Numerical and Statistical Methods (Question Paper) [April – 2017 | 75:25 Pattern]
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The document is a Mumbai University question paper from April 2017 for B.Sc. IT (Semester II) focusing on Numerical and Statistical Methods, consisting of various problems across multiple questions. Students are required to tackle questions involving mathematical modeling, root-finding methods, simultaneous equations, linear programming, probability distributions, and statistical methods. Each question contains specific instructions, marks allocation, and emphasizes the importance of making assumptions and providing proper calculations.
![MUMBAI UNIVERSITY
QUESTION PAPER
NUMERICAL AND STATISTICAL METHODS
(APRIL – 2017 | 75:25 PATTERN)
B.SC.IT
(SEMESTER – II)
KjT MUMBAI B.SC.IT STUDY
FACEBOOK | TWITTER | INSTAGRAM | GOOGLE+ | YOUTUBE | SLIDESHARE | TUMBLR | GITHUB
PAGE
1
Time: 2 ½ Hours Total Marks: 75
N.B.: (1) All Question are Compulsory.
(2) Make Suitable Assumptions Wherever Necessary And State The Assumptions Made.
(3) Answer To The Same Question Must Be Written Together.
(4) Number To The Right Indicates Marks.
(5) Draw Neat Labeled Diagrams Wherever Necessary.
(6) Use of Non – Programmable Calculator is allowed.
Q.1 ATTEMPT ANY THREE QUESTIONS: (15 MARKS)
(A) What is a Mathematical Model? With the help of a flowchart, explain the solving of an engineering
problem.
(5)
(B) Create a Hypothetical Floating-Point number set for a machine that stores information using 7-bit
words. Employ the first bit for the sign of the number, the next three for the sign and the magnitude
of the exponent, and the last three for the magnitude of the mantissa.
(5)
(C) Suppose that you have the task of measuring the lengths of a bridge and a rivet and come up with
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.
(5)
(D) Use zero – through fourth–order Taylor series expansions to approximate the function
𝑓(𝑥) = −0.1𝑥4
− 0.15𝑥3
− 0.5𝑥2
− 0.25𝑥 + 1.2
From 𝑋𝑖 = 0 with h = 1. That is, predict the function’s value at 𝑋𝑖+1 = 1.
(5)
(E) Compute the condition number for 𝑓(𝑥) = tan 𝑥̃ =
𝜋
2
+ 0.1 (
𝜋
2
)
𝑓(𝑥) = tan 𝑥̃ =
𝜋
2
+ 0.01 (
𝜋
2
)
(5)
(F) Explain Blunders, Formulation Errors and Data Uncertainty. (5)
Q.2 ATTEMPT ANY THREE QUESTIONS: (15 MARKS)
(A) Find the roots of the equation 𝑥3
− 12.2𝑥2
+ 7.45𝑥 + 42 = 0 between 11 and 12 using Regula-Falsi
method correct up to 4 decimal places.
(5)
(B) Find the roots of the equation 𝑥 tan 𝑥 = 1 near 4 using Newton Raphson Method correct up to 4
decimal places.
(5)
(C) Use the Secant method to find a solution to 𝑥 = cos 𝑥 correct up to 4 decimal places. (5)
(D) Given log 2 = 0.3010, log 3 = 0.4771, log 5 = 0.6990 andlog 7 = 0.8451. Find the value of log 47. (5)
(E) The table below gives the value of tan 𝜃. Evaluate 𝑡𝑎𝑛 670
20′
𝜃 650
660
670
680
690
tan 𝜃 2.1445 2.2460 2.3559 2.4751 2.6051
(5)
(F) From the table of Bessel function 𝐽 𝑛(1), estimate the value of 𝐽3
2
(1)
𝑛 −1
−
3
4
−
1
2
−
1
4
0 1
4
1
2
3
4
1
𝐽 𝑛(1) −0.4401 0.0447 0.4311 0.6694 0.7652 0.7522 0.6714 0.5587 0.4401
(5)
Q.3 ATTEMPT ANY THREE QUESTIONS: (15 MARKS)
(A) Solve the following simultaneous equations by Gauss – Jordan elimination method.
