AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014 _AT51.pdfAE51/AC51/AT51 ENGINEERING MATHEMATICS-I...
25
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014 © IETE 1 Q2 (a) If u = ( ) ) x / y ( x / y x ψ + φ , prove that 0 y u y y x u xy 2 x u x 2 2 2 2 2 2 2 = ∂ ∂ + ∂ ∂ ∂ + ∂ ∂ Answer
Transcript of AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014 _AT51.pdfAE51/AC51/AT51 ENGINEERING MATHEMATICS-I...
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 1
Q2 (a) If u = ( ) )x/y(x/yx ψ+φ , prove that 0y
uyyxuxy2
xux
2
22
2
2
22 =
∂
∂+
∂∂∂
+∂
∂
Answer
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 2
Q2 (b) Show that the maximum and minimum of the radii vectors of the sections of
the surface 2
2
2
2
2
22222
cz
b
y
ax)zyx( ++=++ by the plane
0zyx =γ+µ+λ are given by the equation-
0rc1
crb1
bra1
a22
22
22
22
22
22=
−
γ+
−
µ+
−
λ
Answer
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 3
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 4
Q3 (a) Evaluate dxdy)yx( 2∫∫ + over the area bounded by the ellipse 1by
ax
2
2
2
2
=+
Answer
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 5
Q3 (b) Evaluate ∫∫∫ dxdydzz2 over the sphere x2+y2 +z2 = 1
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 6
Answer
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 7
Q4 (a) For what values of η , the equations x + y + z =1, x + 2y + 4z = η , x + 4y +
10z = 2η have a solution and solve them completely in each case.
Answer
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 8
Q4 (b) Determine the eigenvalues and the corresponding eigenvectors of the matrix
A =
−−−
−
312132
226
Answer
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 9
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 10
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 11
Q5 (a) Find by Newton – Raphson method, the real root of the equation:
3x = cosx + 1
Nearer to 1, correct to three decimal places.
Answer
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 12
Q5 (b) Apply Runge-Kutta method of fourth order to find approximate value of y
for x = 0.2, in steps of 0.1, if 2y x dxdy
+= , given that y = 1 where x = 0
Answer
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 13
Q6 (a) Solve the equation cos x dy = y (sin x – y) dx
Answer
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 14
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 15
Q6 (b) Find the orthogonal trajectories of the family of curves:
1)b(
y)a(
x2
2
2
2=
λ++
λ+
Where λ is the parameter.
Answer
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 16
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 17
Q7 (a) Find the solution of the equation y4dx
yd2
2+ = 8 cos 2x
given that y = 0 and dxdy = 2 when x = 0
Answer
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 18
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 19
Q7 (b) Solve by the method of variation of parameters:
x2
2
e12y
dxyd
+=−
Answer
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 20
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 21
Q8 (a) Show that )n,m(Ba)ba(
1dx)bxa(
)x1(xnm
1
0nm
1n1m
+=
+
−∫ +
−−
where B (m, n) is Beta function.
Answer
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 22
Q8 (b) Solve following differential equation 2x2xxy4dxdyx2
dxyd 222
2++=+− in
power of x.
Answer
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 23
Q9 (a) Prove that:
)x(xJ)x(J)xnn()x("Jx 1nn22
n2
++−−=
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 24
Answer
Q9 (b) Prove that
0dx'P'P)x1( n
1
1m
2 =−∫+
−
Where m and n are distinct positive integers.
AE51/AC51/AT51 ENGINEERING MATHEMATICS-I JUNE 2014
© IETE 25
Answer
Text Books
1. Higher Engineering Mathematics, Dr. B.S. Grewal, 40th edition 2007, Khanna Publishers, Delhi.
2. Text book of Engineering Mathematics, N.P. Bali & Manish Goyal, 7th edition 2007, Laxmi Publication (P) Ltd.
