ID.No./Seat No. MEHRAN UNIVERSITY OF ENGINEERING AND TECHNOLOGY, JAMSHORO. SECOND SEMESTER (TL) FINAL SEMESTER EXAMINATION 2013 OF 13 – BATCH

LINEAR ALGEBRA AND ANALYTICAL GEOMETRY Dated: 29 – 11 – 2013

Time Allowed: 03 Hours

Maximum Marks 60

NOTE: ATTEMPT ALL QUESTIONS. MARKS ARE SHOWN AGAINST EACH QUESTION. Q.No.

Marks

Q # 01 (a) Find the equation of a straight line passing through the point  2, 1,3 and perpendicular to each of the straight lines: L: M:

x  t , y  1  t , z  2t x  2s, y  3s, z  s

[05]

(b) Two helicopters H1 and H 2 , are travelling together. At time t = 0, they separate and follow different straight line paths given by:

H1 : x  6  40t , y  3  10t , z  3  2t H 2 : x  6  110t , y  3  4t , z  3  t

[07]

Time t is measured in hours and all coordinates are measured in miles. Due to system malfunctions, H 2 stops its flight at  446,13,1 and, in a negligible amount of time, lands at  446,13, 0  . Two hours later, H1 is informed of this fact and it heads towards H 2 at 150 mph. How long will it take H1 to reach H 2 ? Q # 02 (a) A television camera weighing 120 pounds is supported by a tripod (see figure). Represent the force exerted on each leg of the tripod as a vector. Determine only system of linear equations to calculate force (or tension) on each leg? [07]

(b) Find the general equation of the plane passing through the points  2,1,1 ,  0, 4,1 , and  2,1, 4  

[05]

Q # 03 (a) Find the equation of the plane passing through the line of intersection of the planes 2 x  y  3z  0 and x  2 y  2 z  3  0 perpendicular to the xy – plane. [06] (b) Determine the point, if any, common to the straight line: x  1  2t , y  3t , z  1  t and the plane: 2x  y  z  5  [06]

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Q # 04 (a) Identify any two of the following surfaces:

i   sin   2 ii  r  5 iii 

z r

[06]

(b) Find equation of the sphere passing through the points A  3,6,0  , B  2, 5, 1 and C 1, 4, 2  whose centre lies on hypotenuse of the triangle ABC.

[06] 2 2 y 2

Q # 05 (a) Define Double and Triple Integrals and evaluate the integral:

  0

xy 2 dxdy 

[06]

y

4 2

(b) Prove that:

e

x2

dxdy  e4  1

[06]

0 y /2

THE END

---------page 02---------

Question Paper (Final Semester).pdf

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