**[Time: 2 ^{1}/2 Hours.**

Q. 1) Attempt any three of the following:** (15M)**

a) Explain Conservation Laws and Engineering Problem.

b) Explain the following with examples

i) Blunders

ii) Formulation Errors

iii) Data Uncertainty

iv) Total Numerical Errors

c) Explain Floating Point representation and Errors in floating-point arithmetic.

d) Use zero through third-order Taylor series expansions to predict f(3) for

F(X)= 25x^{3}-6x^{2}+7c-88

Using a base point at x=1 Compute the true percent relative error for each approximation

e) Evaluate y=x^{3}-7x^{2}+8x-0.35 at 1.37 use 3 digits and 4 digit arithmetic and find the significant digit lost. Also find the relative error after rounding off

f) Evaluate f (1) using Taylors series for f (x), where, f(x)=x^{3}-3x^{2}+5x-10

Q. 2) Attempt any three of the following:** (15M)**

a) Define and express each of the Δ, ∇, δ, and μ in terms of E.

b) Find the polynomial using Lagrange’s interpolation polynomial which agrees with the table below given values. Hence obtain the value of f (x) at x = 2

x |
0 | 1 | 3 | 4 |

f (x) |
-12 | 0 | 6 | 12 |

c) find the Newton’s forward difference interpolation polynomial which agrees with the table below given values. Hence obtain the value of f (x) at x= 6

x | 0 | 1 | 2 | 3 | 4 | 5 |

f (x) | -5 | -10 | -9 | 4 | 35 | 90 |

d) Obtain the root for each of the following equations using Regula Falsi Method by 5 iterations

F (x) = x^{3}– 8x + 40 = 0 up to 4 decimal place with x_{0}=-5 and x_{1}=4

e) Obtain the root for each of the following equations by Newtons Raphson Method by 5 interaction

F (x) = 2x^{3}+5x^{3}+5x +3 = 0 up to 4 decimal places.

f) Explain BIsection Method. Find the approximate root of x^{3}– x – 4 = 0 by Bisection Method up to 4 decimal place Perform 4 interactions.

Q. 3) Attempt any three of the following:** (15M)**

a) Solve the following system of equations by Gauss-Jordan elimination method.

5x – y + z = 10 , 2x + 3y +18z = 30, x +17y – 2z = 48

b) Solve the following system of equation, correct to four places of decimals by Gauss-Seidal method perform 4 interaction use pivoting if necessary.

30x – 2y + 3z = 75; 2x + 2y + 18z =10; x +17y + 2z = 48

c) Evaluate using Simpson’s one third rule with 8 subintervals.

\int_0^2\log\left(\frac1x\right)dx\;

d) Evaluate by Trapezoidal rule with 10 sub-intervals and find the error.

\int_0^{10}\left(x+\frac1x\right)dx\;

e) Find the values of y(0, 1) and y(0, 2) using Euler’s modified methods with h=0.1 given that. \frac{dy}{dx}+\frac yx=y^2 ,y (1) = 1

f) Find y(0, 2) and y(0, 4) taking h(0, 2) by second order Runge-Kutta method given that

\frac{dy}{dx}=\frac{\displaystyle\left(y-x\right)}{\left(y+x\right)}, y (0) = 1

Q. 4) Attempt any three of the following:** (15M)**

a) An electrical firm manufactures circuit boards in two configurations, say (1) and (2). Each circuit board in configuration (1) requires 1 component of A. 2 components of B and 2 components of C, each circuit board in configuration (2) requires 2 components of A, 2 components of b and 1 component of C. Total components available are 30, 30 and 24 of A, B and respectively. If the profit realized up to sale is Rs. 200 per circuit board in configuration (1) and Rs. 150 per circuit in configuration (2). How many circuit boards of each configuration should be firm manufactured so as to maximize profit? Formulate the problem as linear programming model and solve it graphically.

b) A live diet is to contain at least 400 nits of carbohydrates. 500 units of fat and 300 units of protein. Two foods are available F1 which costs Rs 2 per unit and F2 which costs Rs 4 per unit. A unit of food F1 contain10 units of carbohydrates, 20 units fat, and 15 units of protein, and a unit of food F2 contain 25 units of carbohydrates, 10 units of fat, and 20 units of protein. Find the minimum cost for a diet that consists of a mixture of these two foods and also meets the minimum nutrition requirements. formulate the problems, LInear programming model.

c) Explain the Applications of Linear Programming in Busine and Industry.

d) Find the straight-line approximation to the following data.

X |
71 | 68 | 73 | 69 | 67 | 65 | 66 | 67 |

Y |
69 | 72 | 70 | 70 | 68 | 67 | 68 | 64 |

e) Find the least square polynomial approximation to the following data.

X |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Y |
2 | 6 | 7 | 8 | 10 | 11 | 11 | 10 | 9 |

f) Obtain a regression plane by using multiple regression to fit the following data.

X |
0 | 1 | 2 | 3 | 4 |

Y |
1 | 2 | 3 | 4 | 5 |

Z |
13 | 17 | 19 | 21 | 26 |

Q. 5) Attempt any three of the following:** (15M)**

a) Explain the following.

i) Random Variable

ii) probability density function

iii) Probability mass function

b) The probability distribution function of a discrete of a discrete random variable X is given by

X |
-2 | -1 | 0 | 1 | 2 |

P(X-1) |
0.1 | 0.15 | 0.2 | 0.15 | 0.4 |

Find i P(X<=0) ii P(X>=-1), Also obtain the probability distribution of y = x^{2}

c) A random variable X has the following probability distribution.

X |
1 | 2 | 3 | 4 | 5 | 6 |

P(X=x) |
3C | 5C | 7C | 9C | 11C | 13C |

i) Find C ii) P(X>=2) iii) P(0<X<4)

d) If X-N (μ =30, δ=7 ), Find

i) P(X < 20)

ii) P(33< X <45)

iii) P (15< X <25)

e) It is observed that 20 a/n of he students in a class are vegetarians. If 4 students are selected at random from this class, what is the probability that

i) Exactly one student is vegetarian

ii) At least two of them are vegetarian

f) A senior citizen receives on an average 2.5 telephone calls during his afternoon nap period 400-1405 hrs. Find the probability that on a certain day, he receives

i) No telephone calls ii) Exactly 4 calls during the same period. [ Given e-2.5= 0.0821]

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