Assume that $a_1,a_2,a_3,$cdots,$a_k$ form an arithmetic sequence. If we have $a_5+a_8+a_{11}=10,a_7+a_{10}+a_{13}=12,$ and $a_k=11,$ then $k=$

$($a$)17$

$($b$)19$

$($c$)23$

$($d$)29$

$($e$)31$

Given two lines $L_1$:$x+ay+b=0$ and $L_2$:$x+cy+d=0$ as shown on the right, How many among the real numbers $a,b,c,d,a-c$ are positive $?$

$($a$)1$

$($b$)2$

$($c$)3$

$($d$)4$

$($e$)5$

Let $O$ be the origin of coordinate plane, $A,B$ lie on the upper half plane satisfying $\frac{?}{\text{b}\text{a}\text{r}}(OA)\frac{?}{b}=ar(OB).$ If line $OA$ has slope $1,$ line $OB$ has slope $-7,$ what is the slope of line $AB?$

$($a$)\frac{1}{2}$

$($b$)\frac{1}{3}$

$($c$)-\frac{1}{2}$

$($d$)-\frac{1}{3}$

$($e$)-\frac{1}{4}$

If $f(x)=x^5+x^4-5x^3-2x^2+6x-2,g(x)=x^5-3x^3+2x^2-4x+2.$ How many distinct real numbers a satisfy $f(a)=g(a)=0?$

$($a$)1$

$($b$)2$

$($c$)3$

$($d$)4$

$($e$)0$

If $f(x)=x^3-2x^2-x+5,$ then the remainder of $g(x)=f(f(x))$ divided by $(x-2)$ is

$($a$)3$

$($b$)5$

$($c$)7$

$($d$)9$

$($e$)11$

How many lines on the coordinate plane that has distance $1$ to the origin $O(0,0)$ and distance $2$ to the point $A(3,0)?$

$($a$)1$

$($b$)2$

$($c$)3$

$($d$)4$

$($e$)$ infinitely many

If the maximum and minimum of $f(t)=-t^2+10t+11$ for $1t10$ are $M$ and $m$ respectively, then $M-m$ is

$($a$)9$

$($b$)16$

$($c$)20$

$($d$)25$

$($e$)36$