100001111_questions__set_1_midterm.pdf
TRANSCRIPT
IInnddiiaann IInnssttiittuuttee ooff TTeecchhnnoollooggyy KKhhaarraaggppuurr
QUESTION-CUM-ANSWERSCRIPT
Stamp/Signature of the Invigilator
MID-SEMESTER EXAMINATION SEMSETER ( Autumn-2014 )
Roll Number Section Name
Subject Number M A 1 0 0 0 1 Subject Name Mathematics I
Department/Centre/School Important Instructions and Guidelines for Students
1. You must occupy your seat as per the Examination Schedule/Sitting Plan.
2. Do not keep mobile phones or any similar electronic gadgets with you even in the switched off mode.
3. Loose papers, class notes, books or any such materials must not be in your possession; even if they are irrelevant to the subject you are taking examination.
4. Data book, codes, graph papers, relevant standard tables/charts or any other materials are allowed only when instructed by the paper-setter.
5. Use of instrument box, pencil box and non-programmable calculator is allowed during the examination. 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 page(s) of the answer-script for rough work. Report to the invigilator if the answer-script has torn or distorted page(s).
7. It is your responsibility to ensure that you have signed the Attendance Sheet. Keep your Admit Card/Identity Card on the desk for checking by the invigilator.
8. You may leave the Examination Hall for wash room or for drinking water for a very short period. Record your absence from the Examination Hall in the register provided. Smoking and consumption of any kind of beverages is strictly prohibited inside the Examination Hall.
9. Do not leave the Examination Hall without submitting your answer-script to the invigilator. In any case, you are not allowed to take away the answer-script with you. After the completion of the examination, do not leave your seat until invigilators collect all the answer-scripts.
10. During the 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 βunfair meansβ. Donβt adopt unfair means and also donβt indulge in unseemly behavior.
11. Please see overleaf for more instructions
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 Total
Marks Obtained
Marks Obtained (in words)
Signature of the Examiner Signature of the Scrutineer
A00121 A00124
A00122 A00123
A00121 A00124
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Instructions and Guidelines to the Students appearing in the Examination
1. The question-cum-answer booklet has 28 pages and 7 questions.
2. All questions are compulsory.
3. Answer each question in the space provided below that question only. Otherwise it will
not be checked.
4. No additional answer sheet will be provided.
5. Use the space for rough work given in the booklet only.
6. After the completion of the examination do not leave the examination hall until the
invigilator collects the booklet.
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1(a) Apply mean value theorem to prove that οΏ½1 β 1π₯οΏ½π₯
, π₯ > 0 is an increasing function.
1(b) Is Rolleβs theorem applicable to the function
π(π₯) = οΏ½1 β π₯2, π₯ β€ 0cos π₯ , π₯ > 0
οΏ½
in οΏ½β1, π2οΏ½? Justify your answer with proper arguments. If Rolleβs theorem is applicable, then
find π β οΏ½β1, π2οΏ½ such that πβ²(π) = 0.
[2+2]
6/28
2(a) Expand the function π(π₯, π¦) = π₯π¦ in powers of (π₯ β 1) and (π¦ β 1) upto terms
including 2nd
2(b) Find limπ₯β0
οΏ½ 12π₯β 1
π₯(1+ππ₯)οΏ½ .
order (without remainder). Use the result to compute the approximate value of
(1.1)1.02.
[2+2]
9/28
3(a) Find the intervals for which the function π(π₯) = ln(π₯2 + 1 ) is convex upwards (concave downwards) and convex downwards (concave upwards). Further, find the point of inflexion(s).
3(b) Using π β πΏ approach, show that lim(π₯,π¦)β(1,1)
2π₯2 + 3π¦ = 5.
[2+2]
12/28
4(a) Find the radius of curvature of the curve π₯ = 6π‘2 β 3π‘4, π¦ = 8π‘3 at an arbitrary point π‘. Evaluate its maximum value over π‘ β [0, 1].
4(b) Find the asymptote(s) of the curve π₯3 β π₯2π¦ + ππ₯π¦ + π3 = 0.
[2+2]
15/28
5(a) Is the function
π(π₯, π¦) = οΏ½2π₯3 + 3π¦3
π₯2 + π¦2, (π₯,π¦) β (0, 0)
0, (π₯,π¦) = (0, 0)οΏ½
differentiable at the origin? Justify your answer.
5(b) Is the partial derivative with respect to π₯ of the function
π(π₯,π¦) = οΏ½π₯3π¦
π₯6 + π¦2, (π₯,π¦) β (0, 0)
0 (π₯,π¦) = (0, 0)οΏ½
continuous at the origin? Justify your answer.
[3+2]
19/28
6(a) Let
π’ = sinβ1 οΏ½π₯52 β π¦
52
π₯2 + π¦2οΏ½ , (π₯,π¦) β (0, 0)
Find the value of π₯2π’π₯π₯ + π¦2π’π¦π¦ + 2π₯π¦ π’π₯π¦ as a function of u.
6(b) If π§ = π(π’, π£),π’ = π₯ + 4π¦, π£ = βπ₯ β 4π¦ and π has continuous first and second order partial derivatives, then find the relation between π§π₯π₯ and π§π¦π¦.
[2+2]
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7(a) Using the Lagrange multiplier method, find the minimum value of the function
π(π₯,π¦) = π₯2 + π¦2
subject to
π₯2 β π₯π¦ + π¦2 = 48.
7(b) If the curves π(π₯,π¦) = 0 and π(π₯,π¦) = 0 touch each other at the point P, then evaluate
ππππ₯
ππππ¦
βππππ¦
ππππ₯
at the point P.
[3+2]