Page 1 GATE EC - 1993 Time : 3 hours PART I Maximum Marks : 200 SECTION - A 1. This questions 1.1 to 1.7 below one or more the alternatives are correct. Write the code letter(s), (A,B, C and D) corresponding to the correct alternatives in the answer book. Marks will be given only if all the correct alternatives have been selected and no incorrect alternative is picked up. 1.1 The eigen vector(s) of the matrix 0 0 0 0 0 0 , 0, 0 0 0 a ? ? ? ? ? ? ? ? ? ? ? is/are: (a) ( ) 0,0,a (b) ( ) ,0,0 a (c) ( ) 0,0,1 (d) ( ) 0, ,0 a 1.2 The differential equation, 2 2 sin 0, d y dy y dx dx + + = is: (a) linear (b) non-linear (c) homogeneous (d) of degree two 1.3 Simpson’s rule for integration gives exact result when ( ) f x is a polynomial of degree (a) 1 (b) 2 (c) 3 (d) 4 1.4 Which of the following is (are) valid FORTRAN 77 statement(s)? (a) DO 131 = 1 (b) A = DIM***7 (c) READ = 15.0 (d) GOTO 3 = 10 1.5 Fourier series of the periodic function (period 2p) defined by ( ) 0 0 0 x f x x x p p - < < ? = ? < < ? is 2 1 1 1 cos 1cos cos sin 4 n n nx n nx n p p p p 8 ? ? ? ? + - - ? ? ? ? ? ? ? ? ? By putting x p = in the above, one can deduce that the sum of the series 2 2 2 1 1 1 1 , 3 5 7 + + + +K is (a) 2 4 p (b) 2 6 p (c) 2 8 p (d) 2 12 p 1.6 Which of the following improper integrals is (are) convergent? (a) 1 0 sin 1 cos x dx x - ? (b) 0 cos 1 x dx x 8 + ? ` (c) 2 0 1 x dx x 8 + ? (d) 1 5 0 2 1 cosx dx x - ? Page 2 GATE EC - 1993 Time : 3 hours PART I Maximum Marks : 200 SECTION - A 1. This questions 1.1 to 1.7 below one or more the alternatives are correct. Write the code letter(s), (A,B, C and D) corresponding to the correct alternatives in the answer book. Marks will be given only if all the correct alternatives have been selected and no incorrect alternative is picked up. 1.1 The eigen vector(s) of the matrix 0 0 0 0 0 0 , 0, 0 0 0 a ? ? ? ? ? ? ? ? ? ? ? is/are: (a) ( ) 0,0,a (b) ( ) ,0,0 a (c) ( ) 0,0,1 (d) ( ) 0, ,0 a 1.2 The differential equation, 2 2 sin 0, d y dy y dx dx + + = is: (a) linear (b) non-linear (c) homogeneous (d) of degree two 1.3 Simpson’s rule for integration gives exact result when ( ) f x is a polynomial of degree (a) 1 (b) 2 (c) 3 (d) 4 1.4 Which of the following is (are) valid FORTRAN 77 statement(s)? (a) DO 131 = 1 (b) A = DIM***7 (c) READ = 15.0 (d) GOTO 3 = 10 1.5 Fourier series of the periodic function (period 2p) defined by ( ) 0 0 0 x f x x x p p - < < ? = ? < < ? is 2 1 1 1 cos 1cos cos sin 4 n n nx n nx n p p p p 8 ? ? ? ? + - - ? ? ? ? ? ? ? ? ? By putting x p = in the above, one can deduce that the sum of the series 2 2 2 1 1 1 1 , 3 5 7 + + + +K is (a) 2 4 p (b) 2 6 p (c) 2 8 p (d) 2 12 p 1.6 Which of the following improper integrals is (are) convergent? (a) 1 0 sin 1 cos x dx x - ? (b) 0 cos 1 x dx x 8 + ? ` (c) 2 0 1 x dx x 8 + ? (d) 1 5 0 2 1 cosx dx x - ? GATE EC - 1993 1.7 The function ( ) 2 , 3 2 , f x y x y xy y x = - + + has (a) no local extremum (b) one local minimum but no local maximum (c) one local maximum but no local minimum (d) one local minimum but one local maximum 2. In questions 2.1 to 2.10 below, each blank (_________) is to be suitably filled in. in the answer book write the question number and the answer only. Do not copy the question. Also no explanations for the answers are to be given. 2.1 ( ) ( ) ( ) 0 1 2 cos 1 lim 1 cos x x x e x x x ? - + - - is _________. 2.2 The radius of convergence of the power series ( ) ( ) 3 3 0 3 ! ! m m x m 8 ? is _________ 2.3 If the linear velocity V ur is given by 2 2 ˆ ˆ V x yi xyzj yz k = + - ur the angular velocity ? r at the point (1, 1, -1) is _________. 