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 ‘unfair means’. Don’t adopt unfair means and also don’t 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

2/28

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.

3/28

1(a) Apply mean value theorem to prove that �1 − 1𝑥�𝑥

, 𝑥 > 0 is an increasing function.

1(b) Is Rolle’s theorem applicable to the function

𝑓(𝑥) = �1 − 𝑥2, 𝑥 ≤ 0cos 𝑥 , 𝑥 > 0

�

in �−1, 𝜋2�? Justify your answer with proper arguments. If Rolle’s theorem is applicable, then

find 𝑐 ∈ �−1, 𝜋2� such that 𝑓′(𝑐) = 0.

[2+2]

4/28

5/28

6/28

2(a) Expand the function 𝑓(𝑥, 𝑦) = 𝑥𝑦 in powers of (𝑥 − 1) and (𝑦 − 1) upto terms

including 2nd

2(b) Find lim𝑥→0

� 12𝑥− 1

𝑥(1+𝑒𝑥)� .

order (without remainder). Use the result to compute the approximate value of

(1.1)1.02.

[2+2]

7/28

8/28

9/28

3(a) Find the intervals for which the function 𝑓(𝑥) = ln(𝑥2 + 1 ) is convex upwards (concave downwards) and convex downwards (concave upwards). Further, find the point of inflexion(s).

3(b) Using 𝜖 − 𝛿 approach, show that lim(𝑥,𝑦)→(1,1)

2𝑥2 + 3𝑦 = 5.

[2+2]

10/28

11/28

12/28

4(a) Find the radius of curvature of the curve 𝑥 = 6𝑡2 − 3𝑡4, 𝑦 = 8𝑡3 at an arbitrary point 𝑡. Evaluate its maximum value over 𝑡 ∈ [0, 1].

4(b) Find the asymptote(s) of the curve 𝑥3 − 𝑥2𝑦 + 𝑎𝑥𝑦 + 𝑎3 = 0.

[2+2]

13/28

14/28

15/28

5(a) Is the function

𝑓(𝑥, 𝑦) = �2𝑥3 + 3𝑦3

𝑥2 + 𝑦2, (𝑥,𝑦) ≠ (0, 0)

0, (𝑥,𝑦) = (0, 0)�

differentiable at the origin? Justify your answer.

5(b) Is the partial derivative with respect to 𝑥 of the function

𝑓(𝑥,𝑦) = �𝑥3𝑦

𝑥6 + 𝑦2, (𝑥,𝑦) ≠ (0, 0)

0 (𝑥,𝑦) = (0, 0)�

continuous at the origin? Justify your answer.

[3+2]

16/28

17/28

18/28

19/28

6(a) Let

𝑢 = sin−1 �𝑥52 − 𝑦

52

𝑥2 + 𝑦2� , (𝑥,𝑦) ≠ (0, 0)

Find the value of 𝑥2𝑢𝑥𝑥 + 𝑦2𝑢𝑦𝑦 + 2𝑥𝑦 𝑢𝑥𝑦 as a function of u.

6(b) If 𝑧 = 𝑓(𝑢, 𝑣),𝑢 = 𝑥 + 4𝑦, 𝑣 = −𝑥 − 4𝑦 and 𝑓 has continuous first and second order partial derivatives, then find the relation between 𝑧𝑥𝑥 and 𝑧𝑦𝑦.

[2+2]

20/28

21/28

22/28

23/28

7(a) Using the Lagrange multiplier method, find the minimum value of the function

𝑓(𝑥,𝑦) = 𝑥2 + 𝑦2

subject to

𝑥2 − 𝑥𝑦 + 𝑦2 = 48.

7(b) If the curves 𝑓(𝑥,𝑦) = 0 and 𝜑(𝑥,𝑦) = 0 touch each other at the point P, then evaluate

𝜕𝑓𝜕𝑥

𝜕𝜑𝜕𝑦

−𝜕𝑓𝜕𝑦

𝜕𝜑𝜕𝑥

at the point P.

[3+2]

24/28

25/28

26/28

Space for rough work

27/28

Space for rough work

28/28

Space for rough work