THE INSTITUTE OF ENGINEERS-SRI LANI(A

Engineering certificate course Stage 1 Examination-November 2012

Engineering Mathematics 1

Answer FM Questions only Time Allowed: Three Hours

Question I

(a)Solve for the matr ix ,,r' [:a' * [ I o

))' : oo( (z s))

(b)If A and AB are invertible matrices and B is a square matrix, solve the equation

(AB)-' = 2A-r for B and deduce that B is also invertible.

(c) If B is an mxk matrix, show that the matrix BrB is a kxk symmetric matrix.

(d) use elementary row operations to find inverse of the following matrix A.

(t 2 3\ttA=12 3 0l

[o t2)Question 2

consider the system of iinear equations Ax :b given below, where ), and p are

constants:

(t 2 o) (x\ (2\rtttttA=ls I rlx--lvl u=lzl[r -r 1) [rJ [u.]

(a) Compute the determinant of A, lAl.

(b) Determine for which values of ), and p this system has:

(i) a unique solution

(ii) no solutions

(iii) inflnitely many solutions.

In case (i), use Cramer's rule to find the value of z in terms of )" and p. In case (iii), solve the

system using row operations and express the solution in vector form, x:a*tb where a and b are

column matrices of 3x1 and t is a parameter.

M"*^,u;

Question 3

(a) If 21 and Zz are two complex numbers, show that

-

l+l=fl , una I +) = (2,)*,-(2.)*rlz,l lz,l' \2, ,u,*

(b)Show that (cos 0 + i sin 0)' = (cos 20 + i sin 20) .

Hence work (cos 0 + i sin 0)3 in polar form. Can you generalize this result?

(c) Find the division of following complex numbers.

(1 + z)4

(3 + 4i)3

Question 4

(a) Given that y = x3lnx ,

/::\ r , dY d'Y d'Y(tt) lrnd "' .---:-dx dx'' dx'

(ii) Find the Taylor series expansion of y = x' ln x in ascending powers of (x - 1 ) up to and

including the term in (x * 1)3.

(b) The function f is given, for some number a, by f (x, y) =2xy + xuy'u .

(i) Find, in terms of x, y and a,the partial derivatives * ,* , # ,#

(ii) Now suppose that we know that f satisfies Z*' * ,' ** rr, = 36xy.Ax' ' Ayt

Determine the possible values of a.

Question 5

(a) Find the general solution of ordinary differential equation(ODE)

xy+y2-*y+=odx

(b) Find the general solution of the ODE

dvxvt-,:*-==f/l+X_.dx 1+x'

What is the particular solution if y(0) : 1?

vt)f ,,

I

(c) Use the substitution w(t) = y/ (t) to show that the ODE

d'y _1dy __..dt' t dt J'

can be written as a linear ODE in terms of w(t). Solve this linear ODE for w(t) and hence find the

general solution of the original ODE.

Question 6

Consider the function given by f(x) : 3x5-25x3+60x.

(a) Show that the curve y : f(x) has only one x-intercept and find it.

(b) Find the stationary points of this function and classify them.

(c) Sketch the curve y : f(x).

(d) If the domain of f is restricted to values of x such that-2<x< 2, identify the

global maximum and the global minimum of the function f(x).

(e)What are the global maximum and the global minimum of the function if the

domain of f is restricted to values of x such that-3< x< 3?

Find the point (x) of the disconti

'= (b) evaluate the following limits

Question 7

(a)A function f(x) is defined as

(i) Lim.--

when 0<x<1when x=1

1 for 1<x<2

function and draw its graph.

l-cosxxslnx

[,.f(x)={2

l.**nuity of the

(c)Ify=,fr;sin-r

(ii) Lim.-r 1,*- (iii) Lim_-o

x, prove that (l - *' ,:l = xy + L

*x

By applying Leibnitz's theorem, show thar (l - *,)## - (2n +,,, # - *, 4I!. = g

)417/,"

l-

\{-

Question 8

(a) Prove that the lines 'L1 ,: *-'= Y*l =1119un6 L2 7!-1= Y*3 -z+l are' 2 -3 8 - 1 -4 7

coplanar and find also their point of intersection and the equation of the plane containing them.

(b)Find the equation of the plane through the line *^ ' = +:'-=4 and parallel to the32-2

,. x+l v-l z+2line

- = ' ' : . Hence or otherwise find the shortest distance between the lines.2 - ^ : -----End of the paper-------------

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