2.4 Given the differential equation, y x y = - with the initial condtion ( ) 0 0. y = the value of ( ) 0.1 y calculated numerically up to the third place of decimal by the second order Runge-Kutta method with step size h = 0.1 is _________ 2.5 For X = 4.0, the value of I in the FORTRAN 77 statement 3 2 **2 5.0 * *3 4 I X X = - + + is _________ 2.6 The value of the double integral 1 1 2 0 1 x x x dxdy y + ?? is _________ 2.7 If 1 0 0 1 0 1 0 1 0 0 0 0 0 A i i i ? ? ? ? - - ? ? = ? ? ? ? ? ? - ? ? The matrix 4 , A calculated by the use of Coyley – Mamilton theorem or otherwise, is _________ Page 3 GATE EC - 1993 Time : 3 hours PART I Maximum Marks : 200 SECTION - A 1. This questions 1.1 to 1.7 below one or more the alternatives are correct. Write the code letter(s), (A,B, C and D) corresponding to the correct alternatives in the answer book. Marks will be given only if all the correct alternatives have been selected and no incorrect alternative is picked up. 1.1 The eigen vector(s) of the matrix 0 0 0 0 0 0 , 0, 0 0 0 a ? ? ? ? ? ? ? ? ? ? ? is/are: (a) ( ) 0,0,a (b) ( ) ,0,0 a (c) ( ) 0,0,1 (d) ( ) 0, ,0 a 1.2 The differential equation, 2 2 sin 0, d y dy y dx dx + + = is: (a) linear (b) non-linear (c) homogeneous (d) of degree two 1.3 Simpson’s rule for integration gives exact result when ( ) f x is a polynomial of degree (a) 1 (b) 2 (c) 3 (d) 4 1.4 Which of the following is (are) valid FORTRAN 77 statement(s)? (a) DO 131 = 1 (b) A = DIM***7 (c) READ = 15.0 (d) GOTO 3 = 10 1.5 Fourier series of the periodic function (period 2p) defined by ( ) 0 0 0 x f x x x p p - < < ? = ? < < ? is 2 1 1 1 cos 1cos cos sin 4 n n nx n nx n p p p p 8 ? ? ? ? + - - ? ? ? ? ? ? ? ? ? By putting x p = in the above, one can deduce that the sum of the series 2 2 2 1 1 1 1 , 3 5 7 + + + +K is (a) 2 4 p (b) 2 6 p (c) 2 8 p (d) 2 12 p 1.6 Which of the following improper integrals is (are) convergent? (a) 1 0 sin 1 cos x dx x - ? (b) 0 cos 1 x dx x 8 + ? ` (c) 2 0 1 x dx x 8 + ? (d) 1 5 0 2 1 cosx dx x - ? GATE EC - 1993 1.7 The function ( ) 2 , 3 2 , f x y x y xy y x = - + + has (a) no local extremum (b) one local minimum but no local maximum (c) one local maximum but no local minimum (d) one local minimum but one local maximum 2. In questions 2.1 to 2.10 below, each blank (_________) is to be suitably filled in. in the answer book write the question number and the answer only. Do not copy the question. Also no explanations for the answers are to be given. 2.1 ( ) ( ) ( ) 0 1 2 cos 1 lim 1 cos x x x e x x x ? - + - - is _________. 2.2 The radius of convergence of the power series ( ) ( ) 3 3 0 3 ! ! m m x m 8 ? is _________ 2.3 If the linear velocity V ur is given by 2 2 ˆ ˆ V x yi xyzj yz k = + - ur the angular velocity ? r at the point (1, 1, -1) is _________. 2.4 Given the differential equation, y x y = - with the initial condtion ( ) 0 0. y = the value of ( ) 0.1 y calculated numerically up to the third place of decimal by the second order Runge-Kutta method with step size h = 0.1 is _________ 2.5 For X = 4.0, the value of I in the FORTRAN 77 statement 3 2 **2 5.0 * *3 4 I X X = - + + is _________ 2.6 The value of the double integral 1 1 2 0 1 x x x dxdy y + ?? is _________ 2.7 If 1 0 0 1 0 1 0 1 0 0 0 0 0 A i i i ? ? ? ? - - ? ? = ? ? ? ? ? ? - ? ? The matrix 4 , A calculated by the use of Coyley – Mamilton theorem or otherwise, is _________ GATE EC - 1993 2.8 Given, $ 2 2 2 cos sin x V x yi x e j z yk = + + $ and S the surface of a unit cube with one corner at the origin and edges parallel to the coordinate axes, the value of the integral $ . S V ndS ?? ur is _________ 2.9 The differential equation 0 y y + = is subjected to the boundary conditions ( ) ( ) 0 0 0 y y ? + = . In order that the equation