Download PTU B.Tech 2020 March CSE-IT 4th Sem BTCS402 Mathematics Iii Question Paper

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Roll No. [ ] Total No. of Pages : 02
Total No. of Questions : 18

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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.

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SECTION-A

Write briefly :

1. Define periodic functions.
2. State the sufficient condition for the existence of Laplace transforms.
3. Define analytic and conjugate functions of a complex variable.

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4. Define Homogenous linear partial differential equation.
5. Define critical region of the testing.
6. Define Eigen value and eigen vector of a matrix.
7. Define Binomial and Poisson distributions.
8. Write the Laplace transform of t* sin 2t.

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9. Write the difference between chi-square and t-distributions.
10. Differentiate between Euler’s and modified Euler’s method for solving the ordinary differential equation.

SECTION-B

11. Obtain the Fourier series of x sin x as a cosine series in (0, p). Hence show that
    1/1.3 - 1/3.5 + 1/5.7 - ... = p/4

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12. Using the Laplace transform, prove that
    ?₀⁸ (e^-at - e^-bt)/t dt = log(b/a).
13. Solve the following equation by Gauss elimination method :
    2x+y+z=10; 3x+2y+3z=18; x+4y+9z=16
14. The theory predicts the proportion of beans, in the four groups A, B, C and D should be 9:3:3:1. In an experiment among 1600 beans, the numbers in the four groups were 882, 313, 287 and 118. Does the experimental result support the theory ? (The table value of x² for 3 d.f. at 5% level of significance is 7.81).

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15. Show that f (z) = x³y / (x² + y²), z?0 and f (z) = 0, z = 0 is not analytic at z = 0, although C-R equations are satisfied at the origin.

SECTION-C

16. a) Marks obtained by a number of students are assumed to be normal distributed with mean 50 and variance 36. If 4 students are taken at random, what is the probability that exactly two of them will have marks over 65? Given that ?₀^1.5 F(z)dz = 0.4772 where z is N (0, 1).
    b) The intelligence quotients (IQ) of 16 students from B.Tech. IInd year showed a mean of 107 and a standard deviation of 10, while the 1Qs of 14 students from B.Tech. Ist year showed a mean of 112 and a standard deviation of 8. Is there a significant difference between the 1Qs of the two groups at significance levels of 0.05? Given that critical value of 28 degree of freedom with 5% level of significance is 2.05.
17. Find the largest eigen value and the corresponding eigen vector of the matrix
    

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    2 -1 0
    -1 2 -1
    0 -1 2
18. Solve the following by Euler’s modified method :
    dy/dx = x+y, y(0)=1
    

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    at x = 0.3 with step size 0.1.

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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