This document contains questions for an exam on applied mathematics for engineers. It has three parts: Part A contains 10 short answer questions worth 2 marks each. Questions cover topics like defining terms related to chains and generalized eigenvalues. Part B has two longer answer questions worth 13 marks each. One question involves finding the number of generalized eigenvectors for a given matrix. Part C contains two questions worth 15 marks each. One asks to show a given functional is stationary for given functions, the other asks to derive moments and the moment generating function of the Poisson distribution.

1. I tion Paper Code: 10854 I
ME 1M T h DF.GRF.F. EXAMINATIONS APRHJMAY 2019
CAD CAM
MA 5156 APPLIED MATHEMATICS FOR ENGINEF...RS
(C mmon toME Computer Aided Destgn/M.E Computer Integrated
Manufa turmg/M E Engmeenng Destgn!M.E. Mecltatrorucs Engm nng/
M E Product Destgn and Development)
(Regulation 2017)
Tam Three hour& Max1mum. 100 marks
Answer ALL que&tJone
PART A- (10 x 2 = 20 marks)
J Define the terms cham and length or the cham an the generahU:d e~ n
"'*--
Write down the equatlona ofshtlted QR algonthm
SoJv th Euler s equation for the foUowmg functaonal
FiDd the extremal• of the functwnal J1
;.Y }x
OODU.Duoa&a random vanable X baa the denatty funcboD /(s) pftll bJ
~ FiDel the value of C
131414141251253235353463463473
rwetwetwetewtwetwetwettewt

2. 11.
PART B- (ii x 1:3 =()5 mnrks)
(a) Ddt•rmrrw t lw llllllllH'l' of generalized eigen vectors of each
C'OITE'sponding to A =.J thaL wtll appeHI' in a canonical has1s for
I 2 I 0 0 0
0 •1 -1 0 0 0
A-
0 0 · 4 0 0 0
0 0 0 4 2 0
0 0 0 0 4 0
0 0 0 0 0 7
Or
(b) Construct a QR decomposition for the given matrix
[
25 -131
A= 11 - 41
-4 28
-86]-18 .
0
rank
(13)
(13)
12. (a) Find the path on which a particle in the absence of friction will slide from
one point to another point in the shortest time under the action of
gravity. (13)
Or
(b) Solve the boundary value problem y"-y + x = 0 (0 ~ x ~ 1) given that
y(O) = 0, y(l) = 0 by using Rayleigh-Ritz method. (13)
13. (a) The density function of a continuous random variable X is defined by
f(x) =A(2x- x
2
), 0 ~ x ~ 2. Find (i) Value of A (ii) Mean (iii) Vanance
(iv) nth moment about origin. (13)
Or
(b) (i) State and pi.·ove the memory less property of Exponential
distributwn. (7)
(ii) Find the mean and variance of a binomial random variable. (6)
14 (a) Solve the followjng BVP using Laplace transfm·m technique:
PDE : U1 =lin; 0 < X < l,I _..- 0
(l;l)
BCS: u(O,t) l,u(l,l) =1,1 > 0
JC: u(x,O) 1 -1 sin n:x, 0 < x < I.
Or
2
10854

3. tb) A string i~ stretched and fixed between two points (0,0) and (l,O). Motion
i~ initiated by diRplucing the string in the form u = A. sin(7) and
rcll1:lSl~d from rest at time I =0. Find the displacement at any pomt on
till' sll'ing ai any Lune t. (13)
1!1. (n) Solve the heat conduction problem described by
l>DE , d2y au
: l l - -
2
= - ,0 < X < co, t > 0
dx ot
BC: u.(O,t) = u.0 ,1 ;:::: 0
IC: u(x,O) =0, 0 < x < cc
u and ou both tend to zero as x ~ cc.
ox
Or
(13)
(b) Determine the temperature distribution in the semi-infinite medium
x ;:::: 0, when the end x = 0 is maintained at zero temperature and the
initial temperature distribution is f(x). (13)
PART C- (1 x 15 =15 marks)
16. (a) Show that the functional
'tl2x:y + ( ':)' + (: )}t such that x(O) = 0 x(n/2) = -1, y0) = 0,
y(tr/2) =1 is stationary for x = -sin t, y =sin t.
Or
(b) Derive the first four moments about origin and moment generating
function of (MGF) Poisson distribution.
..
3
10854 .