has non-trivial solution (s), the genral value of ? is _________ 2.10 The Laplace transform of the periodic function ( ) f t described by the curve below i.e. ( ) ( ) ( ) sin if 2 1 2 1,2,3, 0 t n t n n f t p p ? - = = = ? = ? ? ? K Otherwise is _________ SECTION B: PHYSICS 3. In the following questions 3.1 to 3.17 there are some multiple choice questions and some questions where blanks are to be filled in. Answer ALL the questions. All multiple choice questions have ONE or MORE correct answers those suggested. Credit will be given only if every correct alternative(s), and no incorrect alternative, is selected. Write only the letters corresponding to the select alternatives in the answer book. In the fill in the blank type questions, write the answer only in the answer book. Useful data: 34 8 6.63 10 3 10 / sc h J m s - = × = × 3.1. Two particles of masses 1 2 and M M ( ) 1 2 M M > attract each other with a force inversely proportional to the square of the distance between them. The particles are initially at rest and then released. The centre of mass relative to a stationary observer (a) moves towards 1 M (b) moves towards 2 M (c) remains at rest (d) moves with a speed proportional to 1 2 M M f(t) 0 p 2p 3p 4p 5p 6p 7p 8p t Page 4 GATE EC - 1993 Time : 3 hours PART I Maximum Marks : 200 SECTION - A 1. This questions 1.1 to 1.7 below one or more the alternatives are correct. Write the code letter(s), (A,B, C and D) corresponding to the correct alternatives in the answer book. Marks will be given only if all the correct alternatives have been selected and no incorrect alternative is picked up. 1.1 The eigen vector(s) of the matrix 0 0 0 0 0 0 , 0, 0 0 0 a ? ? ? ? ? ? ? ? ? ? ? is/are: (a) ( ) 0,0,a (b) ( ) ,0,0 a (c) ( ) 0,0,1 (d) ( ) 0, ,0 a 1.2 The differential equation, 2 2 sin 0, d y dy y dx dx + + = is: (a) linear (b) non-linear (c) homogeneous (d) of degree two 1.3 Simpson’s rule for integration gives exact result when ( ) f x is a polynomial of degree (a) 1 (b) 2 (c) 3 (d) 4 1.4 Which of the following is (are) valid FORTRAN 77 statement(s)? (a) DO 131 = 1 (b) A = DIM***7 (c) READ = 15.0 (d) GOTO 3 = 10 1.5 Fourier series of the periodic function (period 2p) defined by ( ) 0 0 0 x f x x x p p - < < ? = ? < < ? is 2 1 1 1 cos 1cos cos sin 4 n n nx n nx n p p p p 8 ? ? ? ? + - - ? ? ? ? ? ? ? ? ? By putting x p = in the above, one can deduce that the sum of the series 2 2 2 1 1 1 1 , 3 5 7 + + + +K is (a) 2 4 p (b) 2 6 p (c) 2 8 p (d) 2 12 p 1.6 Which of the following improper integrals is (are) convergent? (a) 1 0 sin 1 cos x dx x - ? (b) 0 cos 1 x dx x 8 + ? ` (c) 2 0 1 x dx x 8 + ? (d) 1 5 0 2 1 cosx dx x - ? GATE EC - 1993 1.7 The function ( ) 2 , 3 2 , f x y x y xy y x = - + + has (a) no local extremum (b) one local minimum but no local maximum (c) one local maximum but no local minimum (d) one local minimum but one local maximum 2. In questions 2.1 to 2.10 below, each blank (_________) is to be suitably filled in. in the answer book write the question number and the answer only. Do not copy the question. Also no explanations for the answers are to be given. 2.1 ( ) ( ) ( ) 0 1 2 cos 1 lim 1 cos x x x e x x x ? - + - - is _________. 2.2 The radius of convergence of the power series ( ) ( ) 3 3 0 3 ! ! m m x m 8 ? is _________ 2.3 If the linear velocity V ur is given by 2 2 ˆ ˆ V x yi xyzj yz k = + - ur the angular velocity ? r at the point (1, 1, -1) is _________. 