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) $x y z = 4$

If $I _ { n } = \int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 2 } } \cot ^ { n } x \, d x$, then
(1) $I _ { 2 } + I _ { 4 } , \left( I _ { 3 } + I _ { 5 } \right) ^ { 2 } , I _ { 4 } + I _ { 6 }$ are in G.P.
(2) $I _ { 2 } + I _ { 4 } , I _ { 3 } + I _ { 5 } , I _ { 4 } + I _ { 6 }$ are in A.P.
(3) $\frac { 1 } { I _ { 2 } + I _ { 4 } } , \frac { 1 } { I _ { 3 } + I _ { 5 } } , \frac { 1 } { I _ { 4 } + I _ { 6 } }$ are in A.P.
(4) $\frac { 1 } { I _ { 2 } + I _ { 4 } } , \frac { 1 } { I _ { 3 } + I _ { 5 } } , \frac { 1 } { I _ { 4 } + I _ { 6 } }$ are in G.P.

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

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

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

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 $S = 2 + \frac { 6 } { 7 } + \frac { 12 } { 7 ^ { 2 } } + \frac { 20 } { 7 ^ { 3 } } + \frac { 30 } { 7 ^ { 4 } } + \ldots$. then $4 S$ is equal to
(1) $\left( \frac { 7 } { 2 } \right) ^ { 2 }$
(2) $\left( \frac { 7 } { 3 } \right) ^ { 3 }$
(3) $\frac { 7 } { 3 }$
(4) $\left( \frac { 7 } { 3 } \right) ^ { 4 }$

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) $\frac{9 \cdot 3^{10} + 1}{2}$

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 $\frac { 6 } { 3 ^ { 12 } } + \frac { 10 } { 3 ^ { 11 } } + \frac { 20 } { 3 ^ { 10 } } + \frac { 40 } { 3 ^ { 9 } } + \ldots + \frac { 10240 } { 3 } = 2 ^ { n } \cdot m$, where $m$ is odd, then $m \cdot n$ is equal to $\_\_\_\_$.

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 $A _ { 1 }$ and $A _ { 2 }$ be two arithmetic means and $G _ { 1 } , G _ { 2 }$ and $G _ { 3 }$ be three geometric means of two distinct positive numbers. Then $G _ { 1 } ^ { 4 } + G _ { 2 } ^ { 4 } + G _ { 3 } ^ { 4 } + G _ { 1 } ^ { 2 } G _ { 3 } ^ { 2 }$ is equal to
(1) $\left( A _ { 1 } + A _ { 2 } \right) ^ { 2 } G _ { 1 } G _ { 3 }$
(2) $2 \left( A _ { 1 } + A _ { 2 } \right) G _ { 1 } G _ { 3 }$
(3) $\left( A _ { 1 } + A _ { 2 } \right) G _ { 1 } ^ { 2 } G _ { 3 } ^ { 2 }$
(4) $2 \left( A _ { 1 } + A _ { 2 } \right) G _ { 1 } ^ { 2 } G _ { 3 } ^ { 2 }$

Let $0 < z < y < x$ be three real numbers such that $\frac { 1 } { x } , \frac { 1 } { y } , \frac { 1 } { z }$ are in an arithmetic progression and $x , \sqrt{2} y , z$ are in a geometric progression. If $x y + y z + z x = \frac { 3 } { \sqrt { 2 } } x y z$, then $3 ( x + y + z ) ^ { 2 }$ is equal to

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

If $(20)^{19} + 2(21)(20)^{18} + 3(21)^{2}(20)^{17} + \ldots + 20(21)^{19} = k(20)^{19}$, then $k$ is equal to $\_\_\_\_$.

For the two positive numbers $a , b$, if $a , b$ and $\frac { 1 } { 18 }$ are in a geometric progression, while $\frac { 1 } { a } , 10$ and $\frac { 1 } { b }$ are in an arithmetic progression, then $16 a + 12 b$ 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 $\_\_\_\_$

The parabolas: $ax^{2} + 2bx + cy = 0$ and $dx^{2} + 2ex + fy = 0$ intersect on the line $y = 1$. If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$ are in G.P., then
(1) $d, e, f$ are in A.P.
(2) $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in G.P.
(3) $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in A.P.
(4) $d, e, f$ are in G.P.

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