QUESTION $1(7$ Marks$)$

Evaluate the following integral

${\displaystyle \underset{0}{\overset{ln2}{}}}\frac{coshx}{\sqrt[\phantom{2}]{sin{h}^{2}x+4sinhx+2}}dx.$

QUESTION $2(7$ Marks$)$

Find ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{ta{n}^{-1}x}{x^2}dx$ using integration $by$ parts. $(7$ marks$)$

QUESTION $3(5$ Marks$)$

Determine the convergence $of$ the integral

${\displaystyle \underset{\frac{}{3}}{\overset{\frac{}{2}}{}}}\frac{tanx}{(ln(cosx){)}^2}dx$

$(5$ marks$)$

QUESTION $4(11$ Marks$)$

Show that the Maclaurin series for $\frac{2e^{-\frac{x}{2}}}{\sqrt[\phantom{2}]{4-x^2}}is1-\frac{x}{2}+\frac{x^2}{4}+$cdots.

$(5$ marks$)$

Hence,

$i.$ evaluate $\underset{x0}{lim}\frac{\frac{2e^{-\frac{x}{2}}}{\sqrt[\phantom{2}]{4-x^2}}-1}{x}.$

$(3$ marks$)$

$ii.$ approximate the definite integral ${\displaystyle \underset{0\mathrm{.}1}{\overset{0\mathrm{.}4}{}}}\frac{2e^{-\frac{x}{2}}}{x\sqrt[\phantom{2}]{4-x^2}}dx$ correct $to4$ decimal

places.

$(3$ marks$)$

QUESTION $5(15$ Marks$)$

Let $A(-1,3,3),B(4,-1,0),C(5,4,-7),$ and $D(2,-1,-1).$ Find

$i.$ the normal vector $to$ the plane ABC.

$ii.$ the parametric equation $of$ the line perpendicular $to$ the plane ABC and

passes through point $B.$

iii. the equation $of$ the plane ABD.

$iv.$ the acute angle between planes ABD and ABC.

QUESTION $6(5$ Marks$)$

Find all the eigenvalues $of$ the matrix $P=([1,0,-2],[0,3,0],[-2,0,1]).$ Hence, obtain

$an$ eigenvector corresponding $to$ the smallest eigenvalue $of$ the matrix $P.$

$(5$ marks$)$

QUESTION $7(15$ Marks$)$

$aA=([1,1,0],[3,-2,2],[2,-1,1]).By$ using adjoint method, find the

inverse $ofA.$ Hence, solve the system $of$ linear equations $Ax=B$ where

$x=([x],[y],[z])$ and $B=([-2],[3],[1]).$

$(7$ marks$)$