2𝑥1 + 6𝑥2 − 𝑥3 = −14
5𝑥1 − 𝑥2 + 2𝑥3 = 29
𝑥3 − 3𝑥1 − 4𝑥2 = 4
(5)
(B) Solve the following simultaneous equations by Gauss – Seidel method:
3𝑥1 − 0.1𝑥2 − 0.2𝑥3 = 7.85
0.1𝑥1 − 7𝑥2 − 0.3𝑥3 = 19.3
0.3𝑥1 − 0.2𝑥2 + 10𝑥3 = 71.4 [TURN OVER]
(5)](https://image.slidesharecdn.com/75-25-nsm-apr-2017-qp-180619012833/85/Numerical-and-Statistical-Methods-Question-Paper-April-2017-75-25-Pattern-1-320.jpg)
![MUMBAI UNIVERSITY
QUESTION PAPER
NUMERICAL AND STATISTICAL METHODS
(APRIL – 2017 | 75:25 PATTERN)
B.SC.IT
(SEMESTER – II)
KjT MUMBAI B.SC.IT STUDY
FACEBOOK | TWITTER | INSTAGRAM | GOOGLE+ | YOUTUBE | SLIDESHARE | TUMBLR | GITHUB
PAGE
2
(C) For the set of points (0, 2), (2, −2), (3, −1), evualate (
𝑑𝑦
𝑑𝑥
)
2
(5)
(D) Evaluate ∫
1−𝑒−𝑥
𝑥
1
0
𝑑𝑥 using Trapezoidal Rule Simpson’s 3/8 rule. (5)
(E) Solve
𝑑𝑦
𝑑𝑥
= 𝑥 + 𝑦; 𝑦(1) = 1for the interval 1 (0.1) 1.2, using method of Taylor series. (5)
(F) Solve
𝑑𝑦
𝑑𝑥
=
𝑦−𝑥
𝑦+𝑥
, where 𝑦(0) = 1, to find 𝑦(0.1) using Runge-Kutta method. (5)
Q.4 ATTEMPT ANY THREE QUESTIONS: (15 MARKS)
(A) Fit a straight line to the x and y values in the two rows:
x 1 2 3 4 5 6 7
y 0.5 2.5 2.0 4.0 3.5 6.0 5.2
(5)
(B) Fit a second degree parabola for the following:
x 2.5 3 3.5 4 4.5 5 5.5
y 4.32 4.83 5.27 5.47 6.26 6.79 7.23
(5)
(C) Fit the function ∫(𝑥; 𝑎0, 𝑎1) = 𝑎0(1 − 𝑒− 𝑎1 𝑥) to the data:
x 0.25 0.75 1.25 1.75 2.25
y 0.28 0.57 0.68 0.74 0.79
Using initial guesses 𝑎0 = 1 and 𝑎1 = 1. (Use Gauss Method)
(5)
(D) Maximize 50𝑥 + 100𝑦 subject to 10𝑥 + 5𝑦 ≤ 2500, 4𝑥 + 10𝑦 ≤ 2000, 𝑥 + 1.5𝑦 ≤ 450 and 𝑥 ≥
0; 𝑦 ≥ 0.
(5)
(E) A firm makes two types of furniture – chairs and tables. The contribution for each product as
calculated by the accounting department is Rs. 20 per chair and Rs. 30 per table. Both products are
processed on three machines 𝑀1, 𝑀2 and 𝑀3. The time required in hours by each product and total
time available in hours per week on each machine are as follows:
MACHINE CHAIR TABLE AVAILABLE TIME
𝑀1 3 3 36
𝑀2 5 2 50
𝑀3 2 6 60
How should the manufacturer schedule his production in order maximize contribution?
(5)
(F) An aged person must receive 4000 units of vitamin, 50 units of minerals and 1400 calories a day. A
dietician advises to thrive on two foods F1 and F2 that cost Rs. 4 and Rs. 2 respectively per unit of
food. It one unit of F1 contains 200 units of vitamins, 1 unit of minerals and 40 calories, formulate a
liner programming model to minimize the cost of diet.
(5)
Q.5 ATTEMPT ANY THREE QUESTIONS: (15 MARKS)
(A) The diameter of an electric cable; say X, is assumed to be 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)
(5)
(B) Define and explain the concept of probability density function. (5)
(C) 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 value 𝑝(0) = 3𝑐3
, 𝑝(1) = 4𝑐 − 10𝑐2
, 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
2
.
(v) Find the smallest x such 𝐹(𝑥) ≥
1
3
.
