###### Board Diploma Examinations

###### IV Semester Examinations

###### Model Question Paper-I

###### Engineering Mathematics-III

###### (C-14 Scheme)

Max Marks: 80 Time: 3 Hrs

__Part-A__

- Answer all questions.
- Each question carries
**three**marks. - Answer should be brief and straight to the point 10×3=30 marks

- Solve the equation
- Solve (D
^{4}– 8D^{2}+ 16) y = 0. - Find the Particular Integral for (D
^{2}+ 4) y = Sin2x + e^{-2x}. - State the change of scale property and first shifting theorem of Laplace Transforms.
- Find the Laplace Transform of 4e
^{2t }+ 6t^{3}-3 Cos 5t + Sin t.

6.Find the inverse Laplace Transform of .

7.Find inverse Laplace Transform of

8.Write the Euler’s formulae for Fourier series of a function f(x) in the Interval [C, C+2π].

9.Find the half range Fourier Sine series of unity in (0, π).

10.State Addition and Multiplication Theorems of Probability for two events.

** Part-B** 10×5 = 50 marks

- Answer
*any***five**questions. - Each question carries
**ten**marks. - Answer should be comprehensive and the criteria for valuation are the content but not the length of the answer.

11 (a) Solve (D^{2 }+ D -6) y = e^{-3x}.

(b) Solve (D^{3} +4D) y = 5 + Cos 2x.

12(a) Solve (D^{2} + 2D +1) y = 2x + x^{2}.

(b) Solve (D^{4}– 16) y = Cos 2x + Sinh 2x.

13 (a) Find the Laplace Transform of *t ^{2}. Cos t.*

* *(b) Find the Laplace Transform of

(b) Find using Convolution theorem.

16 (a) Expand *f(x) = * as Fourier Series in (-2 , 2).

(b) Find the half range Cosine series for *f(x) = x *in (0, 2).

17 (a) When two dice are thrown, find the probability of getting a sum 7.

(b) In a hostel 60% students Telugu news paper, 40% students read English news paper and 20% read both the papers. A student is selected at random, find the probability that the Student reads neither Telugu nor English news paper.

18 (a) Let A and B are independent events with P (A) = 0.3 and P (B) = 0.4. Fin

(b) Box-I contains 8 white , 2 black balls, Box–II contains 5 white , 5 black balls and Box-III contains 4 white , 6 black balls. A box is selected at random and a ball is drawn from it, what is the probability that the ball is white.

###### Board Diploma Examinations

###### IV Semester Examinations

###### Model Question Paper-II

###### Engineering Mathematics-III

###### (C-14 Scheme)

Max Marks: 80 Time: 3 Hrs

__Part-A__

- Answer all questions.
- Each question carries
**three**marks. - Answer should be brief and straight to the point 10×3=30 marks

- Solve the equation
- Solve (D
^{4}– 5D^{2 }+ 4) y = 0. - Find the Particular Integral for (D
^{2 }+ D -6) y = e^{3x}+ e^{-3x}. - Find the Laplace Transform of
*Sin3t.Cos4t.* - Define Convolution of two functions and state the convolution theorem.
- Find the inverse Laplace Transform of
- Find inverse Laplace Transform of
- Write the Euler’s formulae for Fourier series of a function f(x) in the Interval [C, C+2
*l*]. - Find the Fourier Coefficient in
- An integer is picked from 1 to 20 numbers, both inclusive. Find the probability that it is a prime.

** Part-B** 10×5 = 50 marks

- Answer
*any***five**questions. - Each question carries
**ten**marks. - Answer should be comprehensive and the criteria for valuation are the content but not the length of the answer.

11(a) Solve (D^{2 }– 4) y = (1+ e^{x})^{ 2}.

(b) Solve (D^{2} + 4D + 16) y = Sin 4x.

12(a) Solve (D^{2} – 2D +1) y = x^{3}.

(b) Solve (D^{4}– 81) y = Sinh 3x + x.

13(a) Find the Laplace Transform of t.e^{-t} ^{.}Sin 4t*.*

* *(b) Evaluate using Laplace Transforms.

16 (a) Expand *f(x) = * as Fourier series in –π < x < π.

(b) Find the half range Sine series for *f(x) = *

17 (a) When two dice are thrown, find the probability of getting a sum even number.

(b) In a committee of 25 members, each member is proficient either in mathematics or in statistics or in both. If 19 of these are proficient mathematics, 16 in statistics.Find the probability that a person selected from the committee is proficient in both.

18 (a) Let A and B are independent events with P (A) = 0.8 and P (B) = 0.4. Find

(b) Box-I contains 3 red, 4 black balls, Box–II contains 5 red , 6 black balls. one ball is drawn at random from one of the bags and is found to be red. What is the probability that the ball is drawn from box-II.