Let $n$ denote the number of solutions of the equation $z ^ { 2 } + 3 \bar { z } = 0$, where $z$ is a complex number. Then the value of $\sum _ { k = 0 } ^ { \infty } \frac { 1 } { n ^ { k } }$ is equal to
(1) 1
(2) $\frac { 4 } { 3 }$
(3) $\frac { 3 } { 2 }$
(4) 2

Three numbers are in an increasing geometric progression with common ratio $r$. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference $d$. If the fourth term of GP is $3 r ^ { 2 }$, then $r ^ { 2 } - d$ is equal to :
(1) $7 - \sqrt { 3 }$
(2) $7 + 3 \sqrt { 3 }$
(3) $7 - 7 \sqrt { 3 }$
(4) $7 + \sqrt { 3 }$

If $e ^ { \cos ^ { 2 } x + \cos ^ { 4 } x + \cos ^ { 6 } x + \ldots \infty \log _ { e } 2 }$ satisfies the equation $t ^ { 2 } - 9 t + 8 = 0$, then the value of $\frac { 2 \sin x } { \sin x + \sqrt { 3 } \cos x }$, where $0 < x < \frac { \pi } { 2 }$, is equal to
(1) $\frac { 3 } { 2 }$
(2) $\frac { 1 } { 2 }$
(3) $\sqrt { 3 }$
(4) $2 \sqrt { 3 }$

If the sum of an infinite GP, $a , a r , a r ^ { 2 } , a r ^ { 3 } , \ldots$ is 15 and the sum of the squares of its each term is 150 , then the sum of $a r ^ { 2 } , a r ^ { 4 } , a r ^ { 6 } , \ldots$ is:
(1) $\frac { 25 } { 2 }$
(2) $\frac { 9 } { 2 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 5 } { 2 }$

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

Let $A _ { 1 } , A _ { 2 } , A _ { 3 } , \ldots$ be squares such that for each $n \geqslant 1$, 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

If $x = \sum _ { n = 0 } ^ { \infty } a ^ { n } , y = \sum _ { n = 0 } ^ { \infty } b ^ { n } , z = \sum _ { n = 0 } ^ { \infty } c ^ { n }$, where $a , b , c$ are in A.P. and $| a | < 1 , | b | < 1 , | c | < 1 , a b c \neq 0$, then
(1) $x , y , z$ are in A.P.
(2) $x , y , z$ are in G.P.
(3) $\frac { 1 } { x } , \frac { 1 } { y } , \frac { 1 } { z }$ are in A.P.
(4) $\frac { 1 } { x } + \frac { 1 } { y } + \frac { 1 } { z } = 1 - ( a + b + c )$

Let $A_1, A_2, A_3, \ldots\ldots$ be an increasing geometric progression of positive real numbers. If $A_1 A_3 A_5 A_7 = \frac{1}{1296}$ and $A_2 + A_4 = \frac{7}{36}$, then, the value of $A_6 + A_8 + A_{10}$ is equal to
(1) 43
(2) 33
(3) 37
(4) 48

Consider two G.Ps. $2,2 ^ { 2 } , 2 ^ { 3 } , \ldots$ and $4,4 ^ { 2 } , 4 ^ { 3 } , \ldots$ of 60 and $n$ terms respectively. If the geometric mean of all the $60 + n$ terms is $( 2 ) ^ { \frac { 225 } { 8 } }$, then $\sum _ { k = 1 } ^ { n } k ( n - k )$ is equal to:
(1) 560
(2) 1540
(3) 1330
(4) 2600

Let the sum of an infinite G.P., whose first term is $a$ and the common ratio is $r$, be 5 . Let the sum of its first five terms be $\frac { 98 } { 25 }$. Then the sum of the first 21 terms of an AP, whose first term is $10 a r , n ^ { \text {th } }$ term is $a _ { n }$ and the common difference is $10 a r ^ { 2 }$, is equal to
(1) $21 a _ { 11 }$
(2) $22 a _ { 11 }$
(3) $15 a _ { 16 }$
(4) $14 a _ { 16 }$

Let the first term $a$ and the common ratio $r$ of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to
(1) 241
(2) 231
(3) 210
(4) 220

If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296, respectively, then the sum of common ratios of all such GPs is
(1) 7
(2) $\frac{9}{2}$
(3) 3
(4) 14

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be a G.P. of increasing positive numbers. Let the sum of its $6 ^ { \text {th} }$ and $8 ^ { \text {th} }$ terms be 2 and the product of its $3 ^ { \text {rd} }$ and $5 ^ { \text {th} }$ terms be $\frac { 1 } { 9 }$. Then $6 ( a _ { 2 } + a _ { 4 } )( a _ { 4 } + a _ { 6 } )$ is equal to
(1) 3
(2) $3 \sqrt { 3 }$
(3) 2
(4) $2 \sqrt { 2 }$

Let $\mathrm { a } _ { 1 } , \mathrm { a } _ { 2 } , \mathrm { a } _ { 3 } , \ldots$ be a GP of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24 , then $\mathbf { a } _ { 1 } \mathbf { a } _ { 9 } + \mathbf { a } _ { 2 } \mathbf { a } _ { 4 } \mathbf { a } _ { 9 } + \mathbf { a } _ { 5 } + \mathbf { a } _ { 7 }$ is equal to