[TURN OVER]
(5)](https://image.slidesharecdn.com/75-25-nsm-apr-2017-qp-180619012833/85/Numerical-and-Statistical-Methods-Question-Paper-April-2017-75-25-Pattern-2-320.jpg)
Numerical and Statistical Methods (Question Paper) [April – 2017 | 75:25 Pattern]
- 1. MUMBAI UNIVERSITY QUESTION PAPER NUMERICAL AND STATISTICAL METHODS (APRIL – 2017 | 75:25 PATTERN) B.SC.IT (SEMESTER – II) KjT MUMBAI B.SC.IT STUDY FACEBOOK | TWITTER | INSTAGRAM | GOOGLE+ | YOUTUBE | SLIDESHARE | TUMBLR | GITHUB PAGE 1 Time: 2 ½ Hours Total Marks: 75 N.B.: (1) All Question are Compulsory. (2) Make Suitable Assumptions Wherever Necessary And State The Assumptions Made. (3) Answer To The Same Question Must Be Written Together. (4) Number To The Right Indicates Marks. (5) Draw Neat Labeled Diagrams Wherever Necessary. (6) Use of Non – Programmable Calculator is allowed. Q.1 ATTEMPT ANY THREE QUESTIONS: (15 MARKS) (A) What is a Mathematical Model? With the help of a flowchart, explain the solving of an engineering problem. (5) (B) Create a Hypothetical Floating-Point number set for a machine that stores information using 7-bit words. Employ the first bit for the sign of the number, the next three for the sign and the magnitude of the exponent, and the last three for the magnitude of the mantissa. (5) (C) Suppose that you have the task of measuring the lengths of a bridge and a rivet and come up with 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. (5) (D) Use zero – through fourth–order Taylor series expansions to approximate the function 𝑓(𝑥) = −0.1𝑥4 − 0.15𝑥3 − 0.5𝑥2 − 0.25𝑥 + 1.2 From 𝑋𝑖 = 0 with h = 1. That is, predict the function’s value at 𝑋𝑖+1 = 1. (5) (E) Compute the condition number for 𝑓(𝑥) = tan 𝑥̃ = 𝜋 2 + 0.1 ( 𝜋 2 ) 𝑓(𝑥) = tan 𝑥̃ = 𝜋 2 + 0.01 ( 𝜋 2 ) (5) (F) Explain Blunders, Formulation Errors and Data Uncertainty. (5) Q.2 ATTEMPT ANY THREE QUESTIONS: (15 MARKS) (A) Find the roots of the equation 𝑥3 − 12.2𝑥2 + 7.45𝑥 + 42 = 0 between 11 and 12 using Regula-Falsi method correct up to 4 decimal places. (5) (B) Find the roots of the equation 𝑥 tan 𝑥 = 1 near 4 using Newton Raphson Method correct up to 4 decimal places. (5) (C) Use the Secant method to find a solution to 𝑥 = cos 𝑥 correct up to 4 decimal places. (5) (D) Given log 2 = 0.3010, log 3 = 0.4771, log 5 = 0.6990 andlog 7 = 0.8451. Find the value of log 47. (5) (E) The table below gives the value of tan 𝜃. Evaluate 𝑡𝑎𝑛 670 20′ 𝜃 650 660 670 680 690 tan 𝜃 2.1445 2.2460 2.3559 2.4751 2.6051 (5) (F) From the table of Bessel function 𝐽 𝑛(1), estimate the value of 𝐽3 2 (1) 𝑛 −1 − 3 4 − 1 2 − 1 4 0 1 4 1 2 3 4 1 𝐽 𝑛(1) −0.4401 0.0447 0.4311 0.6694 0.7652 0.7522 0.6714 0.5587 0.4401 (5) Q.3 ATTEMPT ANY THREE QUESTIONS: (15 MARKS) (A) Solve the following simultaneous equations by Gauss – Jordan elimination method. 2𝑥1 + 6𝑥2 − 𝑥3 = −14 5𝑥1 − 𝑥2 + 2𝑥3 = 29 𝑥3 − 3𝑥1 − 4𝑥2 = 4 (5) (B) Solve the following simultaneous equations by Gauss – Seidel method: 3𝑥1 − 0.1𝑥2 − 0.2𝑥3 = 7.85 0.1𝑥1 − 7𝑥2 − 0.3𝑥3 = 19.3 0.3𝑥1 − 0.2𝑥2 + 10𝑥3 = 71.4 [TURN OVER] (5)
