Solve any one question form Q.1(a, b) &Q.2(a, b, c)Solve any two question from Q.1(a)(i, ii, iii)
1(a)
i)4d2ydx2+4dydx+y=xe−x/2cosxii)x2d2ydx2−xdydx−3y=x2logxiii)d2ydx2−2dydxexsinx/ (use method of variation of parameters).
8 M
1(b)
Solve the differential equation by using Laplace transform method: dydt+3y(t)+2∫t0y(t)dt=tgiveny(0)=0/
4 M
2(a)
An inductor of 0.25 henry is connected in series with a capacitor of 0.04 farads and a generator having alternative voltage given by 12 sin 10t. Find the charge and current at any time t.
4 M
Solve any one question from Q.2(b) (i, ii)
2(b)
Find the Laplace transform of: ddt(1−costt)./
ii) Find the inverse Laplace transform of: SS4+S2+1./
ii) Find the inverse Laplace transform of: SS4+S2+1./
4 M
2(c)
Find the Laplace transform of:f(t)=(1+2t+3t2)u(t−2)+sin2tδ(t−π4).
4 M
Solve any one question from Q.3(a,b,c) & Q.4(a,b,c)
3(a)
Find the Fourier transform of: f(x)={1−x2,|x|≤10,|x|>1.
4 M
3(b)
Find: z−1{1(z−12)(z−13)For|z|>\[12.\]}/
4 M
3(c)
Find the constants a and b, so that the surface \( \ax^2-byz=\left ( a+2 \right )x)/
will be orthogonal to the surface 4x2y+z3=4/
at the point (1, -1, 2).
will be orthogonal to the surface 4x2y+z3=4/
at the point (1, -1, 2).
4 M
4(a)
Show that the vector field f(r)→r/ is always irrotational and determine f(r) such that the field is solential also.
4 M
Solve any one question from Q.4(b)(i, ii)
4(b)(i)
Prove the following:\nabla^2\left ( \nabla.\frac{\underset{r}{\rightarrow}}{r^2} \right )=\frac{2}{r^4}
4 M
4(b)(ii)
∇×(→a∇1r)=→ar3−(→a.→r)→rr5./
4 M
4(c)
Solve the difference equation: f(k+2)−3f(k+1)+2f(k)=1wheref(0)=0,f(1)=3./
4 M
Solve any one question from Q.5(a,b,c) & Q.6(a,b,c)
5(a)
Evaluate: ∫CˉF.dˉrforˉF=(2y+3)ˉi+xzˉj+(yz−x)ˉk/ along a straight line joining (0, 0, 0) to (3, 1, 1).
4 M
5(b)
Evaluate: ∬S(∇×ˉF).ˆndS/ where S is the curved surface of the paraboloid x2+y2=2z/ bounded by plane z=2 where ˉF=3(x−y)ˉi+2xzˉj+xyˉk./
5 M
5(c)
Evaluate: ∬ˉr.ˆndS/ over the surface of sphere of radius 2 with centre at origin.
4 M
6(a)
Using Green's theorem evaulate: \int _C\left [ \cos y\bar{i} +x\left ( 1-\sin y \right )\right\bar{j} ].d\bar{r}/ where C is the closed curve x2+y2=1,z=0./
4 M
6(b)
Prove that: ∫C(ˉa×ˉr).dˉr=2ˉa.∬SdˉS/ where C is open surface bouded by closed curve C.
4 M
6(c)
Evaluate: ∬SˉF.ˆndSwhereˉF=(2x+3z)ˉi−(xz+y)ˉj(y2+2z)ˉK/ and S is surface of sphere with radius 3.
5 M
Solve any one question from Q.7(a,b,c) & Q.8(a,b,c)
7(a)
If u=log(x2+y2),/ find v such that f(z)=u+iv is analytic. Determine f(z) in terms of z.
5 M
7(b)
Evaluate: ∮Cdzz2/ where C is the circle |z| = 1.
4 M
7(c)
Find the bilinear transformation which maps the points 0, -1, i of z-plane on to the points 2, &infty;,12(5+i)/ of the w-plane.
4 M
8(a)
Show that the map of straight line parallel to x-axis is family of ellipses under the transformation w = sinh (z).
4 M
8(b)
Evaluate: ∮Cz+2z2+1dz/ where C is the circle | z -i| = 12.
4 M
8(c)
Find analytic function f(z)=u+ivwhereu=r3cos3θ+rsinθ./
5 M
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