,~ ,..I~F-~~~ .• Indian Institute of Technology Kharagpur.'f\~ V.)iGl

~ ~~ QUESTION-CUM-ANSWERSCRIPT Stamp/Signature of the Invigilator- ..

END-SEMESTEREXAMINATION (SPRING 2018 ) I Full marks: 50 Time: 3 hrs

Roll Number I I Section Name I

Subject Number M A 1 0 0 0 2 Subject Name Mathematics II

Department/Centre/School

Important Instructions and Guidelines for Students1. You must occupy your seat as per the Examination Schedule/Sitting Plan.

2. Mobile phones or any such electronic gadgets, even in the switched off mode, are strictly banned.

3. Loose papers, class notes, books or any such materials must not be in your possession; even if they areirrelevant to the subject you are taking examination.

4. Data book, codes, graph papers, relevant standard tables/charts or any other materials are allowed onlywhen instructed by the paper-setter.

5. Use of instrument box, pencil box and non-programmable calculator is allowed during the examination.r However, the exchange of these items or any other papers (including question papers) is not permitted.

6. Write on both sides of the answer-script and do not tear off any page. Use last pagels] of the answer-scriptfor rough work. If the answer-script contains torn or distorted page(s), report to the invigilator.

7. It is your responsibility to ensure that you have signed the Attendance Sheet. Keep your Admit Card andIdentity Card on the desk for checking by the invigilators.

8. During examination, you may leave the Examination Hall for a very short period for wash room or drinking

~water. Record your absence from the Examination Hall in the register provided. Smoking and theconsumption of any kind of beverages are strictly prohibited inside the Examination Hall.

9. Do not leave the Examination Hall without submitting your answer-script to the invigilator. In any case, youare not allowed to take away the answer-script with you. After the completion of the examination, do notleave your seat until the invigilators collect all the answer-scripts.

:10. During examination, either inside or outside the Examination Hall, gathering information from any kind of

sources or exchanging information with others or any such attempt will be treated as 'unfair means'. Don't

I adopt unfair means and also don't indulge in unseemly behavior.

Violation of any of the above instructions may lead to severe punishment.Signature of the Student

To be Filled by the Examiner

Question Number 1 2 3 4 5 6 7 8 Total

Marks Obtained

Marks Obtained (in words) Signature of the Examiner Signature of the Scrutineer

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Instructions and Guidelines to the Students

1. The question-cum-answer booklet has 44 pages and 8 questions.

2. Before start writing you must ensure that all 8 questions are printed in

the booklet.

3. All questions are compulsory.

4. Answer each question in the space provided below to that question only. Otherwise

it will not be checked.

5. No additional answer sheet will be provided.

6. Use the space for rough work given in the booklet only.

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1. (a) Using Cayley-Hamilton theorem only, find A100 and A-l, if exists, and justify

the reason if doesn't exist, for the following matrix A:

(OW W2)

A = W 0 w2 ,where W = -~ - V;i.w2 w 0

[3 marks]

(b) Find the matrix A of the linear transformation T : ]R4--1 M2x2(]R) defined by

T(Xl' X2, X3, X4) = (Xl X2) relative to the ordered basesX3 X4

51 = {(I, I, I, I), (I, I, -1,0), (2,0,3, I), (-1,2, -I, In of]R4 and

S2 ~ {G ~),(~ ~),(~ ~),(~ ~)} of M2x2 (R)

[NO MARKS IF THE BASES ARE TAKEN IN DIFFERENT ORDER] [3 marks]

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2. (a) Consider the equation x2 - 4x + 3 = O. Which of the following iteration equa-

tions, (i) and (ii) below, should be chosen so that fixed point iteration process

converges to a root of the above equation for any initial guess Xo in the interval

(2,4) ? Justify your answer.

(i) x = x2:3 (ii) x = J4x - 3 [2 marks]

(b) Applying Newton's forward interpolation formula, determine the number of

students who obtained less than 45 marks from the data given below:

Marks 30 - 40 40 - 50 50 - 60 60 -70 70 - 80

umber of students 31 42 51 35 31

[4 marks]

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3. (a) Using Simpson's 1/3rd rule, evaluate the integral fl\3x2+1) loge(x) dx, (correct

up to 3 decimal places) by dividing the interval [1,4] into six equal parts.

[3 marks]

(b) For a function f (x) given that

x 0 1 1 3 14 "2 4

f(x) a b 3 -4

where a and b are real numbers. If approximate values of f01f(x)dx are equal to

100 and 101 by using the trapezoidal rule and Simpson's 1/3rd rule respectively,

by taking four equal length subintervals for both. then find a and b ?[4 marks]

112 + sin(nx)For the improper integral ( ) dx,o 1- x p

(i) find all possible value of p so that the integral converges

(ii) find all possible values of p 0 that the integral diverges.

(SHOW YOUR TESTS SEPARATELY FOR (i) A D (ii))

100 t3(b) Using Beta function evaluate the integral ( )7dt.o 1+ t

4. (a)

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[3 marks]

[3 marks]

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5. (a) Using double integration, find the volume of the solid bounded above by the

surface z = xy and bounded below by the region enclosed by y = 4 - x2, X =1, x = 2 and the x-axis. (sketch and shade the region of integration) [3 marks]

(b) Use the transformation x = u-v, y = u+v to evaluate JL (x+y)dA, where

R is the region bounded by y = x, y = 3x, x + y = 4. [3 marks]

(c) Using Leibnitz rule, find (~~t=l'where I = 12Q ax dx. [1 mark]

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6. (a) Find the surface area of the paraboloid 2z = x2 + y2 inside the cylinder

x2 + y2 = 1. [3 marks]

(b) The volume of the region D, bounded by the paraboloid z = x2 + y2 and the

plane z = 2y, is given by J Jl dx dz dy. Then

(i) write the limits of the above integration without changing the order

(ii) find the volume of the region D. [3 marks]

7. (a) Evaluate Jis xz2 dS, where 5 is the portion of the cylinder y = 2X2 + 1 in the

first octant bounded by x = O.x = 2, Z = 4 and z = 8. [3 marks]

(b) Find the directional derivative of f(x, y, z) = x2yz+4xz2 at the point (I, -2, -1)

in the direction of the maximum rate of change. [3 marks]

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8. (a) Using the Green's theorem, evaluate i(x2 - y2) dx + (2y - x) dy, where C

consists of the boundary of the region in the first quadrant that is bounded by[3 marks]

(b) Consider the force field F = (2xy + z3)i + x2j + 3xz2k,

(i) Verify whether F is a conservative force field or not. If so, find the

potential function of F.(ii) Find the work done in moving an object by this field F from (1, -2, 1)

to (3,1,4). [3 marks]