Computer Engineering (Semester 3)
Total marks: 80
Total time: 3 Hours
INSTRUCTIONS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Draw neat diagrams wherever necessary.
1.a.
If Laplace transform of ![\Large erf($\sqrt{t}$) = $\dfrac{1}{s \sqrt{s+1} }$, then find L{ $e^{t}$ .erf(2$\sqrt{t}$)}
(5 marks)
00
1.b.
Find the Orthogonal Trajectory of the family of curves given by $e^{-x}$ .
cos y +
x.y =
c
(5 marks)
00
1.c.
Find Complex Form of Fourier Series for . $e^{2x}$ ; 0 < x < 2
(5 marks)
00
1.d.
If the two regression equations are 5x - 6y + 90, 15x - 8y -180 = 0, find the means of x and y, the Correlation Coefficient and Standard deviation of x if variance of Y is 1
(5 marks)
00
2.a.
Show that the function is Harmonic and find the Harmonic Conjugate v = $e^{x}$ .cos y + $x^{3}$ - 3x $y^{2}$
(6 marks)
00
2.b.
Find Laplace Transform of
(6 marks)
00
2.c.
Find Fourier Series expansion of f(x) = x - $x^{2}$, -1 < x < 1
(8 marks)
00
3.a.
Find the Analytics function f(z) = u + iv if v = log$\large \left ( x^{2} + y^{2} \right)$ + x - 2y
(6 marks)
00
3.b.
Find Inverse Z Transform of
(6 marks)
00
3.c.
Solve the Differential Equation
using Laplace Transform
(8 marks)
00
4.a.
Find
(6 marks)
00
4.b.
FInd the spearman's Rank correlation coefficient between X and Y.
| X | 60 | 30 | 37 | 30 | 42 | 37 | 55 | 45 |
| Y | 50 | 25 | 33 | 27 | 40 | 33 | 50 | 42 |
(6 marks)
00
4.c.
Find Inverse Laplace transform of i) $\dfrac{3s + 1}{ (s+1)^{4} }$ ii) $\dfrac{e^{4-3s}}{ (s+4)^{5/2}}$
(8 marks)
00
5.a
Find Inverse Laplace Transform using Convolution theorem $\dfrac{1}{ (s-4)^{2}(s+3)}$
(6 marks)
00
5.b.
Show that the functions
are Orthogonal on (-1,1). Determine the constants
a,b such that functions f(x) = -1 +
ax +
b$x^{2}$ is Orthogonal to both
on the (-1,1)
(6 marks)
00
5.c.
Find the Laplace transform of
(8 marks)
00
6.a.
Fit a second degree parabola to the given data.
| X | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 |
| Y | 1.1 | 1.3 | 1.6 | 2 | 2.7 | 3.4 | 4.1 |
(6 marks)
00
6.b
Find the image of
under the transformation
w = $\dfrac{3 - z}{z - 2}$
(6 marks)
00
6.c
Find Half Range Cosine Series for f(x) = xsinx in (0,$\small \pi$) and hence find $\dfrac{1}{1.3}$ - $\dfrac{1}{3.5}$ + $\dfrac{1}{5.7}$ - ....... = $\dfrac{\small \pi - 2}{4}$
(8 marks)
00