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Total No. of Questions: 11
[Total No. of Printed Pages: 8
MN-467
M.A/M.Sc. IIIrd Semester (Reg./PVT./ATKT)
Examination, 2023-24
Mathematics
Paper - X
Integral transform-I
Time: 3 Hours]
[Maximum Marks: Reg.= 85
Pvt.= 100
Note:- Attempt all the questions.
SECTION - 'A'
Objective Type Questions
10x1½=15
1. Choose the correct answer:
(i) L (sin at) =
(a)
(b)
(c)
(d)
(ii) The Laplace transform of the function f(x) = 1 is:
(a) 1/p, p > 0
(b) 1/p+1, p > 0
(c) p, p > 1
(d) 1/p-1, p > 1
(iii) If L (F(t) = f(p), then L {F' (t)} =
(a) pf (p) + F (o)
(b) f (p) + pF (o)
(c) pf (p) F (o)
(d) f (p) – p F (o)
(iv)
(a) 0
(b) π / 4
(c) π / 2
(d) π
(v) If (D² - 2D + 2) y = 0 and y = 1 Dy at t = 0 then y =
(a) eᵗ cos t
(b) eᵗ sin t
(c) e⁻ᵗ cos t
(d) e⁻ᵗ sin t
(vi) L
(a)
(b) 0
(c) 1
(d) - 1
(vii) L⁻¹ (1/p-2) =
(a) e⁻²ᵗ
(b) e²ᵗ
(viii) Equation of vibrating membrane is.
(a)
(b)
(c)
(d)
(ix) Equation of electric circuit without condenser is:
(a)
(b)
(c)
(d)
(x) Two dimensional heat conduction equation is :
(a)
(b)
(c)
(d)
SECTION - 'B'
Short Answer Type Questions
5x5=25
2. Evaluate L⁻¹
OR
Find the Laplace transform of √t.
3. Solve (D² - 3D + 2) y = 1 - e⁻²ᵗ, y = 1, D (y) = 0 when t = 0
OR
Solve ∫₀ᵗ
4. Find the solution of two dimensional Laplace's equation in polar form.
OR
A tightly stretched string with fixed end - points x = 0 and x = l is initially displaced in a sinusoidal arch of height y₀ and then released from rest. Find the displacement y at any distance x from one and at time t.
5. An inductor of 3 henrys is in series with a resistance of 30 Ohms and an e.m.f. of 150 volts. Assuming that t = 0 the current is zero find the current at time t > 0.
OR
Write the boundary conditions for a beam when its ends are clamped, hinged and free.
6. Find the temperature u (x, t) in a bar whose ends x = 0, x = l are kept of temperature zero and whose initial temperature f (x) = sin xπ/l.
OR
A semi infinite solid x > 0 initially at temperature zero. At time t = 0, a constant temperature u₀ > 0 is applied and maintained at the face x = 0. Find the temperature at any point of the solid at any time t > 0.
SECTION - 'C'
Long Answer Type Questions
9x5=45
7. Prove that
OR
State & prove the convolution theorem.
8. Solve
y (0) = 1, y' (0) = 1
OR
Solve
(D-2)x - (D+1)y = 6e³ᵗ
(2D-3)x + (D-3)y = 6e³ᵗ
x (0) = 3, y (0) = 0
9. Determine the steady state temperature in a thin plate bounded by the lines x = 0, x = l, y = 0 y = b, the edges x = 0, x = l, y = 0 are maintained at temperature zero, where as the edge y = b is kept at temperature F (x).
OR
A string is stretched between two fixed points (0,0) and (l,0). If it is displaced into curve y = b sin (πx/c) and released from rest in that position at time t = 0. Find its displacement at any time t and at any point 0 < x < c.
10. An inductor of 3 henrys is in series with a resistance of 30 ohms and an e.m.f. of 150 sin 20 t volts. Assuming that at t = 0 the current is zero, find the current at any time t > 0.
OR
A beam which is simply supported at its ends x = 0 and x = l carries a uniform load W₀ per unit length. Find the deflection at any point.
11. To find the temperature u (x, t) in a bar of length l which is perfectly insulated also at the ends x = 0 and x = l assuming that the initial temperature of bar is u (x, 0) = f(x)
OR
A semi - infinite solid x > 0 initially at temperature zero. At time t = 0, a constant temperature u₀ > 0 is applied and maintained at the face x = 0. Find the temperature at any point of the solid at any time t > 0.