[Question Paper] Mathematics – II (Old Course) [September / 2013]
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This is a Question Papers of Mumbai University for B.Sc.IT Student of Semester - II [Mathematics – II] (Old Course). [Year - September / 2013] . . .Solution Set of this Paper is Coming soon..
![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 ( S e p t e m b e r – 2 0 1 3 ) [ 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) Attempt any four questions from Question Nos. 2 to 7.
(3) Figure to the right indicate Full Marks.
Q.1 Attempt The Following Question: (20 Marks)
(A) Find Laplace Transform of the following:
(i) 𝑡2
− 2𝑡 + 3
(ii) cos 5𝑡 sin 3𝑡
(10)
(B) Prove that ∫
𝑒−𝑎𝑥−𝑒−𝑏𝑥
𝑥
∞
0
𝑑𝑥 = log (
𝑏
𝑎
) (10)
Q.2 Attempt The Following Question: (20 Marks)
(A) Express sin 𝜃 and cos 𝜃 in terms of sin 𝜃 and cos 𝜃 (10)
(B) Using De Moivre’s Theorem prove that sin 7𝜃 = 7 sin 𝜃 − 56 𝑠𝑖𝑛 3
𝜃 + 112 𝑠𝑖𝑛5
𝜃 −
64 𝑠𝑖𝑛5
𝜃 − 64 𝑠𝑖𝑛2
𝜃
(10)
Q.3 Attempt The Following Question: (20 Marks)
(A) Evaluate ∫ ∫ (𝑥 + 𝑦)
√4−𝑥
0
3
0
𝑑𝑦 𝑑𝑥 (10)
(B) Evaluate ∫ ∫
1
√𝑥2+𝑦2
𝑥
𝑥2
1
0
𝑑𝑥𝑑𝑦 (10)
Q.4 Attempt The Following Question: (20 Marks)
(A)
Evaluate
(1+𝑖√2)
14
(√2−1)
15
(10)
(B) Obtain the Fourier Series for 𝑓(𝑥) = 𝑒−𝑥
in the interval 0 < 𝑥 < 2Π (10)
Q.5 Attempt The Following Question: (20 Marks)
(A) Evaluate ∫ 𝑒−4𝑥2∞
0
𝑑𝑥 (10)
(B) Find the volume of solid S, where S is the interior of sphere 𝑥2
+ 𝑦2
+ 𝑧2
= 𝑎2 (10)](https://image.slidesharecdn.com/m-ii-qpoldcoursesep-2013-170714203203/85/question-paper-mathematics-ii-old-course-september-2013-1-320.jpg)
![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 ( S e p t e m b e r – 2 0 1 3 ) [ 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 Question: (20 Marks)
(A)
Evaluate
Where C is the circle |𝑧| = 2.
(10)
(B)
Prove that ∫ 𝑒−𝑥2∞
0
𝑑𝑥 = √
Π
2
(10)
Q.7 Attempt The Following Question: (20 Marks)
(A)
∫
𝑥 𝛼−𝑥 𝛽
log 𝑥
1
0
𝑑𝑥 by using D.U.I.D (𝛼, 𝛽 ≥ 0) (10)
(B) Evaluate ∫ 𝑧21+𝑖
0
𝑑𝑧 along the paths
(i) parabola 𝑦 = 𝑥2
(ii) line 𝑦 = 𝑥
(10)](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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[Question Paper] Mathematics – II (Old Course) [September / 2013]
- 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 ( S e p t e m b e r – 2 0 1 3 ) [ 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) Attempt any four questions from Question Nos. 2 to 7. (3) Figure to the right indicate Full Marks. Q.1 Attempt The Following Question: (20 Marks) (A) Find Laplace Transform of the following: (i) 𝑡2 − 2𝑡 + 3 (ii) cos 5𝑡 sin 3𝑡 (10) (B) Prove that ∫ 𝑒−𝑎𝑥−𝑒−𝑏𝑥 𝑥 ∞ 0 𝑑𝑥 = log ( 𝑏 𝑎 ) (10) Q.2 Attempt The Following Question: (20 Marks) (A) Express sin 𝜃 and cos 𝜃 in terms of sin 𝜃 and cos 𝜃 (10) (B) Using De Moivre’s Theorem prove that sin 7𝜃 = 7 sin 𝜃 − 56 𝑠𝑖𝑛 3 𝜃 + 112 𝑠𝑖𝑛5 𝜃 − 64 𝑠𝑖𝑛5 𝜃 − 64 𝑠𝑖𝑛2 𝜃 (10) Q.3 Attempt The Following Question: (20 Marks) (A) Evaluate ∫ ∫ (𝑥 + 𝑦) √4−𝑥 0 3 0 𝑑𝑦 𝑑𝑥 (10) (B) Evaluate ∫ ∫ 1 √𝑥2+𝑦2 𝑥 𝑥2 1 0 𝑑𝑥𝑑𝑦 (10) Q.4 Attempt The Following Question: (20 Marks) (A) Evaluate (1+𝑖√2) 14 (√2−1) 15 (10) (B) Obtain the Fourier Series for 𝑓(𝑥) = 𝑒−𝑥 in the interval 0 < 𝑥 < 2Π (10) Q.5 Attempt The Following Question: (20 Marks) (A) Evaluate ∫ 𝑒−4𝑥2∞ 0 𝑑𝑥 (10) (B) Find the volume of solid S, where S is the interior of sphere 𝑥2 + 𝑦2 + 𝑧2 = 𝑎2 (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 ( S e p t e m b e r – 2 0 1 3 ) [ 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 Question: (20 Marks) (A) Evaluate Where C is the circle |𝑧| = 2. (10) (B) Prove that ∫ 𝑒−𝑥2∞ 0 𝑑𝑥 = √ Π 2 (10) Q.7 Attempt The Following Question: (20 Marks) (A) ∫ 𝑥 𝛼−𝑥 𝛽 log 𝑥 1 0 𝑑𝑥 by using D.U.I.D (𝛼, 𝛽 ≥ 0) (10) (B) Evaluate ∫ 𝑧21+𝑖 0 𝑑𝑧 along the paths (i) parabola 𝑦 = 𝑥2 (ii) line 𝑦 = 𝑥 (10)