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- Applied Mathematics-1 - M-1
- 2018
- PYQ
**Biju Patnaik University of Technology Rourkela Odisha - BPUT**- Civil Engineering
- B.Tech
**2301 Views**- 17 Offline Downloads
- Uploaded 10 months ago

Registration No : Total Number of Pages : 02 B.Tech PAM1A001 1st Semester Back Examination 2018-19 APPLIED MATHEMATICS-I BRANCH : AEIE, AERO, AUTO, BIOMED, BIOTECH, CHEM, CIVIL, CSE, ECE, EEE, EIE, ELECTRICAL, ENV, ETC, FAT, IEE, IT, MANUFAC, MANUTECH, MECH, METTA, MINERAL, MINING, MME, PE, PLASTIC, PT, TEXTILE Max time : 3 Hours Max Marks : 100 Q.CODE : E684 Answer Question No.1 (Part-1) which is compulsory, any EIGHT from Part-II and any TWO from Part-III. The figures in the right hand margin indicate marks. Q1 a) b) c) d) e) f) g) h) i) j) Q2 a) b) c) d) e) f) g) h) Part- I Short Answer Type Questions (Answer All-10) Find the asymptotes parallel to X-axis of the curve y3+x2y+2xy2-y+1=0 Finds the points on the parabola y=x2 where the radius of curvature is 4. What do you mean by integrating factor? How it helps to solve differential equations? What is the Wronskian? What role does it play in getting solution of a differential equation? What do you mean by general solution and particular solution of a differential equation? What is the practical significance of these two concepts? Define Cauchy’s homogeneous linear equation. What is the rank of a matrix? Write its basic importance. Define Legendre equation and Legendre polynomial. Define a Unitary matrix and give examples. If is an eigen value of an orthogonal matrix,then find its eigen value. Part- II Focused-Short Answer Type Questions- (Answer Any Eight out of Twelve) 1 2 3 Find the rank of the matrix A= 3 4 5 . 4 6 8 Show that the radius of curvature of any point of the asteroid = = , is equal to three times the length of the perpendicular from the origin to the tangent. Show that the eight points of intersection of the curve ( − )+ + = , and its asymptotes lie on a circle whose center is at the origin. If = sin show that + = tan . √ √ Obtain Taylors series expansion oftan ( ) about (1,1) up to and including the second degree. A rectangular box with square base and open top is to be made from 12 sq. feet of cardboard. What is the maximum possible volume of such a box? Solve the differential equation + 2 = For what value of , the equation + + =1 +2 +4 = + 4 + 10 = Has a solution and solve them completely in each case. i) Solve: j) Solve the differential equation − = cos . cosh ′′ + = ,by using variation of parameter method. (2 x 10) (6 x 8)

k) l) =0 Part-III Long Answer Type Questions (Answer Any Two out of Four) Find all the asymptotes of the cubic polynomial − 2 + (2 − ) + ( − ) + 1 = 0 and show that cut the curve in three point which lie on the straight line − + 1 = 0 Q3 Q4 Q5 State and prove Rodrigues’s Formula. Solve: ( + + 1) + ( + 3 + 2) a) b) Q6 (16) Find the basis of eigenvectors and diagonalize the following matrix 18 0 0 24 −4 0 42 −12 −2 (16) Solve ( Solve ( (16) − 9) = + + 5 + 6) = − sin 2 cosh 2 The Legendre polynomials a) ∫ ( ) ( ) b) ∫ ( ) = = 0, ≠ ( )satisfy the following orthogonal property: (16)

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