The number of 5 digit numbers which are divisible by 4, with digits from the set $\left\{1,2,3,4,5\right\}$and the repetition of digits is allowed, is ________.

The value of ${\left({\left({\left(\log \right)}_{2}9\right)}^2\right)}^{\frac{1}{{\left(\log \right)}_2\left({\left(\log \right)}_{2}9\right)}}\times {\left(\sqrt{7}\right)}^{\frac{1}{{\left(\log \right)}_{4}7}}$is ________.

Let $a,b,c$be three non-zero real numbers such that the equation $\sqrt{3}a\cos x+2b\sin x=c,x\in \left[-\frac{\pi }{2},\frac{\pi }{2}\right]$, has two distinct real roots $\alpha$and $\beta$with $\alpha +\beta =\frac{\pi }{3}$. Then, the value of $\frac{b}{a}$is _______.

Let $f:R\to R$and $g:R\to R$be two non-constant differentiable functions. If $f^{prime}\left(x\right)=\left(e^{\left(f\left(x\right)-g\left(x\right)\right)}\right)g^{prime}\left(x\right)$for all $x\in R$, and $f\left(1\right)=g\left(2\right)=1$, then which of the following statement(s) is (are) TRUE?$f\left(2\right)<1-{\left(\log \right)}_{e}2$(b) $f\left(2\right)>1-{\left(\log \right)}_{e}2$(c) $g\left(1\right)>1-{\left(\log \right)}_{e}2$(d) $g\left(1\right)<1-{\left(\log \right)}_{e}2$

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that $\vec{a}.\vec{b}=0$ For some $x,y\in R$, let $\vec{c}=x\vec{a}+y\vec{b}+(\vec{a}\times \vec{b}$ If $(\left|\vec{c}\right|=2$ and the vector $\vec{c}$ is inclined at same angle $\alpha$ to both $\vec{a}$ and $\vec{b}$ then the value of $8{\cos }^2\alpha$ is

For every twice differentiable function $f:R\to \left[-2,2\right]$with ${\left(f\left(0\right)\right)}^2+{\left(f^{prime}\left(0\right)\right)}^2=85$, which of the following statement(s) is (are) TRUE?There exist $r,s\in R$where $r<s$, such that $f$is one-one on the open interval $\left(r,s\right)$(b) There exists $x_0\in \left(-4,0\right)$such that $\left|f^{prime}\left(x_0\right)\right|\le 1$(c) ${\left(\lim \right)}_{x\to \infty }f\left(x\right)=1$(d) There exists $\alpha \in \left(-4,4\right)$such that $f\left(\alpha \right)+f\left(\alpha \right)=0$and $f^{prime}\left(\alpha \right)\ne 0$

For a non-zero complex number $z$, let $arg\left(z\right)$denote theprincipal argument with $\pi <arg\left(z\right)\le \pi$Then, whichof the following statement(s) is (are) FALSE?$arg\left(-1,-i\right)=\frac{\pi }{4},$where $i=\sqrt{-1}$(b) The function $f:R\to \left(-\pi ,\pi \right],$defined by $f\left(t\right)=arg\left(-1+it\right)$for all $t\in R$, iscontinuous at all points of $\mathbb{R}$, where $i=\sqrt{-1}$(c) For any two non-zero complex numbers $z_1$and $z_2$, $arg\left(\frac{z_1}{z_2}\right)-arg\left(z_1\right)+arg\left(z_2\right)$is an integer multiple of $2\pi$(d) For any three given distinct complex numbers $z_1$, $z_2$and $z_3$, the locus of the point $z$satisfying the condition $arg\left(\frac{\left(z-z_1\right)\left(z_2-z_3\right)}{\left(z-z_3\right)\left(z_2-z_1\right)}\right)=\pi$, lies on a straight line

For each positive integer $n$, let $y_n=\frac{1}{n}{\left(\left(n+1\right)\left(n+2\right)\dot{n+n}\right)}^{\frac{1}{n}}$For $x\in R$let $\left[x\right]$be the greatest integer less than or equal to $x$. If ${\left(\lim \right)}_{n\to \infty }{y}_n=L$, then the value of $\left[L\right]$is ______.

Let $f:\left[0,\infty \right)\to R$be a continuous function such that $f\left(x\right)=1-2x+{\int }_0^x{e}^{x-t}f\left(t\right)dt$for all $x\in \left[0,\infty \right)$. Then, which of the following statement(s) is (are) TRUE?The curve $y=f\left(x\right)$passes through the point $\left(1,2\right)$(b) The curve $y=f\left(x\right)$passes through the point $\left(2,-1\right)$(c) The area of the region $\left\{\left(x,y\right)\in \left[0,1\right]\times R:f\left(x\right)\le y\le \sqrt{1-x^2}\right\}$is $\frac{\pi -2}{4}$(d) The area of the region $\left\{\left(x,y\right)\in \left[0,1\right]\times R:f\left(x\right)\le y\le \sqrt{1-x^2}\right\}$is $\frac{\pi -1}{4}$

Let $X$be the set consisting of the first 2018 terms of the arithmetic progression $1,6,11,\ddot{,}$and $Y$be the set consisting of the first 2018 terms of the arithmetic progression $9,16,23,\ddot{}$. Then, the number of elements in the set $X\cup Y$is _____.