CIATE

QUESTION ONE

$($a$)$ Integrate ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{e}^{x}cosxdx-$

$[5$ marks$]$

$($b$)$ Find the area between the line $y2x=28$ and the curve $y{x}^2=36.$

$[5$ marks$]$

$($c$)$ Solve the differential by using homogeneous method

$xsin(\frac{y}{x}dy)=(ysin(\frac{y}{x})-x)dx$

$[5$ marks$]$

$($d$)$ Expand sin $x$ in powers of $(x-\frac{}{4})$ using taylor series up to power $5.$

$[5$ marks$]$

$[$Total: $20$ marks$]$

QUESTION TWO

$($a$)$ Integrate ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{x}^2\sqrt[3]{1x}dx.$

$[6$ marks$]$

$($b$)$ Find the directional derivative of $x^2{y}^2{z}^2$ at $(2,1,2)$ in the direction $i2j.$

$[4$ marks$]$

$($c$)$ Solve the differential equation uning the Linaar form $x\frac{dy}{dx}2y=x^{3}lnx.$

$[5$ marks$]$

$($d$)$ Expand cos $\sqrt[\phantom{2}]{x}$ using the Maclaurin's series. Expand until the power $6.$

$[5$ marks$]$

$[$Total: $20$ marks$]$

QUESTION THREE

$($a$)$ Use Variable Separable to solve the differential equation $\frac{dy}{dx}=\frac{x(2lnx1)}{sinyycosy}$

$[6$ marks$]$

$($b$)$ Find the volume generated by revolving the area in the first quadrant bounded by $y^2=8x,x=2,y=0$ the $y-$axis.

$[4$ marks$]$

$($c$)$ fvec$(A)=(xyz)i(3x^{2}y)j(x{z}^2-y^{2}z)k.$ Find curl vec$(A)$ at $(1,-2,1).$

$[4$ marks$]$

$($d$)$ Find the Eigen Values and Eigen Vector for $=3$ in the given matrix

$A=([1,-1,0],[-1,2,-1],[0,-1,1])$

$[6$ marks$]$

$[$Total: $20$ marks$]$