THE INSTITUTE OF ENGINEERS-SRI LANKA

Engineering certificate course stage I Examination - November 2011

Engineering Mathematics I

Answer f'M Questions only Time Allowed: Three Hours

Question 1

(tz 3) (r -3 2)(i) If o:l , 4 s | *a

": I-, 1 -, lorconsideringABorotherwisefindqbsuchthat[rs6) [, ab)'

A:B-l

(iD If P = QRQ-I, show that Pn = QRoQ-l where n is a positive integer.

(iii) Let. : (| *l) *aa: (1 l) . sr'"* that e-rpe : (? !) . n",,"" nnd pn.

Question 2

(i) Transformz:2(l + iy'3 ) to polar form and indicate following complex numbers on an

Argand diagram

(a) z+) @)n, .

(ii) Express ,+cJ+tstne intheformof x+iyandshowthat x2 *y2:4x-3.

(iii) Find the locus of P of a complex number z, if lLz-41= 4

x+y+z - 6

x -y+22 -53x+Y+2 *8

2x -2y *32 :7

(b) Hence or otherwise solve them.

(ii) Vibrations of a mechanical system is represented by

/ 1, -2 0 \ /rr\ /xt\(; :, -:)l:):*l::)Find the maximum frequency of vibrations ro.

,rv Question 4

(i) Describe what are two skew lines in 3-D space.

(ii) Find the line ofintersection with the planes x - 2y *32 :0 and 2x + 3y - 4z: 5.

(iii) Examine whether the two lines given uy t tl = 'l' ='!-' uod *

-2 =Y --4 =' =6 *"'3s7ls7skew.

Question 5

(i) Solve the following first order differential equations:

(a)

v2 -(xv+x\!=o+o)dx

d'y , 4"

-f-ctxt **'=x2

+4

(ii)

(a) rindf| if u=log(x2+y2) ,*=r[+t,y=Jl-t@) If Y:

'sin-t *' show that'

h- *r\fu - xdY -v = o.\' * ld*, dx /-vs

Expand y to a series in ascending powers of x , as far as the term containing x3.

Question 6

(i) A metal plate of sides 4 by 2 has squares cut from its corners, and is then folded up to make

an open box. What is the size of the box that has maximum volume?

(ii) Find turning points of y - xa + 4x3 -2x2 -l2x and determine the nature of the above points.

(iii) If ! = x* f + 4 find maximum and minimum values of y and sketch the graph of y.

Question 7

The position vectors measured relative to an origin O, of four points A, B, C, and D are given

by 4i+57+ h, -3i+k, 3i+97+ 4k, -4i+4j +4ft, respectively.

(i) Writedown thevectors 78,7e ,fr interms of i,j,b_.(ii) Prove that the points A, B, C, and D, lie in the same plane.

'!r' (a) Find the equations of lines AB and CD and find the point of intersection of these two

lines.

(b) Find also equation of the plane passing through line AB and normal to the plane ABCD.

Question 8

(i) If in tunction f(x, y), x : s2 + f and y : 2st, nna ff ^d Xin terms of X,H,s and t.

Findff,and use symmetry to write an expressi onforff,

ff#.#:0, simpli* #*ff,(ii) Using the hyperbolic functions cosh 0 and sinh 0, show that cosh0 : cos i0 and isinh 0 : sin i0.

v. Ifu(x, y):sinxcoshy; findv(x,y) suchttrat #=#, v=0 aty:0.

By taking w: u(x,y) + iv(x,y), express w as a function of z, where z: x* iy.