engineering mathematics stage 01
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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-------------
vzxk'
v
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