2.4 Given the differential equation, y x y = - with the initial condtion ( ) 0 0. y = the value of ( ) 0.1 y calculated numerically up to the third place of decimal by the second order Runge-Kutta method with step size h = 0.1 is _________ 2.5 For X = 4.0, the value of I in the FORTRAN 77 statement 3 2 **2 5.0 * *3 4 I X X = - + + is _________ 2.6 The value of the double integral 1 1 2 0 1 x x x dxdy y + ?? is _________ 2.7 If 1 0 0 1 0 1 0 1 0 0 0 0 0 A i i i ? ? ? ? - - ? ? = ? ? ? ? ? ? - ? ? The matrix 4 , A calculated by the use of Coyley – Mamilton theorem or otherwise, is _________ GATE EC - 1993 2.8 Given, $ 2 2 2 cos sin x V x yi x e j z yk = + + $ and S the surface of a unit cube with one corner at the origin and edges parallel to the coordinate axes, the value of the integral $ . S V ndS ?? ur is _________ 2.9 The differential equation 0 y y + = is subjected to the boundary conditions ( ) ( ) 0 0 0 y y ? + = . In order that the equation has non-trivial solution (s), the genral value of ? is _________ 2.10 The Laplace transform of the periodic function ( ) f t described by the curve below i.e. ( ) ( ) ( ) sin if 2 1 2 1,2,3, 0 t n t n n f t p p ? - = = = ? = ? ? ? K Otherwise is _________ SECTION B: PHYSICS 3. In the following questions 3.1 to 3.17 there are some multiple choice questions and some questions where blanks are to be filled in. Answer ALL the questions. All multiple choice questions have ONE or MORE correct answers those suggested. Credit will be given only if every correct alternative(s), and no incorrect alternative, is selected. Write only the letters corresponding to the select alternatives in the answer book. In the fill in the blank type questions, write the answer only in the answer book. Useful data: 34 8 6.63 10 3 10 / sc h J m s - = × = × 3.1. Two particles of masses 1 2 and M M ( ) 1 2 M M > attract each other with a force inversely proportional to the square of the distance between them. The particles are initially at rest and then released. The centre of mass relative to a stationary observer (a) moves towards 1 M (b) moves towards 2 M (c) remains at rest (d) moves with a speed proportional to 1 2 M M f(t) 0 p 2p 3p 4p 5p 6p 7p 8p t GATE EC - 1993 3.2. The temperature of an ideal gas is held constant while its volume is increased. The pressure exerted by the gas on the walls of the container decreases because its molecules (a) strike the walls with smaller force (b) strike the walls with lower velocities (c) strike the walls less frequently (d) collide with each other more frequently 3.3. Although a laser beam is highly directional, its beam width increase with propagation. This increase is due to (a) coherence (b) diffraction (c) polarization (d) interference 3.4. A plane electromagnetic wave of the form ( ) ( ) 14 1 6 1 0 cos2 5 10 sec 2.5 10 E yE t m x p - - ? ? = × - × ? ? ur Where 0 E is a constant and y is the unit vector along y-direction) represents a wave propagating along (a) + x direction (b) + y direction (c) - x direction (d) - y direction 3.5. While you are listening to a programme from a radio, if a near by electric light bulb is switched on or switched off, you hear a momentary noise in your radio. This is due to electromagnetic radiation emitted by _____. 3.6. Nuclear fusion reactions required very high temperatures so as to overcome (a) nuclear forces (b) van der waals forces (c) coulomb forces (d) gravitational forces 3.7. In radioactive decay, the disintegration rate of the nuclei is: (a) constant at all times (b) inversely proportional to half-life of the nuclei (c) inversely proportional to the number of nuclei at any time (d) directly proportional to the number of nuclei at any time 3.8. In an hydrogen atom 10.2 eV is given out as radiation when an electron is de- excited to the ground state. The principal quantum number of the excited state is ___________. Page 5 GATE EC - 1993 Time : 3 