B.E-B.E 2nd Sem Engineering Mathematics -I(University of Pune, Pune-2013)

UNIVERSITY OF PUNE
[4361-1]
F.E.-Maths-I
Engineering Mathematics -I
(2008 pattern)

Time : 3 Hours                                                                                                  Max. Marks : 100

[Total No. of Question=12] [Total no. of printed pages= 5]

Instructions:

(1)Answer 3 questions from Section-I and 3 questions from Section-II.
(2)Answers to the two sections should be written in separate answer books.
(3)Neat diagrams must be drawn whenever necessary.
(4)Figures to the right indicate full marks.
(5)Use of logarithmic tables slide rule,Mollier charts,electronic pocket calculator
and steam tables is allowed.

SECTION-I
Q.1 (a)Reduce matrix A=[ 1 2 −1 2
−2 5 3 0
1 0 1 10] to it's Normal form and Hence determine
it's RANK. (5)
(b)Determine the values of 'k' for which the system of equations,
x+y+z=1
x+2y+4z=k
x4y10z=k 2 is consistent.Find the solution for k=1. (6)
(c)Verify cayley-Hawiltou's theorem foe the matrix A=[2 1 1
0 1 0
1 1 2] and Hence find
matrix for A4−5 A38 A2 . (6)
OR
Q.2 (a)Find eigue values and eigen vectors for A=[ 4 2 −2
−5 3 2
−2 4 1 ] . (6)
(b)Examine the vectors. (6)
X1=(1,2,3,−2)x2=( 2,−2,1,3) X 3=(3,0,4,1)
For Linear dependance or independence.
(c)Is the matrix A=13
[ 2 2 1
−2 1 2
1 −2 2] orthogonal? If not can, it be converted to
orthogonal? (5)
Q.3 (a)Show that (1+ i √3)8+(1− i √3)8=−28 . (5)
(b)If | i + z|=|i – z| then show that z is real quantity. (5)
(c)If cosec( π
4 +ix)=u+iv ,where u,v,x are real then show that u2v 22=2u2−v2
(6)
OR
Q.4 (a)If ii =i  ,then show that 22=e−4m1  . (5)
(b)Solve by usign De-Moivre's theorem z3=i  z−13 . (5)
(c)If z1 , z 2 and origin '0' represent on the Argand's diagram,vertices of an
equivalent triangle then show that (6)
1
z1 2  1
z2 2 = 1
z1 z2
Q.5 (a)If y=cosh4x cos 3x.then find yn . (6)
(b)If x= sin  ,y= sin 2 then prove that (1−x2) yn+2−(2n+1) xyn+̄1−(n2−4) y n=0
(6)
(c)Discuss convergence or divergence(any one ). (5)
(i) Σn
=1
∞ n1n
n!
(ii) Σn
=r
∞  n31− n3
Q.6 (a)If f(x)= tanx then show that (6)
f n(0)−nc2 f n−2(0)+uC4 f x−4(0) ....=sin ( nπ
2 )
(b)If y= x 4
 x−1 x−2 , then find yn . (6)
(c)Discuss convergence or deconvergence.(any one) (5)
(i) 1
2  1
 x2
3 2
 x4
4  3
....
(ii) Σn
=1
∞ 5na
3nb (a>0,b>0)
SECTION-II
Q.7 (a)Expand 1x x in the power of x up to x5 . (6)
(b)Expand x4−3x32x2−x1 in the powers of 'x-3' (5)
(c)Attempt any one. (6)
(i)Find the constant a,b so that lim
x0
x 1acosx−bsinx
x3 =1
(ii)Evaluate lim
x∞[ a1/ xb1/ xc1/ x
3 ]x
OR
Q.8 (a)Show that
tan−1{ p−qx
q px }=tan−1 p
q −{x− x3
3 x5
5 −−.....} (6)
(b)Expand logcosx in the powers of ' x−
3
' by using taylor's series. (5)
(c)Attempt any one. (6)
(i)Evaluate : lim
x0
cot x−1
x
x
(ii)Evaluate: lim
x0 [ 
4x− 
2x  e x1 ]
Q.9 Attempt any two.
(a)If u=f(r),where x=rcos  , y=rcos  then show that
∂2 u
∂ x2∂2 u
∂ y2= f ' ' r 1r
f '  r . (8)
(b)If u=cosec−1  x1/ 2y1/ 2
x1/ 3y1/ 3 then
show that x2 uxx+2xyuxy+ y2 uyy=tanu
12 {13
12 + tan2 u
12 } . (8)
(c)If z= f(x,y),where x=eucos v , y=eusin v ,then show that y ∂ z
∂ ux ∂ z
∂ v=e2u ∂ z
∂ y
(8)
OR
Q.10 (a)If u=  x2y2m
2m2m−1
xf  y
x  y
x  then find x2 uxx2xyuxyy2 uyy . (8)
(b)If x2
a2 y2
b2 z2
c2=1 and lx +my+nz=0then find dy
dx and dy
dz . (8)
(c)If u=x y+ y x then show that ∂2 u
∂ x ∂ y= ∂2 u
∂ y ∂ x . (8)
Q.11 (a)Show that the functions u=x+y+z, v=x2y2z2 ,w=xy+yz+zx are
functionally dependent .Hence ,find the relation between u,v and w. (6)
(b)Use Langrange's Methods to find minimum distance from origin to find plane
3x +2y+z=12. (6)
(c)The resonant frequency in a series electrical circuit is given by f = 1
2 LC .If
measurement of L and C are in error by 2 % and -1 % respectively.find Percentage
error in calculated value of f. (5)
OR
Q.12 (a)If x=eucos v , y=eusin v then show that JJ'=1. (6)
(b)Determine maximum or minimum values of x3 y21−x− y . (6)
(c)If x=u + v, y=v2+w2 , z=w3+u3 find ∂u
∂ x . (5)