This is a Question Papers of Mumbai University for B.Sc.IT Student of Semester - II [Mathematics – II] (Old Course). [Year - June / 2014] . . .Solution Set of this Paper is Coming soon..

1. M a t h e m a t i c s – I I
Q u e s t i o n P a p e r ( J u n e – 2 0 1 4 ) [ O l d C o u r s e ]
1 | Page
M u m b a i B . S c . I T S t u d y
F a c e b o o k | T w i t t e r | I n s t a g r a m | G o o g l e + | Y o u T u b e
– Kamal T.
Time: 3 Hours Total Marks: 100
N.B.: (1) Questions No. 1 is Compulsory.
(2) Figure any four questions from Question Nos. 2 to 7.
(3) All questions carry equal marks..
Q.1 Attempt Both The Question: (10 Marks)
(A) Find the Laplace Transform of the following: –
(i) (𝑡 +
1
𝑡
)
3
(ii) 𝑡2
sin 3𝑡
(5)
(B) Evaluate: –
(i) ∫ ∫ 𝑥𝑦 𝑑𝑦 𝑑𝑥
𝑥
0
3
0
(ii) ∫ ∫ 𝑟23
0
𝑥
0
sin 𝜃 𝑑𝑟 𝑑𝜃
(5)
Q.2 Attempt The Following Questions: (20 Marks)
(A) State and prove De Moivre’s Theorem for Complex Number. (10)
(B) Find the inverse Laplace Transform of the following function: –
(i)
1
𝑆2+3𝑆+10
(ii)
𝑠
𝑠2+2𝑠+3
(10)
Q.3 Attempt The Following Questions: (20 Marks)
(A) Find curl and divergence of 𝑓(𝑥) − 3𝑥3
𝑦 𝑖 + 5𝑥𝑦𝑧𝑗 + 𝑥𝑦𝑧2
𝑘 at the point (1, 2, 3). (10)
(B) Evaluate:
(i) ∫ 7−3𝑥2∞
0
𝑑𝑥 (ii) ∫ 𝑥3(1 − 𝑥2)41
0
𝑑𝑥
(10)
Q.4 Attempt The Following Questions: (20 Marks)
(A) State and prove 𝐶 − 𝑅 equations for Analytic function. (10)
(B) Prove that 𝑓(𝑧) = 𝑒 𝑧
sin 𝑧 is Analytic on and hence find 𝑓1(𝑧). (10)
Q.5 Attempt The Following Questions: (20 Marks)
(A) Evaluate ∬(𝑥 + 𝑦) 𝑑𝑦 𝑑𝑥 over the region bounded by 𝑦 = 𝑥, 𝑥 − 𝑎𝑥𝑖𝑠 and 𝑥 = 1. (10)
(B) Find the Fourier series of 𝑓(𝑥) = 𝑥 cos 𝑥 in (−𝜋, 𝜋). (10)

2. M a t h e m a t i c s – I I
Q u e s t i o n P a p e r ( J u n e – 2 0 1 4 ) [ O l d C o u r s e ]
2 | Page
M u m b a i B . S c . I T S t u d y
F a c e b o o k | T w i t t e r | I n s t a g r a m | G o o g l e + | Y o u T u b e
– Kamal T.
Q.6 Attempt The Following Questions: (20 Marks)
(A) Evaluate ∭ 𝑥2
𝑦𝑧 𝑑𝑥 𝑑𝑦 𝑑𝑧 over the volume bounded by the planes 𝑥 = 0, 𝑦 =
0, 𝑧 = 0 and 𝑥 + 𝑦 + 𝑧 = 1.
(10)
(B) Evaluate ∫ 𝑠𝑖𝑛3𝜋
0
𝜃(1 + cos 𝜃)2
𝑑𝜃. (10)
Q.7 Attempt The Following Questions: (20 Marks)
(A) Evaluate ∭(𝑥2
+ 𝑦2
+ 𝑧2)2
𝑑𝑥 𝑑𝑦 𝑑𝑧 over the first octant of the sphere 𝑥2
+ 𝑦2
+
𝑧2
= 25.
(10)
(B) Express 𝑓(𝑥) = 𝑥3
as Half Range Since Series in (0, 3). (10)