hours PART I Maximum Marks : 200 SECTION - A 1. This questions 1.1 to 1.7 below one or more the alternatives are correct. Write the code letter(s), (A,B, C and D) corresponding to the correct alternatives in the answer book. Marks will be given only if all the correct alternatives have been selected and no incorrect alternative is picked up. 1.1 The eigen vector(s) of the matrix 0 0 0 0 0 0 , 0, 0 0 0 a ? ? ? ? ? ? ? ? ? ? ? is/are: (a) ( ) 0,0,a (b) ( ) ,0,0 a (c) ( ) 0,0,1 (d) ( ) 0, ,0 a 1.2 The differential equation, 2 2 sin 0, d y dy y dx dx + + = is: (a) linear (b) non-linear (c) homogeneous (d) of degree two 1.3 Simpson’s rule for integration gives exact result when ( ) f x is a polynomial of degree (a) 1 (b) 2 (c) 3 (d) 4 1.4 Which of the following is (are) valid FORTRAN 77 statement(s)? (a) DO 131 = 1 (b) A = DIM***7 (c) READ = 15.0 (d) GOTO 3 = 10 1.5 Fourier series of the periodic function (period 2p) defined by ( ) 0 0 0 x f x x x p p - < < ? = ? < < ? is 2 1 1 1 cos 1cos cos sin 4 n n nx n nx n p p p p 8 ? ? ? ? + - - ? ? ? ? ? ? ? ? ? By putting x p = in the above, one can deduce that the sum of the series 2 2 2 1 1 1 1 , 3 5 7 + + + +K is (a) 2 4 p (b) 2 6 p (c) 2 8 p (d) 2 12 p 1.6 Which of the following improper integrals is (are) convergent? (a) 1 0 sin 1 cos x dx x - ? (b) 0 cos 1 x dx x 8 + ? ` (c) 2 0 1 x dx x 8 + ? (d) 1 5 0 2 1 cosx dx x - ? GATE EC - 1993 1.7 The function ( ) 2 , 3 2 , f x y x y xy y x = - + + has (a) no local extremum (b) one local minimum but no local maximum (c) one local maximum but no local minimum (d) one local minimum but one local maximum 2. In questions 2.1 to 2.10 below, each blank (_________) is to be suitably filled in. in the answer book write the question number and the answer only. Do not copy the question. Also no explanations for the answers are to be given. 2.1 ( ) ( ) ( ) 0 1 2 cos 1 lim 1 cos x x x e x x x ? - + - - is _________. 2.2 The radius of convergence of the power series ( ) ( ) 3 3 0 3 ! ! m m x m 8 ? is _________ 2.3 If the linear velocity V ur is given by 2 2 ˆ ˆ V x yi xyzj yz k = + - ur the angular velocity ? r at the point (1, 1, -1) is _________. 2.4 Given the differential equation, y x y = - with the initial condtion ( ) 0 0. y = the value of ( ) 0.1 y calculated numerically up to the third place of decimal by the second order Runge-Kutta method with step size h = 0.1 is _________ 2.5 For X = 4.0, the value of I in the FORTRAN 77 statement 3 2 **2 5.0 * *3 4 I X X = - + + is _________ 2.6 The value of the double integral 1 1 2 0 1 x x x dxdy y + ?? is _________ 2.7 If 1 0 0 1 0 1 0 1 0 0 0 0 0 A i i i ? ? ? ? - - ? ? = ? ? ? ? ? ? - ? ? The matrix 4 , A calculated by the use of Coyley – Mamilton theorem or otherwise, is _________ GATE EC - 1993 2.8 Given, $ 2 2 2 cos sin x V x yi x e j z yk = + + $ and S the surface of a unit cube with one corner at the origin and edges parallel to the coordinate axes, the value of the integral $ . S V ndS ?? ur is _________ 2.9 The differential equation 0 y y + = is subjected to the boundary conditions ( ) ( ) 0 0 0 y y ? + = . In order that the equation has non-trivial solution (s), the genral value of ? is _________ 2.10 The Laplace transform of the periodic function ( ) f t described by the curve below i.e. ( ) ( ) ( ) sin if 2 1 2 1,2,3, 0 t n t n n f t p p ? - = = = ? = ? ? ? K Otherwise is _________ SECTION B: PHYSICS 3. In the following questions 3.1 to 3.17 there are some multiple choice questions and some questions where blanks are to be filled in. Answer ALL the questions. All multiple choice questions have ONE or MORE correct answers those suggested. Credit will be given only if every