Page 1 EC? GATE Paper 2007 Q.1 – Q.20 Carry One Mark Each 1. If E denotes expectation, the variance of a random variable X is given by (A) 2 2 E X E X ? ? - ? ? ? ? ? ? (B) 2 2 E X E X ? ? + ? ? ? ? ? ? (C) 2 E X ? ? ? ? (D) 2 E X ? ? ? ? 2. The following plot shows a function y which varies linearly with x . The value of the integral 2 1 I y dx = ? is (A) 1.0 (B) 2.5 (C) 4.0 (D) 5.0 3. For 1, x << ( ) coth x can be approximated as (A) x (B) 2 x (C) 1 x (D) 2 1 x 4. ( ) 0 sin 2 lim ? ? ? ? is: (A) 0.5 (B) 1 (C) 2 (D) not defined 5. Which one of the following functions is strictly bounded? (A) 2 1 x (B) x e (C) 2 x (D) 2 x e - 6. For the function , x e - the linear approximation around 2 x = is: (A) ( ) 2 3 x e - - (B) 1 x - (C) ( ) 2 3 2 2 1 2 x e - ? ? + - + ? ? (D) 2 e - 7. An independent voltage source in series with an impedance s s s Z R jX = + delivers a maximum average power to a load impedance L Z when (A) L s s Z R jX = + (B) L s Z R = (C) L s Z jX = (D) L s s Z R jX = - 1 2 3 x 1 2 3 y -1 Page 2 EC? GATE Paper 2007 Q.1 – Q.20 Carry One Mark Each 1. If E denotes expectation, the variance of a random variable X is given by (A) 2 2 E X E X ? ? - ? ? ? ? ? ? (B) 2 2 E X E X ? ? + ? ? ? ? ? ? (C) 2 E X ? ? ? ? (D) 2 E X ? ? ? ? 2. The following plot shows a function y which varies linearly with x . The value of the integral 2 1 I y dx = ? is (A) 1.0 (B) 2.5 (C) 4.0 (D) 5.0 3. For 1, x << ( ) coth x can be approximated as (A) x (B) 2 x (C) 1 x (D) 2 1 x 4. ( ) 0 sin 2 lim ? ? ? ? is: (A) 0.5 (B) 1 (C) 2 (D) not defined 5. Which one of the following functions is strictly bounded? (A) 2 1 x (B) x e (C) 2 x (D) 2 x e - 6. For the function , x e - the linear approximation around 2 x = is: (A) ( ) 2 3 x e - - (B) 1 x - (C) ( ) 2 3 2 2 1 2 x e - ? ? + - + ? ? (D) 2 e - 7. An independent voltage source in series with an impedance s s s Z R jX = + delivers a maximum average power to a load impedance L Z when (A) L s s Z R jX = + (B) L s Z R = (C) L s Z jX = (D) L s s Z R jX = - 1 2 3 x 1 2 3 y -1 EC? GATE Paper 2007 8. The RC circuit shown in the figure is (A) a low-pass filter (B) a high-pass filter (C) a band-pass filter (D) a band-reject filter 9. The electron and hole concentrations in an intrinsic semiconductor are i n per 3 cm at 300 K. Now, if acceptor impurities are introduced with a concentration of A N per 3 cm ( ) where A i N n >> , the electron concentration per 3 cm at 300 K will be: (A) i n (B) i A n N + (C) A i N n - (D) 2 i A n N 10. In a p n + junction diode under reverse bias, the magnitude of electric field is maximum at (A) the edge of the depletion region on the p-side (B) the edge of the depletion region on the n-side (C) the p n + junction (D) the centre of the depletion region on the n-side 11. The correct full wave rectifier circuit is: (A) (B) R C R C i V o V + - + - Input Output Input Output Page 3 EC? GATE Paper 2007 Q.1 – Q.20 Carry One Mark Each 1. If E denotes expectation, the variance of a random variable X is given by (A) 2 2 E X E X ? ? - ? ? ? ? ? ? (B) 2 2 E X E X ? ? + ? ? ? ? ? ? (C) 2 E X ? ? ? ? (D) 2 E X ? ? ? ? 