- 2. MUMBAI UNIVERSITY QUESTION PAPER NUMERICAL AND STATISTICAL METHODS (APRIL – 2017 | 75:25 PATTERN) B.SC.IT (SEMESTER – II) KjT MUMBAI B.SC.IT STUDY FACEBOOK | TWITTER | INSTAGRAM | GOOGLE+ | YOUTUBE | SLIDESHARE | TUMBLR | GITHUB PAGE 2 (C) For the set of points (0, 2), (2, −2), (3, −1), evualate ( 𝑑𝑦 𝑑𝑥 ) 2 (5) (D) Evaluate ∫ 1−𝑒−𝑥 𝑥 1 0 𝑑𝑥 using Trapezoidal Rule Simpson’s 3/8 rule. (5) (E) Solve 𝑑𝑦 𝑑𝑥 = 𝑥 + 𝑦; 𝑦(1) = 1for the interval 1 (0.1) 1.2, using method of Taylor series. (5) (F) Solve 𝑑𝑦 𝑑𝑥 = 𝑦−𝑥 𝑦+𝑥 , where 𝑦(0) = 1, to find 𝑦(0.1) using Runge-Kutta method. (5) Q.4 ATTEMPT ANY THREE QUESTIONS: (15 MARKS) (A) Fit a straight line to the x and y values in the two rows: x 1 2 3 4 5 6 7 y 0.5 2.5 2.0 4.0 3.5 6.0 5.2 (5) (B) Fit a second degree parabola for the following: x 2.5 3 3.5 4 4.5 5 5.5 y 4.32 4.83 5.27 5.47 6.26 6.79 7.23 (5) (C) Fit the function ∫(𝑥; 𝑎0, 𝑎1) = 𝑎0(1 − 𝑒− 𝑎1 𝑥) to the data: x 0.25 0.75 1.25 1.75 2.25 y 0.28 0.57 0.68 0.74 0.79 Using initial guesses 𝑎0 = 1 and 𝑎1 = 1. (Use Gauss Method) (5) (D) Maximize 50𝑥 + 100𝑦 subject to 10𝑥 + 5𝑦 ≤ 2500, 4𝑥 + 10𝑦 ≤ 2000, 𝑥 + 1.5𝑦 ≤ 450 and 𝑥 ≥ 0; 𝑦 ≥ 0. (5) (E) A firm makes two types of furniture – chairs and tables. The contribution for each product as calculated by the accounting department is Rs. 20 per chair and Rs. 30 per table. Both products are processed on three machines 𝑀1, 𝑀2 and 𝑀3. The time required in hours by each product and total time available in hours per week on each machine are as follows: MACHINE CHAIR TABLE AVAILABLE TIME 𝑀1 3 3 36 𝑀2 5 2 50 𝑀3 2 6 60 How should the manufacturer schedule his production in order maximize contribution? (5) (F) An aged person must receive 4000 units of vitamin, 50 units of minerals and 1400 calories a day. A dietician advises to thrive on two foods F1 and F2 that cost Rs. 4 and Rs. 2 respectively per unit of food. It one unit of F1 contains 200 units of vitamins, 1 unit of minerals and 40 calories, formulate a liner programming model to minimize the cost of diet. (5) Q.5 ATTEMPT ANY THREE QUESTIONS: (15 MARKS) (A) The diameter of an electric cable; say X, is assumed to be 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) (5) (B) Define and explain the concept of probability density function. (5) (C) 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 value 𝑝(0) = 3𝑐3 , 𝑝(1) = 4𝑐 − 10𝑐2 , 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 2 . (v) Find the smallest x such 𝐹(𝑥) ≥ 1 3 . [TURN OVER] (5)
- 3. MUMBAI UNIVERSITY QUESTION PAPER NUMERICAL AND STATISTICAL METHODS (APRIL – 2017 | 75:25 PATTERN) B.SC.IT (SEMESTER – II) KjT MUMBAI B.SC.IT STUDY FACEBOOK | TWITTER | INSTAGRAM | GOOGLE+ | YOUTUBE | SLIDESHARE | TUMBLR | GITHUB PAGE 3 (D) 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. (5) (E) 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. (5) (F) 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? (5)