Download PTU B-Tech CSE-IT 2020 Dec 3rd Sem 70808 Mathematics Iii Question Paper

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Roll No. [ ] [ ] [ ] [ ]

Total No. of Pages : 02

Total No. of Questions : 18

B.Tech. (CSE /IT) (2012 to 2017)

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(Sem.=3)

MATHEMATICS - II

Subject Code : BTAM-302

M.Code : 70808

Time : 3 Hrs. Max. Marks : 60

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INSTRUCTIONS TO CANDIDATES :

1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks each.
2. SECTION-B contains FIVE questions carrying FIVE marks each and students have to attempt any FOUR questions.
3. SECTION-C contains THREE questions carrying TEN marks each and students have to attempt any TWO questions.

SECTION-A

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Answer briefly :

1. State and prove second shifting theorem for Laplace transforms.
2. Show that |z|² is not analytic at any other point except z = 0.
3. Discuss modified Euler’s method.
4. Find the half-range cosine series for the function f(x) = (x — 1) in the interval 0 < x < 1.

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5. Solve pg = p + q.
6. Evaluate L (e^at sin bt).
7. Find the inverse Laplace transform of (6 + s) / (s² + 6s + 13).
8. Write Cauchy-Riemann equations in polar form.
9. In a normal distribution, 31% of the items are under 45 and 8% are over 64. Find the mean and standard deviation of the distribution.

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10. State Cayley-Hamilton theorem.

SECTION-B

1. Find Fourier series expansion of f(x) = x + x² in the interval —p < x < p. Hence show that
   1/1² - 1/2² + 1/3² - 1/4² + ... = p²/12
2. Show that if L{f(t)} = F(s) then L{tⁿ f(t)} = (-1)ⁿ dⁿ/dsⁿ F(s) where n=1,2,3, ........
   

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   Hence evaluate L{t e^-t}.
3. If f(z) is an analytic function of z, prove that :
   (?²/?x² + ?²/?y²) |f(z)|² = 4 |f'(z)|²
4. Solve
   4x-3y-9z+6w=0
   

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   2x+3y+3z+6w=0
   4x-21y-39z-6w=-24
5. The following table shows the distribution of digits in numbers chosen at random from a telephone directory :
   

   Digits	0	1	2	3	4	5	6	7	8	9
   Frequency	1026	1107	997	966	1075	933	1107	972	964	853

   
   Test whether the digits may be taken to occur equally frequently in the directory.

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SECTION-C

1. Solve (x² — 2yz — z²) p + (y² + zx — x²) q = xy — zx.
2. Find the eigen values and the corresponding eigen vectors of
   | 1 1 1 |
   | 0 0 1 |
   

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   | 0 0 1 |
3. Evaluate y (0.8) using Runge’s method of order four, given that dy/dx = x+y; y(0)= 0.4 (Take h =0.2).

NOTE : Disclosure of Identity by writing Mobile No. or Making of passing request on any page of Answer Sheet will lead to UMC against the Student.

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