MANIPAL INSTITUTE OF TECHNOLOGY MANIPAL UNIVERSITY, MANIPAL

Transcription

1 Reg.No MANIPAL INSTITUTE OF TECHNOLOGY MANIPAL UNIVERSITY, MANIPAL SECOND S EMES TER B.E DEGREE END S EMES TER EXAMINATION 009 SUB: ENGINEERING MATHEMATICS II ( MAT 0) (REVISED CREDIT SYSTEM) Time : 3 Hrs. Ma.Marks : 50 Note : a) Answer an FIVE full questions. b) All questions carr equal marks. A. Epand f(, ) = sin with center at, terms. B. Change the order of integration and evaluate upto second degree - 0 d. C. Define a maimal linearl independent set of vectors in a vector space V. Show that it forms a basis of V. ( ) A. Find the etreme values of B. Find the area of the region enclosed b = 4 and = 4 4. C. Use Gram Schmi process to obtain an orthonormal set of vectors from the vectors (,0,), (0,, ), (,, 3). ( ) 3A. Solve : ( + )d + ( ( +) 3 ) = 0. 3B. Find the volume bounded b z a and + = a 3C. Solve the following sstem of equations b Gauss elimination method + z = + + z = + z = 7 ( ) Page of

2 4A. Solve : (4 + 3 ) + ( + )d = 0 4B. Find L tsinh t cos t L e s s s 4C. Find the inverse of the following matri b elementar row operations. 5A. Solve : cos e. ( ) 5B. Epress the following function interms of unit step functions and hence find its Laplace transform t, 0 t f(t) t 7, t 3 8, t 3 5C. Solve the following differential equation b Laplace transform method (t) 4 (t) 4(t) 4cost, (0)=, (0)=5 ( ) 6A. Solve : (D ) e. 6B. Solve : 3 5t d 3 e t 6C. Evaluate the following integrals using Beta and Gamma functions 0 m 0 cot d n (log ) where m > and n, positive integer ********** ( ) Page of

3 Reg.No MANIPAL INSTITUTE OF TECHNOLOGY MANIPAL UNIVERSITY, MANIPAL SECOND SEMESTER B.E DEGREE END SEMESTER MAKEUP EXAMINATION 009 SUB: ENGINEERING MATHEMATICS II ( MAT 0) (REVISED CREDIT SYSTEM) Time : 3 Hrs. Ma.Marks : 50 Note : a) Answer an FIVE full questions. b) All questions carr equal marks. A. Epand e log ( +) about (0, 0) upto 3 rd degree terms. B. Change into polar co-ordinates and evaluate 0 0 d C. Solve : 5 d 4 ( ) A. Find the volume of the greatest rectangular parallelepiped that can be z inscribed in the ellipsoid a b c B. Solve : (D 3D + ) = cos (e ) C. Test whether the following set of vectors from basis of E 3, if so epress (,, 3) in terms of basis vectors (,, 0), (, 0, ), (,, ). ( ) 3A. Solve : sin ( + sin) + cosd = 0. 3B. Find the volume bounded b + = a, + = az and z = 0 Page of

4 3C. Solve b Gauss Elimination method given 4 5 +z = z = z = 5 + 5z = 9 4A. Solve : ( + + ) + ( ) d = 0. ( ) 4B. Find t L e.t.cos.t L s s a s b 4C. Define the maimal linearl independent set of vectors in a vector space. Show that it forms a basis. ( ) 5A. Solve : d d e sin 5B. Find the Laplace transform of Esin t, 0 t f (t), f t+ f (t) 0, t 5C. In the RL circuit, let the switch be closed at t =0. At some later time t = t0, the direct current element, the constant E, is to be removed from the circuit, which remains closed. Find the current for all t > 0. ( ) 6A. Using double integrals, find the area ling inside r = a sin and outside r = a( cos ). 6B. Solve d d 4 log. 6C. Evaluate : sin d 0 0 d sin ( ) ********* Page of

5 Reg.No MANIPAL INSTITUTE OF TECHNOLOGY MANIPAL UNIVERSITY, MANIPAL SECOND SEMESTER B.E DEGREE END SEMESTER EXAMINATION Ma, 009 SUB: ENGINEERING MATHEMATICS II ( MAT 0) (REVISED CREDIT SYSTEM) Time : 3 Hrs. Ma.Marks : 50 Note : a) Answer an FIVE full questions. b) All questions carr equal marks. A. Epand sin about, up to 3rd degree terms. B. Change into polar co-ordinates and evaluate a a+ a 4a d 0 a a C. Solve : 3 5t d 3 e t ( ) A. Find the minimum distance from the origin to the surface + z + z = 3a. B. Solve : (D D ) e e. C. Use Gram Schmi process to find a set of orthonormal vectors from (,, ), (,, ), (,, 3) in E 3. ( ) 3A. Solve cos.sin + (cos cos )d = 0. 3B. Find the volume inside z a and + = a. Page of

6 3C. Solve b Gauss Elimination method, given ( ) 4A. Solve : ( + 4) (3 6 )d = 0 when = 4, =. 4B. Find t cos t L e t L e s s s s 5 4C. Find the inverse of the following matri b elementar row transformations 3 5A. Solve 6 0 D 4D 3 e sin ( ) 5B. Rewrite interms of unit step functions and hence find its Laplace transform sin t, 0 t F(t) t, t 6, t 5C. A spring with the spring constant 0.75 lb per ft is attached to a support. A 6 lb weight is attached to a spring and is at rest at the equilibrium position. A.5 lb force is applied to the support along the line of action of the spring for 4 seconds and is then removed. Described the motion of the spring. ( ) 6A. Find the area bounded b = 4 and =. 6B. Solve : d d sin(log ) 3 log 6C. Evaluate, using Beta and Gamma functions 0 0 e e ********** ( ) Page of