) $4l^2 m n$
(2) $4lm^2 n$
(3) $4lmn^2$
(4) $4l^2 m^2 n^2$

If $m$ is the A.M. of two distinct real numbers $l$ and $n$ $(l, n > 1)$ and $G _ { 1 } , G _ { 2 }$ and $G _ { 3 }$ are three geometric means between $l$ and $n$, then $G _ { 1 } ^ { 4 } + 2 G _ { 2 } ^ { 4 } + G _ { 3 } ^ { 4 }$ equals
(1) $4 l ^ { 2 } m ^ { 2 } n ^ { 2 }$
(2) $4 l ^ { 2 } m n$
(3) $4 l m ^ { 2 } n$
(4) $4 l m n ^ { 2 }$

For any three positive real numbers $a, b$ and $c$. If $9(25a^2 + b^2) + 25(c^2 - 3ac) = 15b(3a + c)$. Then
(1) $b,\ c$ and $a$ are in G.P.
(2) $b,\ c$ and $a$ are in A.P.
(3) $a,\ b$ and $c$ are in A.P.
(4) $a,\ b$ and $c$ are in G.P.

For any three positive real numbers $a$, $b$ and $c$, $9 ( 25 a ^ { 2 } + b ^ { 2 } ) + 25 ( c ^ { 2 } - 3 a c ) = 15 b ( 3 a + c )$. Then:
(1) $b$, $c$ and $a$ are in G.P.
(2) $b$, $c$ and $a$ are in A.P.
(3) $a$, $b$ and $c$ are in A.P.
(4) $a$, $b$ and $c$ are in G.P.

If $b$ is the first term of an infinite G.P whose sum is five, then $b$ lies in the interval.
(1) $( - \infty , - 10 )$
(2) $( 10 , \infty )$
(3) $( 0,10 )$
(4) $( - 10,0 )$

If $b$ is the first term of an infinite geometric progression whose sum is five, then $b$ lies in the interval
(1) $[ 10 , \infty )$
(2) $( - \infty , - 10 ]$
(3) $( - 10,0 )$
(4) $( 0,10 )$

The sum of the first 20 terms of the series $1 + \frac { 3 } { 2 } + \frac { 7 } { 4 } + \frac { 15 } { 8 } + \frac { 31 } { 16 } + \ldots$ is
(1) $39 + \frac { 1 } { 2 ^ { 19 } }$
(2) $38 + \frac { 1 } { 2 ^ { 20 } }$
(3) $38 + \frac { 1 } { 2 ^ { 19 } }$
(4) $39 + \frac { 1 } { 2 ^ { 20 } }$

If $\alpha , \beta$ and $\gamma$ are three consecutive terms of a non-constant G.P. Such that the equations $\alpha x ^ { 2 } + 2 \beta x + \gamma = 0$ and $x ^ { 2 } + x - 1 = 0$ have a common root, then $\alpha ( \beta + \gamma )$ is equal to:
(1) $\beta \gamma$
(2) $\alpha \beta$
(3) $\alpha \gamma$
(4) 0

The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P. Then the sum of the original three terms of the G.P. is:

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$, be a G.P. such that $a _ { 1 } < 0 , a _ { 1 } + a _ { 2 } = 4$ and $a _ { 3 } + a _ { 4 } = 16$. If $\sum _ { i = 1 } ^ { 9 } a _ { i } = 4 \lambda$, then $\lambda$, is equal to.
(1) - 513
(2) - 171
(3) 171
(4) $\frac { 511 } { 3 }$

The sum of the first three terms of G.P is $S$ and their products is 27. Then all such $S$ lie in
(1) $(-\infty, -9] \cup [3, \infty)$
(2) $[-3, \infty)$
(3) $(-\infty, -3] \cup [9, \infty)$
(4) $(-\infty, 9]$

Let $a _ { n }$ be the $n ^ { \text {th } }$ term of a G.P. of positive terms. If $\sum _ { n = 1 } ^ { 100 } a _ { 2 n + 1 } = 200$ and $\sum _ { n = 1 } ^ { 100 } a _ { 2 n } = 100$, then $\sum _ { n = 1 } ^ { 200 } a _ { n }$ is equal to:
(1) 300
(2) 225
(3) 175
(4) 150

The product $2 ^ { \frac { 1 } { 4 } } \bullet 4 ^ { \frac { 1 } { 16 } } \bullet 8 ^ { \frac { 1 } { 48 } } \bullet 16 ^ { \frac { 1 } { 128 } } \bullet \ldots$ to $\infty$ is equal to:
(1) $2 ^ { \frac { 1 } { 2 } }$
(2) $2 ^ { \frac { 1 } { 4 } }$
(3) 1
(4) 2

If $|x| < 1, |y| < 1$ and $x \neq 1$, then the sum to infinity of the following series $(x + y) + (x^{2} + xy + y^{2}) + (x^{3} + x^{2}y + xy^{2} + y^{3}) + \ldots$ is
(1) $\frac{x + y - xy}{(1 + x)(1 + y)}$
(2) $\frac{x + y + xy}{(1 + x)(1 + y)}$
(3) $\frac{x + y - xy}{(1 - x)(1 - y)}$
(4) $\frac{x + y + xy}{(1 - x)(1 - y)}$

If the sum of the second, third and fourth terms of a positive term G.P. is 3 and the sum of its sixth, seventh and eighth terms is 243, then the sum of the first 50 terms of this G.P. is:
(1) $\frac{1}{26}\left(3^{49}-1\right)$
(2) $\frac{1}{26}\left(3^{50}-1\right)$
(3) $\frac{2}{13}\left(3^{50}-1\right)$
(4) $\frac{1}{13}\left(3^{50}-1\right)$

If $x = \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \tan ^ { 2 } \theta$ and $y = \sum _ { n = 0 } ^ { \infty } \cos ^ { 2 n } \theta$, for $0 < \theta < \frac { \pi } { 4 }$, then:
(1) $x ( 1 + y ) = 1$
(2) $y ( 1 - x ) = 1$
(3) $y ( 1 + x ) = 1$
(4) $x ( 1 - y ) = 1$