2. The following plot shows a function y which varies linearly with x . The value of the integral 2 1 I y dx = ? is (A) 1.0 (B) 2.5 (C) 4.0 (D) 5.0 3. For 1, x << ( ) coth x can be approximated as (A) x (B) 2 x (C) 1 x (D) 2 1 x 4. ( ) 0 sin 2 lim ? ? ? ? is: (A) 0.5 (B) 1 (C) 2 (D) not defined 5. Which one of the following functions is strictly bounded? (A) 2 1 x (B) x e (C) 2 x (D) 2 x e - 6. For the function , x e - the linear approximation around 2 x = is: (A) ( ) 2 3 x e - - (B) 1 x - (C) ( ) 2 3 2 2 1 2 x e - ? ? + - + ? ? (D) 2 e - 7. An independent voltage source in series with an impedance s s s Z R jX = + delivers a maximum average power to a load impedance L Z when (A) L s s Z R jX = + (B) L s Z R = (C) L s Z jX = (D) L s s Z R jX = - 1 2 3 x 1 2 3 y -1 EC? GATE Paper 2007 8. The RC circuit shown in the figure is (A) a low-pass filter (B) a high-pass filter (C) a band-pass filter (D) a band-reject filter 9. The electron and hole concentrations in an intrinsic semiconductor are i n per 3 cm at 300 K. Now, if acceptor impurities are introduced with a concentration of A N per 3 cm ( ) where A i N n >> , the electron concentration per 3 cm at 300 K will be: (A) i n (B) i A n N + (C) A i N n - (D) 2 i A n N 10. In a p n + junction diode under reverse bias, the magnitude of electric field is maximum at (A) the edge of the depletion region on the p-side (B) the edge of the depletion region on the n-side (C) the p n + junction (D) the centre of the depletion region on the n-side 11. The correct full wave rectifier circuit is: (A) (B) R C R C i V o V + - + - Input Output Input Output EC? GATE Paper 2007 (C) (D) 12. In a transconductance amplifier, it is desirable to have (A) a large input resistance and a large output resistance (B) a large input resistance and a small output resistance (C) a small input resistance and a large output resistance (D) a small input resistance and a small output resistance 13. X = 01110 and Y = 11001 are two 5-bit binary numbers represented in two’s complement format. The sum of X and Y represented in two’s complement format using 6 bits is: (A) 100111 (B) 001000 (C) 000111 (D) 101001 14. The Boolean function Y AB CD = + is to be realized using only 2-input NAND gates. The minimum number of gates required is: (A) 2 (B) 3 (C) 4 (D) 5 15. If the closed-loop transfer function of a control system is given as ( ) ( )( ) 5 , 2 3 s T s s s - = + + then it is (A) an unstable system (B) an uncontrollable system (C) a minimum phase system (D) a non-minimum phase system 16. If the Laplace transform of a signal ( ) y t is ( ) ( ) 1 , 1 Y s s s = - then its final value is: (A) -1 (B) 0 (C) 1 (D) unbounded 17. If ( ) R t is the autocorrelation function of a real, wide-sense stationary random process, then which of the following is NOT true? (A) ( ) ( ) R R t t = - (B) ( ) ( ) 0 R R t = (C) ( ) ( ) R R t t = - - (D) The mean square value of the process is ( ) 0 R Input Output Input Output Page 4 EC? GATE Paper 2007 Q.1 – Q.20 Carry One Mark Each 1. If E denotes expectation, the variance of a random variable X is given by (A) 2 2 E X E X ? ? - ? ? ? ? ? ? (B) 2 2 E X E X ? ? + ? ? ? ? ? ? (C) 2 E X ? ? ? ? (D) 2 E X ? ? ? ? 2. The following plot shows a function y which varies linearly with x . The value of the integral 2 1 I y dx = ? is (A) 1.0 (B) 2.5 (C) 4.0 (D) 5.0 3. For 1, x << ( ) coth x can be approximated as (A) x (B) 2 x (C) 1 x (D) 2 1 x 4. ( ) 0 sin 2 lim ? ? ? ? is: (A) 0.5 (B) 1 (C) 2 (D) not defined 5. Which one of the following functions is strictly bounded? (A) 2 1 x (B) x e (C) 2 x (D) 2 x e - 6. For the function , x e - the linear approximation around 2 x = is: (A) ( ) 2 3 x e - - (B) 1 x - (C) ( ) 2 3 2 2 1 2 x e - ? ? + - + ? ? (D) 2 e - 7. An independent voltage source in series with an impedance s s s Z R jX = + delivers a maximum average power to a load impedance L Z when (A) L s s Z R jX = + (B) L s Z R = (C) L s Z jX = (D) L s s Z R jX = - 1 2 3 x 1 2 3 y -1 