QUESTION $1$

a$.$ Using an appropriate test for convergence, determine whether or not the series ${\displaystyle \underset{n=1}{\overset{}{}}}\frac{n^2}{(n+2)!}$ converge. Show your workings to justify your response.

$[4$ marks$]$

b$.$ Evaluate: ${\displaystyle \underset{0}{\overset{}{}}}3e^{-2x}dx$ and state if this integral converges or diverges.

$[3$ marks$]$

c$.$ Given the vectors $\frac{?}{b}=ar()-\frac{3}{b}ar()\frac{+}{b}ar(k)$ and $\frac{?}{\text{b}\text{a}\text{r}}(b)=\frac{2}{b}ar()\frac{+}{b}ar()-\frac{2}{b}ar(k),$ find the:

i$.,$ scalar $($dot$)$ product, $\frac{*}{b}ar(b)$;

ii$.,$ vector $($cross$)$ product, $\frac{}{b}ar(b)$;

iii. angle between the vectors, expressed in degrees, to two decimal places.

$[2+3+4$ marks$]$

d$.$ A function is defined by $f(x)=\frac{3x}{1+x^2}.$

Find:

i$.,$ the intercepts $(x$ and $y)$;

ii$.$ asymptotes $($vertical and$/$or horizontal$).$

$[2+2$ marks$]$

$[$Total marks $20]$