Let $\left\{ a _ { k } \right\}$ and $\left\{ b _ { k } \right\} , k \in \mathbb { N }$, be two G.P.s with common ratio $r _ { 1 }$ and $r _ { 2 }$ respectively such that $\mathrm { a } _ { 1 } = \mathrm { b } _ { 1 } = 4$ and $\mathrm { r } _ { 1 } < \mathrm { r } _ { 2 }$. Let $\mathrm { c } _ { \mathrm { k } } = \mathrm { a } _ { \mathrm { k } } + \mathrm { b } _ { \mathrm { k } } , \mathrm { k } \in \mathbb { N }$. If $\mathrm { c } _ { 2 } = 5$ and $\mathrm { c } _ { 3 } = \frac { 13 } { 4 }$ then $\sum _ { \mathrm { k } = 1 } ^ { \infty } \mathrm { c } _ { \mathrm { k } } - \left( 12 \mathrm { a } _ { 6 } + 8 \mathrm {~b} _ { 4 } \right)$ is equal to $\_\_\_\_$

Let the positive numbers $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 }$ and $a _ { 5 }$ be in a G.P. Let their mean and variance be $\frac { 31 } { 10 }$ and $\frac { m } { n }$ respectively, where $m$ and $n$ are co-prime. If the mean of their reciprocals is $\frac { 31 } { 10 }$ and $a _ { 3 } + a _ { 4 } + a _ { 5 } = 14$, then $m + n$ is equal to $\_\_\_\_$ .

Let $\alpha$ and $\beta$ be the roots of the equation $px^2 + qx - r = 0$, where $p \neq 0$. If $p, q$ and $r$ be the consecutive terms of a non-constant G.P and $\frac{1}{\alpha} + \frac{1}{\beta} = \frac{3}{4}$, then the value of $\alpha - \beta^2$ is:
(1) $\frac{80}{9}$
(2) 9
(3) $\frac{20}{3}$
(4) 8

Let $a$ and $b$ be two distinct positive real numbers. Let $11^{\text{th}}$ term of a GP, whose first term is $a$ and third term is $b$, is equal to $p^{\text{th}}$ term of another GP, whose first term is $a$ and fifth term is $b$. Then $p$ is equal to
(1) 20
(2) 25
(3) 21
(4) 24

If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to
(1) 7
(2) 4
(3) 5
(4) 6

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $\frac { 70 } { 3 }$ and the product of the third and fifth terms is 49 . Then the sum of the $4 ^ { \text {th} } , 6 ^ { \text {th} }$ and $8 ^ { \text {th} }$ terms is equal to : (1) 96 (2) 91 (3) 84 (4) 78

Let $a , a r , a r ^ { 2 } , \quad$ be an infinite G.P. If $\sum _ { n = 0 } ^ { \infty } a r ^ { n } = 57$ and $\sum _ { n = 0 } ^ { \infty } a ^ { 3 } r ^ { 3 n } = 9747$, then $a + 18 r$ is equal to
(1) 46
(2) 38
(3) 31
(4) 27

If each term of a geometric progression $\mathrm { a } _ { 1 } , \mathrm { a } _ { 2 } , \mathrm { a } _ { 3 } , \ldots$ with $\mathrm { a } _ { 1 } = \frac { 1 } { 8 }$ and $\mathrm { a } _ { 2 } \neq \mathrm { a } _ { 1 }$, is the arithmetic mean of the next two terms and $\mathrm { S } _ { \mathrm { n } } = \mathrm { a } _ { 1 } + \mathrm { a } _ { 2 } + \ldots + \mathrm { a } _ { \mathrm { n } }$, then $\mathrm { S } _ { 20 } - \mathrm { S } _ { 18 }$ is equal to
(1) $2 ^ { 15 }$
(2) $- 2 ^ { 18 }$
(3) $2 ^ { 18 }$
(4) $- 2 ^ { 15 }$

If $2 \tan ^ { 2 } \theta - 5 \sec \theta = 1$ has exactly 7 solutions in the interval $0 , \frac { \mathrm { n } \pi } { 2 }$, for the least value of $\mathrm { n } \in \mathrm { N }$ then $\sum _ { \mathrm { k } = 1 } ^ { \mathrm { n } } \frac { \mathrm { k } } { 2 ^ { \mathrm { k } } }$ is equal to :
(1) $\frac { 1 } { 2 ^ { 15 } } 2 ^ { 14 } - 14$
(2) $\frac { 1 } { 2 _ { 1 } ^ { 14 } } 2 ^ { 15 } - 15$
(3) $1 - \frac { 15 } { 2 ^ { 13 } }$
(4) $\frac { 1 } { 2 ^ { 13 } } 2 ^ { 14 } - 15$

Let $ABC$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $ABC$ and the same process is repeated infinitely many times. If P is the sum of perimeters and $Q$ is the sum of areas of all the triangles formed in this process, then:
(1) $\mathrm { P } ^ { 2 } = 6 \sqrt { 3 } \mathrm { Q }$
(2) $\mathrm { P } ^ { 2 } = 36 \sqrt { 3 } \mathrm { Q }$
(3) $P = 36 \sqrt { 3 } Q ^ { 2 }$
(4) $\mathrm { P } ^ { 2 } = 72 \sqrt { 3 } \mathrm { Q }$

If three successive terms of a G.P. with common ratio $r$ ($r > 1$) are the lengths of the sides of a triangle and $[r]$ denotes the greatest integer less than or equal to $r$, then $3[r] + [-r]$ is equal to: