FYBSC IT ( SEM 2 ) NUMERICAL METHODS

 FYBSC IT ( SEM 2 ) NUMERICAL METHODS

IMPORTANT QUESTIONS 

UNIT 1 ( 15 MARKS )

1.State the characteristics of typical mathematical models of physical world. Explain with example.

2.Discuss the conservation laws and engineering with respect to mathematical models.

3.Define significant figures, accuracy ,precision , truncation error.

4.Define absolute,relative,percentage and roundoff errors

5.For x = 3.4537 is correct upto 4 significant figures find relative and percentage error

6.Suppose that you have the task of measuring the lengths of a bridge and a rivet and come upwith 9999 and 9 cm, respectively. If the true values are 10,000 and 10 cm, respectively,compute (i) the true error and (ii) the true percent relative error for each case

7.Let p = 0.54617 and q = 0.54601. Use four-digit arithmetic to approximate p − q and determine the absolute and relative errors using (i) rounding and (ii) chopping.

8.Use zero- through third-order Taylor series expansions to predict f (3) for 𝑓 (𝑥) = 25𝑥3 − 6𝑥2 + 7𝑥 – 88 using a base point at x = 1.

9. Create a hypothetical floating-point number set for a machine that stores information using7-bit words. Employ the first bit for the sign of the number, the next three for the sign andthe magnitude of the exponent, and the last three for the magnitude of the mantissa

10. Compute the condition number for 𝑓(𝑥) = 𝑡𝑎𝑛𝑥 𝑓𝑜𝑟 𝑥̃ =𝜋/2+ 0.1 (𝜋/2) & 

𝑓(𝑥) = 𝑡𝑎𝑛𝑥 𝑓𝑜𝑟 𝑥̃ =𝜋/2+ 0.01 (𝜋/2)

11. Explain blunders, formulation errors and data uncertainty.

12.Use Taylor series expansions with n = 0 to 6 to approximate 𝑓 (𝑥) = 𝑐𝑜𝑠𝑥 at x = π/3 on the basis of the value of f (x) and its derivatives at xi =π/4.

13. Evaluate and interpret the condition number for𝑓 (𝑥) =𝑠𝑖𝑛𝑥/1 + 𝑐𝑜𝑠𝑥 𝑓𝑜𝑟 𝑥 = 1.0001𝜋

14. What are total numerical errors? Discuss stability and condition of a mathematical Problem.

UNIT 2 ( 15 MARKS )

1. Use the Bisection method to find solutions accurate to within 10−2 for

 𝑥3 − 7𝑥2 + 14𝑥 − 6 = 0 in the interval [3.2, 4].

2.The fourth-degree polynomial 𝑓 (𝑥) = 230𝑥 4 + 18𝑥3 + 9𝑥 2 − 221𝑥 – 9 in

[0, 1] correct up to 4 decimal places using Regula-Falsi method.

3.Find the root of 4𝑥2 − 𝑒𝑥 − 𝑒−𝑥 = 0 using Newton Raphson correct upto 4 decimal places using initial value as 1.

4.Given the cube of integers in the following table. Find the values of (53. 5 ) and 153 using Newton’s interpolation formula.

5.Find f (0.9) if f (0.6) = −0.17694460, f (0.7) = 0.01375227, f (0.8) = 0.22363362,

f(1.0) = 0.65809197 using Lagrange’s Interpolation formula.

6.Using appropriate interpolation formula find f(4.25) from the table:

X     4.0      4.1      4.2    4.3      4.4     4.5

f(x) 27.21 30.18 33.35 36.06 40.73 54.01

7.Determine the real root of 𝑓(𝑥) = −26 + 85𝑥 − 91𝑥 2 + 44𝑥3 − 91𝑥2 + 𝑥5 between 0.5 and 1.0 correct up to 3 decimal places using bisection method.

8.Determine the positive real root of ln(𝑥4) = 0.7 between 0.5 and 2 using method of false position.

9.Solve: 𝑥 − 0.8 − 0.2𝑠𝑖𝑛𝑥 = 0 using Newton Raphson method correct upto 4 decimal places starting with initial value 0.

10.Using the necessary interpolation formula find f(1) and f(1.5) from the table:

X    –1  0  2  3

f(x) –8  3  1 12

                                               UNIT 3 ( 15 MARKS )

1.Solve the following system by using the Gauss-Jordan elimination method.

𝑎 + 𝑏 + 2𝑐 = 1, 2𝑎 − 𝑏 + 𝑑 = −2

𝑎 − 𝑏 − 𝑐 − 2𝑑 = 4 , 2𝑎 − 𝑏 + 2𝑐 − 𝑑 = 0

2.Solve the following system by using the Gauss-Seidel iterative method.

10𝑎 − 𝑏 + 2𝑐 = 6 , −𝑎 + 11𝑏 − 𝑐 + 3𝑑 = 25

2𝑎 − 𝑏 + 10𝑐 − 𝑑 = −11 , 3𝑏 − 𝑐 + 8𝑑 = 15

3.Evaluate ∫ √1 − 8𝑥2𝑑𝑥 with limits {0, 0.3} using Simpson’s 3/8th rule.

4.Apply Taylor’s method of order two with N = 10 to the initial-value problem

𝑦′ = 𝑦 − 𝑡 2 + 1, 0 ≤ 𝑡 ≤ 2 ≤, 𝑦 (0) = 0.5

5.Using modified Euler’s method find the solution of

𝑦′ = 𝑐𝑜𝑠2𝑡 + 𝑠𝑖𝑛3𝑡), 0 ≤ 𝑡 ≤ 1; 𝑦(0) = 1 𝑤𝑖𝑡ℎ ℎ = 0.25

Given log 280 = 2.4472, log 281 = 2.4487, log 283 = 2.4518, log 286 = 2.4564. Find

[ 𝑑/𝑑𝑥 (𝑙𝑜𝑔𝑥)] 𝑥=280

Evaluate the following using Simpson’s 3/8th rule.

∫𝑠𝑖𝑛2𝜃/5 + 4𝑐𝑜𝑠𝜃𝑑𝜃 for limits (0 𝜋)

Use Euler’s method to approximate the solution for

𝑦′ = 𝑡−2(𝑠𝑖𝑛2𝑡 − 2𝑡𝑦), 1 ≤ 𝑡 ≤ 2, 𝑦 (1) = 2 𝑤𝑖𝑡ℎ ℎ = 0.5

Solve 𝑦′ = 𝑦 − 𝑡2 + 1, 𝑦(0) = 0.5, 0 ≤ 𝑡 ≤ 2 using Runge Kutta 4th order method with h = 0.5

UNIT 4 ( 15 MARKS )

1.Fit an exponential model to:

x 0.4  0.8   1.2   1.6     2.0     2.3

y 800 975 1500 1950 2900 3600

2.Find the least square polynomial approximation of degree two to the data

x   0   1 2  3   4

y –4 –1 4 11 20

3.Find the best-fit values of a and b so that y = a + bx fits the data given in the table.

x 0  1    2     3   4

y 1 1.8 3.3 4.5 6.3

4.A painter has exactly 32 units of yellow dye and 54 units of green dye. He plans to mix as manyGallons as possible of colour A and colour B. Each gallon of colour A requires 4 units of yellow Dye and 1 unit of green dye. Each gallon of colour B requires 1 unit of yellow dye and 6 units of Green dye. Find the maximum number of gallons he can mix graphically.

5.Rita wants to buy x oranges and y peaches from the store. She must buy at least 5 oranges andThe number of oranges must be less than twice the number of peaches. An orange weighs150 Grams and a peach weighs 100 grams. Joanne can carry not more than 3.6 kg of fruits home. I) Write 3 inequalities to represent the information given above.ii) Plot the inequalities on the Cartesian grid and show the region that satisfies all theInequalities. Label the region S. iii) Oranges cost ₹ 0.70 each and peaches cost ₹ 0.90 each. Find the maximum that Rita Can spend buying the fruits

Fit a second order polynomial to the data given below:

x 0      1     2      3       4      5

y 2.1 7.7 13.6 27.2 40.9 61.1

Fit a straight line to the given data regarding x as the independent variable.

x   1       2      3      4     5    6

y 1200 900 600 200 110 50

Consider the data below: Use linear least-squares regression to determine a function of the form y = bemx for the given data by specifying b and m.

x 1 2  3   4

y 1 7 11 21

A farmer can plant up to 8 acres of land with wheat and barley. He can earn ₹ 5,000 forevery acre he plants with wheat and ₹ 3,000 for every acre he plants with barley. Hisuse of a necessary pesticide is limited by federal regulations to 10 gallons for his entire 8 acres. Wheat requires 2 gallons of pesticide for every acre planted and barley requires just 1 gallon per acre. What is the maximum profit he can make? Solve graphically.

The Bead Store sells material for customers to make their own jewelry. Customer canselect beads from various bins. Grace wants to design her own Halloween necklace fromorange and black beads. She wants to make a necklace that is at least 12 inches long, butno more than 24 inches long. Grace also wants her necklace to contain black beads thatare at least twice the length of orange beads. Finally, she wants her necklace to have atleast 5 inches of black beads.Find the constraints, sketch the problem and find the vertices (intersection points).

UNIT 5

The mileage C in thousands of miles which car owners get with a certain kind of tyre is a Random variable having probability density function

𝑓(𝑥) = 1/20 𝑒− 𝑥20 𝑓𝑜𝑟 𝑥 > 0

= 0, 𝑓𝑜𝑟 𝑥 ≤ 0

Find the probabilities that one of these tyres will last

i. At most 10000 miles

ii. Anywhere from 16000 to 24000 miles

iii. At least 30000 miles

b. A petrol pump is supplied with petrol once a day. If its daily volume X of sales in thousands

Of litres is distributed by

𝑓(𝑥) = 5(1 − 𝑥)4, 0 ≤ 𝑥 ≤ 1

what must be the capacity of its tank in order that the probability that its supply will be

Exhausted in a given day shall be 0.01?

c. A continuous random variable X has a p.d.f.

𝑓(𝑥) = 3𝑥2, 0 ≤ 𝑥 ≤ 1

Find a and b such that

i. P(X ≤ a) = P(X > a) and

ii. P(X > b) = 0.05

What is the probability of getting a total of 9 (i) twice and (ii) at least twice in 6 tosses of a

Pair of dice?

In a precision bombing attack there is a 50% chance that any one bomb will strike the target.

Two direct hits are required to destroy the target completely. How many bombs must be?

Dropped to give a 99% chance or better of completely destroying the target?

A car hire firm has two cars which it fires out day by day. The number of demands for a car

On each day is distributed as Poisson variety with mean 1.5. Calculate the proportion of days

On which (i) neither car is used (ii) some demand is refused.

The diameter of an electric cable; say X, is assumed to be a continuous random variable with p.d. f. 𝑓 ( 𝑥 ) = 6𝑥 ( 1 − 𝑥), 0 ≤ 𝑥 ≤ 1.

(i) Check that above is p.d.f,

(ii) Determine a number b such that P (X < b) = P (X> b)

Define and explain the concept of probability density function.

The probability mass function of a random variable X is zero except at the points 𝑖 = 0, 1,2.

At these points it has the values 𝑝 (0) = 3𝑐3, 𝑝(1) = 4𝑐 − 10𝑐 2, 𝑝(2) = 5𝑐 − 1 for some 𝑐 > 0.

(i) Determine the value of c.

(ii) Compute the following probabilities, 𝑃 (𝑋 < 2) 𝑎𝑛𝑑 𝑃 (1 < 𝑋 ≤ 2).

(iii) Describe the distribution function and draw its graph.

(iv) Find the largest x such that 𝐹 (𝑥) < 1

(v) Find the smallest x such that 𝐹 (𝑥) ≥ 1

What is exponential distribution? Suppose the time till death after infection with Cancer, is

exponentially distributed with mean equal to 8 years. If X represents the time till death after infection with Cancer, then find the percentage of people who die within five years after infection with Cancer.

The price for a litre of whole milk is uniformly distributed between Rs. 45 and Rs. 55 during July in Mumbai. Give the equation and graph the pdf for X, the price per litre of whole milk during July. Also determine the percent of stores that charge more than Rs. 54 per litre.

The monthly worldwide average number of airplane crashes of commercial airlines is 2.2. What is the probability that there will be (i) more than 2 such accidents in the next month? (ii) more than 4 such accidents in the next 2 months?