ENGINEERING MATHEMATICS
Note : Attempt any five questions. All questions carry equal marks.
Q.1.show that function eX(cosy=isiny)is an analytic function and find its derivative ?Q.2.find the nature of quadratic eqn form x2+5y2+z2+2xy+2yz+6zx and reduce it to canonical form ?
Q.3.by using newton raphson method find the roots of x4-x-10=0, nearer to x=2 correct to 3 decimal places ?
Q.4.solve equations yb gauss siedal method ?
27x+6y-z=85
6x+15y+2z=42
x+y-2z=101
27x+6y-z=85
6x+15y+2z=42
x+y-2z=101
Q.5.define vector space show that vectors (2,1,4),(1,1,2),(3,1,2) forms a basis R3 ?
Q.6.using regula fals imethod find a real root of the equation xlogx-1.2=0 correct up to 3 places
of decimals ?
of decimals ?
Q.7.use bessal formula to find value of y if x=3.75,given ?
| X | 2.5 | 3.0 | 3.5 | 4.0 | 4.5 | 5.0 |
| Y | 24.125 | 22.043 | 20.22. | 18.645 | 17.234 | 16.544 |
Applied Mathematics-II
UNIT-I
1. (a) State the De-Moiver’s theorem.
(b) Find all the roots of the equation
(i) cos z=2
(ii) tanh z=2
(c) Separate sin-1 (cosᶿ + I sinᶿ) into real and imaginary parts, where ө is a positive acute angle.
(d) Sum the series:
n sin a +n (n+1) sin 2an (n+1) (n+2) sin 3a+………….∞
1.2 1.2.3
UNIT-II
2. (a) Explain briefly the method of variation of parameters.
(b) Solve:
(D2+2) y=x2e3X+eX cos2x.
(c) Solve the equation:
(1+x)2d2y+(1+x)dy+y=sin[2log(1+x)]
dx2 dx
(d) Solve the simultaneous equation:
dx +2y=et and dy- 2x=e-t
dt dt
UNIT- III
3. (a) Write the relation between Beta and Gamma function.
(b) Evaluate the integral by changing the order of integration.
a a
ʃ ʃ y2 dxdy
0 √ax √y4- a2x2
(c) Evaluate:
∞ ∞
ʃ ʃ e—(x2+y2) dxdy
0 0
by changing iro polar co-ordinates.
Hence show that:
∞
ʃ e—x2 dx= √∏
0 2
(d) Find by double integration, the area lying between the parabola
Y=4x-x2 and the line y = x.
UNIT- IV
4. (a) State the Green’s theorem in the plane.
(b) Prove that:
A.V (B.V 1)=3(A.R)(B.R.) _ A.B
r r5 r3
Where A and B is constant vectors.
R=xI + yJ + K and r = √x2 + y2 +z2
(c) A vector field is given by:
F=(x2 – y2 + x)I – (2xy +y)J
Show that the Field is irrotational and find it’s scalar potential. Hence evaluate the line integral
from (1,2) to (2,1).
(d) Verify stokes theorem for the vector field
F=(2x – y) I – yz2J – y2zK on the upper half surface of x2 + y1 + z2= 1 bounded by its projection on the x y palne.
UNIT- V
5. (a) Write the relation between roots and coefficient of the equation.
a0xn + a1xn-1 + a2xn-2 +……….+ an-1x + an = 0
(b) If aβϒ be the roots of x3 + px + q = 0
Show that:
(i) a3 + β3 +ϒ3 = 5 aβϒ ∑ aβ
(ii) 3∑a2∑a5= 5∑ a∑ a4
(c) Show that the equation x4 – 10x3 + 23x2 – 6x ─15= 0 can be transformed into reciprocal equation by diminishing the roots by 2. Hence solve the equation.
(d) Solve the cardon’s method, the equation:
27x3 + 54x2 + 198x – 73= 0
1. (a) If y = (sin-1 x)2, show that:
(1-x2)yn+2-(2n+1)xyn+1-n2yn=0
(b) Using Taylor’s theorem expresses the polynomial
2x3 + 7x2 4x—6 in powers of (x-1).
(c) Test the convergence or divergence of the series:
∞
∑ [√ n2=1 - n]n=1
UNIT-II
2. (a) Find the reduction formula for : I„ = ʃ sec” xdx
(b) Find the area of the ellipse: x2 + y2
a2 b2
(c) Find the volume of a sphere of radius.
UNIT-III
3. (a) If u=exyz, find the value of ∂3z
∂x∂y∂z
(b) If u=sin-1 X2+y2 prove that.
x+y
X ∂u + ∂u = tan u
∂x ∂y
(c) Find the volume of the greatest rectangular parallelepiped that can be inscribed in the ellipsoid
X2 + y2 + z2 = 1
a2 b2 c2
UNIT- IV
4. (a) Change the order of integration in:
I = ʃ 01 ʃ22-x x y dx dy.
(b) Find the volume of the tetrahedron bounded by co-ordinate planes and the plane.
x + y + z = 1.
a b c
(c) Show that ʃ0x/2 √sinᶿdᶿ * ʃ0x/2 dᶿ = ∏.
√sinᶿ
UNIT- V
5. (a) Show that: V2 (rn) =n (n+1) rn-2.
(b) Verify stokes theorem for F = (x2+y2)j-2xyj taken around the rectangle bounded by the lines.
X=±a, y=0, y=h.
(c) If is any closed surface enclosing a volume V and
F = axi+byj+czk. Prove that fS FNdS=(a+b+c)V.
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