The number of real solutions of the equation $$\sin ^ { - 1 } \left( \sum _ { i = 1 } ^ { \infty } x ^ { i + 1 } - x \sum _ { i = 1 } ^ { \infty } \left( \frac { x } { 2 } \right) ^ { i } \right) = \frac { \pi } { 2 } - \cos ^ { - 1 } \left( \sum _ { i = 1 } ^ { \infty } \left( - \frac { x } { 2 } \right) ^ { i } - \sum _ { i = 1 } ^ { \infty } ( - x ) ^ { i } \right)$$ lying in the interval $\left( - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$ is $\_\_\_\_$. (Here, the inverse trigonometric functions $\sin ^ { - 1 } x$ and $\cos ^ { - 1 } x$ assume values in $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ and $[ 0 , \pi ]$, respectively.)

Let
$$\alpha = \sum _ { k = 1 } ^ { \infty } \sin ^ { 2 k } \left( \frac { \pi } { 6 } \right) .$$
Let $g : [ 0,1 ] \rightarrow \mathbb { R }$ be the function defined by
$$g ( x ) = 2 ^ { \alpha x } + 2 ^ { \alpha ( 1 - x ) }$$
Then, which of the following statements is/are TRUE ?
(A) The minimum value of $g ( x )$ is $2 ^ { \frac { 7 } { 6 } }$
(B) The maximum value of $g ( x )$ is $1 + 2 ^ { \frac { 1 } { 3 } }$
(C) The function $g ( x )$ attains its maximum at more than one point
(D) The function $g ( x )$ attains its minimum at more than one point

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) $x ( 1 - y ) = 1$

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 the sequence $\left\{ a _ { n } \right\} ( n = 1,2,3 , \cdots )$ be an arithmetic progression satisfying
$$a _ { 2 } = 2 , \quad a _ { 6 } = 3 a _ { 3 } .$$
Then, consider the series $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } } { r ^ { a _ { n } } }$, where $r$ is a positive real number.
(1) When we denote the first term of $\left\{ a _ { n } \right\}$ by $a$, and the common difference by $d$, we have
$$a = \mathbf { A B } , \quad d = \mathbf { C } .$$
(2) The series $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } } { r ^ { a _ { n } } }$ is an infinite geometric series where the first term is $\square r^{\mathbf{E}}$, and the common ratio is $\dfrac { \mathbf { F } } { r^{\mathbf{G}} }$. Hence, this series converges when
$$r > 3 ^ { \frac { \mathbf { H } } { \mathbf{I} } } ,$$
and its sum $S$ is
$$S = \frac { \square \mathbf { J } \, r ^ { \mathbf { K } } } { r ^ { \mathbf { L } } - \mathbf { M } } .$$
(3) This sum $S$ is minimized at
$$r = \mathbf { N } ^ { \frac { \mathbf { O } } { 2 } } .$$

You are given that
$$S = 4 + \frac { 8 k } { 7 } + \frac { 16 k ^ { 2 } } { 49 } + \frac { 32 k ^ { 3 } } { 343 } + \cdots + 4 \left( \frac { 2 k } { 7 } \right) ^ { n } + \cdots$$
The value for $k$ is chosen as an integer in the range $- 5 \leq k \leq 5$
All possible values for $k$ are equally likely to be chosen.
What is the probability that the value of $S$ is a finite number greater than 3 ?

Let $R$ be the set of real numbers. For every natural number n,
$$\begin{aligned} & f _ { n } : [ n \pi , ( n + 1 ) \pi ] \rightarrow R \\ & f _ { n } ( x ) = \frac { 1 } { 5 ^ { n } } | \sin x | \end{aligned}$$
What is the sum of the areas of the regions between the functions defined in this form and the x-axis in square units?
A) $\frac { 7 } { 5 }$
B) $\frac { 8 } { 5 }$
C) $\frac { 9 } { 5 }$
D) $\frac { 3 } { 2 }$
E) $\frac { 5 } { 2 }$