# Model Paper IV SEM ||Engg.Mathematics-III||

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

Max Marks:  80                                                                              Time: 3 Hrs

Part-A

• Each question carries three marks.
• Answer should be brief and straight to the point                            10×3=30 marks
1. Solve the equation
2. Solve (D 4– 8D2+ 16) y = 0.
3. Find the Particular Integral for (D2+ 4) y = Sin2x + e-2x.
4. State the change of scale property and first shifting theorem of Laplace Transforms.
5. Find the Laplace Transform of 4e2t + 6t3-3 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

•  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 (D2 + D -6) y = e-3x.

(b) Solve (D3 +4D) y = 5 + Cos 2x.

12(a) Solve (D2 + 2D +1) y = 2x + x2.

(b) Solve (D4– 16) y = Cos 2x + Sinh 2x.

13 (a) Find the Laplace Transform of t2. Cos t.

(b) Find the Laplace Transform of

14 (a) Find  .

(b) Find  using Convolution theorem.

1. Find Fourier series for the function

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.

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

Max Marks:  80                                                  Time: 3 Hrs

Part-A

• Each question carries three marks.
• Answer should be brief and straight to the point                            10×3=30 marks
1. Solve the equation
2. Solve (D 4– 5D2 + 4) y = 0.
3. Find the Particular Integral for (D2 + D -6) y = e3x+ e-3x.
4. Find the Laplace Transform of Sin3t.Cos4t.
5. Define Convolution of two functions and state the convolution theorem.
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+2l].
9. Find the Fourier Coefficient in
10. 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

• 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 (D2 – 4) y = (1+ ex) 2.

(b) Solve (D2 + 4D + 16) y =  Sin 4x.

12(a) Solve (D2 – 2D +1) y = x3.

(b) Solve (D4– 81) y = Sinh 3x + x.

13(a) Find the Laplace Transform of t.e-t .Sin 4t.

(b) Evaluate   using Laplace Transforms.

14 (a) Find

(b) Find

1. Find Fourier series for the function

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.

Engineering Mathematics