EC? GATE Paper 2007 8. The RC circuit shown in the figure is (A) a low-pass filter (B) a high-pass filter (C) a band-pass filter (D) a band-reject filter 9. The electron and hole concentrations in an intrinsic semiconductor are i n per 3 cm at 300 K. Now, if acceptor impurities are introduced with a concentration of A N per 3 cm ( ) where A i N n >> , the electron concentration per 3 cm at 300 K will be: (A) i n (B) i A n N + (C) A i N n - (D) 2 i A n N 10. In a p n + junction diode under reverse bias, the magnitude of electric field is maximum at (A) the edge of the depletion region on the p-side (B) the edge of the depletion region on the n-side (C) the p n + junction (D) the centre of the depletion region on the n-side 11. The correct full wave rectifier circuit is: (A) (B) R C R C i V o V + - + - Input Output Input Output EC? GATE Paper 2007 (C) (D) 12. In a transconductance amplifier, it is desirable to have (A) a large input resistance and a large output resistance (B) a large input resistance and a small output resistance (C) a small input resistance and a large output resistance (D) a small input resistance and a small output resistance 13. X = 01110 and Y = 11001 are two 5-bit binary numbers represented in two’s complement format. The sum of X and Y represented in two’s complement format using 6 bits is: (A) 100111 (B) 001000 (C) 000111 (D) 101001 14. The Boolean function Y AB CD = + is to be realized using only 2-input NAND gates. The minimum number of gates required is: (A) 2 (B) 3 (C) 4 (D) 5 15. If the closed-loop transfer function of a control system is given as ( ) ( )( ) 5 , 2 3 s T s s s - = + + then it is (A) an unstable system (B) an uncontrollable system (C) a minimum phase system (D) a non-minimum phase system 16. If the Laplace transform of a signal ( ) y t is ( ) ( ) 1 , 1 Y s s s = - then its final value is: (A) -1 (B) 0 (C) 1 (D) unbounded 17. If ( ) R t is the autocorrelation function of a real, wide-sense stationary random process, then which of the following is NOT true? (A) ( ) ( ) R R t t = - (B) ( ) ( ) 0 R R t = (C) ( ) ( ) R R t t = - - (D) The mean square value of the process is ( ) 0 R Input Output Input Output EC? GATE Paper 2007 18. If ( ) S f is the power spectral density of a real, wide-sense stationary random process, then which of the following is ALWAYS true? (A) ( ) ( ) 0 S S f = (B) ( ) 0 S f = (C) ( ) ( ) S f S f - = - (D) ( ) 0 S f df 8 -8 = ? 19. A plane wave of wavelength ? is traveling in a direction making an angle 30° with positive x-axis and 90° with positive y-axis. The E u r field of the plane wave can be represented as ( ) is constant o E (A) µ 3 0 j t x z E yE e p p ? ? ? ? ? - - ? ? ? ? ? ? = u r (B) µ 3 0 j t x z E yE e p p ? ? ? ? ? - - ? ? ? ? ? ? = u r (C) µ 3 0 j t x z E yE e p p ? ? ? ? ? + + ? ? ? ? ? ? = u r (D) µ 3 0 j t x z E yE e p p ? ? ? ? ? - + ? ? ? ? ? ? = u r 20. If C is a closed curve enclosing a surface S, then the magnetic field intensity , H u r the current density J r and the electric flux density D u r are related by (A) . . s c D H ds J dl t ? ? ? = + ? ? ? ? ? ? ? ?? ? u r u r r r r Ñ (B) . . c s D H dl J d ds t ? ? ? = + ? ? ? ? ? ? ? ? ?? u r r u r r r Ò (C) . . s c D H ds J dl t ? ? ? = + ? ? ? ? ? ? ? ?? ? u r u r r r r Ò (D) . . c s D H dl J ds t ? ? ? = + ? ? ? ? ? ? ? ? ?? u r u r r r r Ñ Q.21 – Q.75 Carry Two Marks Each 21. It is given that 1 2 , ,.... M X x X are M non-zero, orthogonal vectors. The dimension of the vector space spanned by the 2M vectors 1 2 1 2 , ,... , , ,... M M X X X X X