Let $A = \{x \in (0, \pi) - \{\frac{\pi}{2}\} : \log_{(2/\pi)}|\sin x| + \log_{(2/\pi)}|\cos x| = 2\}$ and $B = \{x \geq 0 : \sqrt{x}(\sqrt{x} - 4) - 3|\sqrt{x} - 2| + 6 = 0\}$. Then $n(A \cup B)$ is equal to:

Find the simplest form of the following switching circuit.

Let $A$ be the set of first $101$ terms of an $A$.$P$.,whose first term is $1$ and the common difference is $5$, and let $B$ be the set of first $71$ terms of an $A$.$P$.,whose first term is $9$ and the common difference is $7$. Then, the number of elements in $A \cap B$ which are divisible by $3$ is:

Let $S = \{(m, n): m, n \in \{1, 2, 3, \ldots, 50\}\}$. If the number of elements $(m, n)$ in $S$ such that $6^{m} + 9^{n}$ is a multiple of $5$ is $p$ and the number of elements $(m, n)$ in $S$ such that $m + n$ is a square of a prime number is $q$, then $p + q$ is equal to :

The number of polynomials $p(x)$ with integer coefficients such that the curve $y=p(x)$ passes through $(2,2)$ and $(4,5)$ is

Let $x_k$ be real numbers such that $x_k \geq k^4+k^2+1$ for $1 \leq k \leq 2018$. Denote $N=\sum_{k=1}^{2018} k$. Consider the following inequalities.
$I$. $\left(\sum_{k=1}^{2018} k x_k\right)^2 \leq N\left(\sum_{k=1}^{2018} k x_k^2\right)$
$II$. $\left(\sum_{k=1}^{2018} k x_k\right)^2 \leq N\left(\sum_{k=1}^{2018} k^2 x_k^2\right)$
Then,