Download PTU B.Tech 2020 March CSE-IT 3rd Sem BTAM 302 Mathematics Iii Question Paper

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Roll No. EEEEEEEEEEN Total No. of Pages : 03

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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.

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3. SECTION-C contains THREE questions carrying TEN marks each and students have to attempt any TWO questions.

SECTION-A
Answer briefly :

1. Write Euler’s formula of Fourier series.
2. Define Laplace transforms.

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3. Define the Homogeneous partial differential equations.
4. Define analytic functions and write its Cauchy-Riemann equations.
5. Define Binomial and Poisson distributions.
6. Define Null and Alternative hypothesis.
7. What is the difference between Euler’s and Runge-Kutta methods for solving the differential equations?

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8. Write the difference between chi-square and t-distributions.
9. Write the Laplace transform of # sin 2t
10. Define eigen value.

SECTION-B

11. Express f(x) = x as a half-range cosine series in 0 < x < 2.

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12. Using the Laplace transform, evaluate ? te^-t sin(2t) dt from 0 to 8
13. Solve the following equation ?²z/?x² - 3 ?²z/?x?y + 4 ?²z/?y² = 0
14. a) Service calls come to a maintenance center, according to a Poisson process and, on the average, 2.7 calls come per minute. Find the probability that (a) no more than 4 calls come in any minute ; (b) fewer than 2 calls came in any minute.
    b) Find the value of c such that P (|X— 25| < c) = 0.9544 where X ~ N (25, 36). Given that P (Z < -2) = 0.0228 and P (Z < - 1.69) = 0.0456, Z being a standard normal variate.
15. A survey of 240 families with 4 children each revealed the following distribution :
    

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    No. of boys 4 3 2 1 0
    No. of families 10 55 105 58 12
    Is the result consistent with the hypothesis that male and female births are equally probable? Use chi-square value for 4 & 5 d.f. at 5% level of significance is 9.49 & 11.07 respectively.

SECTION-C

16. Prove that the function f (z) define by f(z) = (x³ - y³) / (x² + y²), z?0 and f(0) =0 is continuous and the Cauchy-Riemann equations are satisfied at the origin, yet f'(0) does not exist.

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17. Determine the largest eigen value and the corresponding eigen vector of the matrix
    2 -1 0
    -1 2 -1
    0 -1 2
    using the power method. Take [1, 0, 0]^T as initial eigen vector.

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18. a) Using Euler’s method, find an approximate value of y corresponding to x = 0.5 given that dy/dx = x + y, and y = 1, where x = 0. Use step size 0.1
    b) Apply Gauss elimination method to solve the equations
    x + 4y - z = -5
    x + y - 6z = -12
    3x - y - z = 4.

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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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