STUPIDSID
Answer any one question from Q1 and Q2
1 (a)
Solve (any two): i) (D2+9)y=x3−cos3xii) (D2+2D+1)y=e−xlogxiii) (2x−1)2d2ydx2−6(2x−1)dydx+16=8(2x+1)2.
8 M
1 (b)
Find Fourier sine transform of f(x)=e-x cos x, x>0.
4 M
2 (a)
A resistance of 50 ohms, an inductor of 2H and a 0.005 Farad capacitor are connected in series with e.m.f. of 40 volts and an open switch. Find the instantaneous charged and current after the switch is closed at t=0, assuming that at the time charge on capacitor is 4 Coulomb.
4 M
2 (b)
Solve (any one):
i) Find z-transform of f(k)=sinakk, k>0 ii) Find inverse z-transform of 3z2+2zz2+3z+2,1<|z|<2
i) Find z-transform of f(k)=sinakk, k>0 ii) Find inverse z-transform of 3z2+2zz2+3z+2,1<|z|<2
4 M
2 (c)
Solve difference equation:
f(k+2) -3f(k+1) + 2f(k)=0, f(0)=0, f(1)=1.
f(k+2) -3f(k+1) + 2f(k)=0, f(0)=0, f(1)=1.
4 M
Answer any one question from Q3 and Q4
3 (a)
The first four moments of a distribution about the value 5 ae 2, 20, 40 and 50. Obtain the first four central moments, mean, standard deviation and coefficient of skewness and kurtosis.
4 M
3 (b)
A manufacture of electronic goods has 4% of his product defective. He sells the articles in packets of 300 and guarantees 90% good quality. Determine the probability that a particular packet will violet the guarantee.
4 M
3 (c)
Find the directional derivative of xy2+yz3 at (2, -1, 1) along the line 2(x-2)=(y+1)=(z-1).
4 M
4 (a)
in an intelligence test administered to 1000 students the average score was 42 and standard deviation 24. Find the number of students with score lying between 30 and 54.
(Given: For z=0.5, area = 0.1915).
(Given: For z=0.5, area = 0.1915).
4 M
4 (b)
Prove (any one): i) ∇2(¯a⋅¯br)=0ii) ∇×(¯aׯrr)=¯ar+(¯a⋅¯r)¯rr3
4 M
4 (c)
Show that F = r2r is conservative. Obtain the scalar potential associated with it.
4 M
Answer any one question from Q5 and Q6
5 (a)
Evaluate: ∫c¯F⋅d\overlinrwhere ¯F=(2x+y2)\overlieni+(3y?4x)¯j and C is the parabolic arc y=x2 joining (0, 0) and (1, 1).
4 M
5 (b)
Using Strokes theorem, evaluate: ∫c(x+y)dx+(2x?z)dy+(y+z)dz) Where C is the curve given by
x2 + y2 + z2 - 2ax =0, x+y=2a.
x2 + y2 + z2 - 2ax =0, x+y=2a.
5 M
5 (c)
Use divergence theorem to evaluate: ∬s(x¯i−2y2¯j+z2¯k)⋅d¯s where s is the surface bounded by the region x2 + y2=1 and z=0 and z=1.
4 M
6 (a)
Apply Green's theorem to evaluate: ∫c(2x2−y2)dx+(x2+y2)dy where C is the boundary of the area enclosed by the x-axis and the upper-half of the circle x2 + y2=16.
4 M
6 (b)
Using Strokes theorem, evaluate: ∬s(∇ׯF)⋅d¯where ¯F=3y¯i−xz¯j+yz2¯k and 's' is the surface of the paraboloid 2z=x2+y2 bounded by z=2.
5 M
6 (c)
Show that: ∭v2rdv=∬s¯s⋅ˆnrds
4 M
Answer any one question from Q7 and Q8
7 (a)
Find the imaginary part of the analytic function whose real part is x3-3xy2+3x2 - 3y2.
4 M
7 (b)
Evaluate: ∮cz2+1z2−1dz where C is the circle: |z-1|=1.
4 M
7 (c)
Find the bilinear transformation which maps the points
z=-1, 0, 1 on to the points w=0, i, 3i respectively.
z=-1, 0, 1 on to the points w=0, i, 3i respectively.
4 M
8 (a)
Show that analytic function f(z) with constant amplitude is constant.
4 M
8 (b)
Evaluate the following integral using residue theorem: ∮c4−3zz(z−1)(z−2)dz where C is the circle: |z|=32
4 M
8 (c)
Find the image of the straight line y=3x under the transformation w=z−1z+1
5 M
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