Download PTU B.Tech 2021 Jan ECE 3rd Sem 76448 Mathematics Iii Question Paper

Roll No. Total No. of Pages : 03

Total No. of Questions: 18

B.Tech.(ECE) (2018 Batch) (Sem.-3)

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MATHEMATICS III

Subject Code: BTAM-303-18

M.Code: 76448

Time: 3 Hrs. Max. Marks: 60

INSTRUCTIONS TO CANDIDATES :

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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

Write briefly :

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1. If a random variable has a Poisson distribution such that P(1) = P(2). Find the mean of the distribution.
2. Find the Laplace transform of \( t e^{-t} \).
3. Represent \( f(t) = \begin{cases} \sin 2t, & 2\pi < t < 4\pi \\ 0, & \text{otherwise} \end{cases} \) in terms of unit step function.
4. State convolution theorem of Fourier transform.
5. Find the Z-transform of \( e^t \sin 2t \).

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6. Write the relation between Fourier and Laplace transforms.
7. Define discrete and continuous random variables.
8. Define Rank correlation.
9. State initial and final value theorems of Z-transform.
10. Define Binomial and Poisson distribution functions.

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SECTION-B

11. Evaluate : \[ L \left\{ e^{-t} \int_0^t \frac{\sin t}{t} dt \right\} \]
12. Find the Fourier transform of: \[ e^{-2(x-3)^2} \]
13. Using the Z-transform, solve : \[ u_{n+2} + 4u_{n+1} + 3u_n = 3^n \] with \( u_0 = 0, u_1 = 1 \).
14. The two regression equations of the variables x and y are \( x = 19.13 - 0.87y \) and \( y = 11.64 - 0.50x \). Find (i) mean of x and y (ii) the correlation co-efficient between x and y.

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15. 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 IQs of 14 students from B. Tech. 1st year showed a mean of 112 and a standard deviation of 8. Is there a significant difference between the IQs of the two groups at significance levels of 0.05? Given that critical value at 28 degree of freedom with 5% level of significance is 2.05.

SECTION-C

16. a) Apply Convolution theorem to evaluate \[ L^{-1} \left[ \frac{1}{(s^2+1)(s^2+9)} \right] \] b) Find the inverse Laplace transform of \[ \frac{se^{-s\pi/2} + \pi e^{-s\pi}}{s^2 + \pi^2} \]
17. If f(x)=sinx, \( 0 \leq x \leq \pi \) and f(x) = 0, \( -\pi \leq x \leq 0 \), Prove that \[ f(x) = \frac{1}{\pi} + \frac{\sin x}{2} - \frac{2}{\pi} \sum_{n=1}^{\infty} \frac{\cos 2nx}{4n^2 - 1} \] Hence show that : \[ \frac{1}{1.3} + \frac{1}{3.5} + \frac{1}{5.7} + ... = \frac{\pi - 2}{4} \]
18. Find the coefficient of correlation and obtain the lines of regression from the given data

    X	62	64	65	69	70	71	72	74
    Y	126	125	139	145	165	152	180	208

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