correct alternative(s), and no incorrect alternative, is selected. Write only the letters corresponding to the select alternatives in the answer book. In the fill in the blank type questions, write the answer only in the answer book. Useful data: 34 8 6.63 10 3 10 / sc h J m s - = × = × 3.1. Two particles of masses 1 2 and M M ( ) 1 2 M M > attract each other with a force inversely proportional to the square of the distance between them. The particles are initially at rest and then released. The centre of mass relative to a stationary observer (a) moves towards 1 M (b) moves towards 2 M (c) remains at rest (d) moves with a speed proportional to 1 2 M M f(t) 0 p 2p 3p 4p 5p 6p 7p 8p t GATE EC - 1993 3.2. The temperature of an ideal gas is held constant while its volume is increased. The pressure exerted by the gas on the walls of the container decreases because its molecules (a) strike the walls with smaller force (b) strike the walls with lower velocities (c) strike the walls less frequently (d) collide with each other more frequently 3.3. Although a laser beam is highly directional, its beam width increase with propagation. This increase is due to (a) coherence (b) diffraction (c) polarization (d) interference 3.4. A plane electromagnetic wave of the form ( ) ( ) 14 1 6 1 0 cos2 5 10 sec 2.5 10 E yE t m x p - - ? ? = × - × ? ? ur Where 0 E is a constant and y is the unit vector along y-direction) represents a wave propagating along (a) + x direction (b) + y direction (c) - x direction (d) - y direction 3.5. While you are listening to a programme from a radio, if a near by electric light bulb is switched on or switched off, you hear a momentary noise in your radio. This is due to electromagnetic radiation emitted by _____. 3.6. Nuclear fusion reactions required very high temperatures so as to overcome (a) nuclear forces (b) van der waals forces (c) coulomb forces (d) gravitational forces 3.7. In radioactive decay, the disintegration rate of the nuclei is: (a) constant at all times (b) inversely proportional to half-life of the nuclei (c) inversely proportional to the number of nuclei at any time (d) directly proportional to the number of nuclei at any time 3.8. In an hydrogen atom 10.2 eV is given out as radiation when an electron is de- excited to the ground state. The principal quantum number of the excited state is ___________. GATE EC - 1993 3.9 Typical current voltage characteristic of a solar cell is given in the following figure by (a) curve A (b) curve B (c) curve C (d) curve D 3.10. Consider a solid sphere and a hollow sphere, both of mass M, radius R and initially at rest, which start rolling down the same inclined plane without slipping. At the bottom of the inclined plane, the ratio of speeds / solid hollow V V is: (a) 1 (b) 12 7 (c) 10 7 (d) 25 21 [Note: The moment of inertia about any diameter for a solid sphere is (2/5) MR 2 , and for a hollow sphere (2/3) MR 2 ]. 3.11. An optical fibre consists of a cylindrical dielectric rod of refractive index 1 n , surrounded by another dielectric of refractive index 2 n , where 2 1 , n n < as shown in the following figure. If a ray is incident from air at an angle i to the axis, then it undergoes total internal reflection at the interface AB if (a) 1 2 2 1 2 sin i n n - = - (b) 1 1 2 sin i n n - = - (c) 1 2 2 1 2 sin i n n - = - (d) 1 1 2 sin i n n - = - 3.12. For a uniformly charged sphere of radius R and charge density p, the ratio of magnitude of electric fields at distance 2 R and 2R from the centre, i.e., ( ) 2 2 R E r E r R ? ? = ? ? ? ? = is _______. Current Voltage A B C D Air A B n 2 n 1 iRead More

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