X - - - is: (A) 2M (B) M + 1 (C) M (D) dependent on the choice of 1 2 , ,... M X X X 22. Consider the function ( ) 2 2. f x x x = - - The maximum value of ( ) f x in the closed interval 4,4 - ? ? ? ? is: (A) 18 (B) 10 (C) -2.25 (D) indeterminate 23. An examination consists of two papers, Paper 1 and Paper 2. The probability of failing in Paper 1 is 0.3 and that in Paper 2 is 0.2. Given that a student has failed in Paper 2, the probability of failing in Paper 1 is 0.6. The probability of a student failing in both the papers is: (A) 0.5 (B) 0.18 (C) 0.12 (D) 0.06 Page 5 EC? GATE Paper 2007 Q.1 – Q.20 Carry One Mark Each 1. If E denotes expectation, the variance of a random variable X is given by (A) 2 2 E X E X ? ? - ? ? ? ? ? ? (B) 2 2 E X E X ? ? + ? ? ? ? ? ? (C) 2 E X ? ? ? ? (D) 2 E X ? ? ? ? 2. The following plot shows a function y which varies linearly with x . The value of the integral 2 1 I y dx = ? is (A) 1.0 (B) 2.5 (C) 4.0 (D) 5.0 3. For 1, x << ( ) coth x can be approximated as (A) x (B) 2 x (C) 1 x (D) 2 1 x 4. ( ) 0 sin 2 lim ? ? ? ? is: (A) 0.5 (B) 1 (C) 2 (D) not defined 5. Which one of the following functions is strictly bounded? (A) 2 1 x (B) x e (C) 2 x (D) 2 x e - 6. For the function , x e - the linear approximation around 2 x = is: (A) ( ) 2 3 x e - - (B) 1 x - (C) ( ) 2 3 2 2 1 2 x e - ? ? + - + ? ? (D) 2 e - 7. An independent voltage source in series with an impedance s s s Z R jX = + delivers a maximum average power to a load impedance L Z when (A) L s s Z R jX = + (B) L s Z R = (C) L s Z jX = (D) L s s Z R jX = - 1 2 3 x 1 2 3 y -1 EC? GATE Paper 2007 8. The RC circuit shown in the figure is (A) a low-pass filter (B) a high-pass filter (C) a band-pass filter (D) a band-reject filter 9. The electron and hole concentrations in an intrinsic semiconductor are i n per 3 cm at 300 K. Now, if acceptor impurities are introduced with a concentration of A N per 3 cm ( ) where A i N n >> , the electron concentration per 3 cm at 300 K will be: (A) i n (B) i A n N + (C) A i N n - (D) 2 i A n N 10. In a p n + junction diode under reverse bias, the magnitude of electric field is maximum at (A) the edge of the depletion region on the p-side (B) the edge of the depletion region on the n-side (C) the p n + junction (D) the centre of the depletion region on the n-side 11. The correct full wave rectifier circuit is: (A) (B) R C R C i V o V + - + - Input Output Input Output EC? GATE Paper 2007 (C) (D) 12. In a transconductance amplifier, it is desirable to have (A) a large input resistance and a large output resistance (B) a large input resistance and a small output resistance (C) a small input resistance and a large output resistance (D) a small input resistance and a small output resistance 13. X = 01110 and Y = 11001 are two 5-bit binary numbers represented in two’s complement format. The sum of X and Y represented in two’s complement format using 6 bits is: (A) 100111 (B) 001000 (C) 000111 (D) 101001 14. The Boolean function Y AB CD = + is to be realized using only 2-input NAND gates. The minimum number of gates required is: (A) 2 (B) 3 (C) 4 (D) 5 15. If the closed-loop transfer function of a control system is given as ( ) ( )( ) 5 , 2 3 s T s s s - = + + then it is (A) an unstable system (B) an uncontrollable system (C) a minimum phase system (D) a non-minimum phase system 16. If the Laplace transform of a signal ( ) y t is ( ) ( ) 1 , 1 Y s s s = - then its final value is: (A) -1 (B) 0 (C) 1 (D) unbounded 17. If ( ) R t is the autocorrelation function of a real, wide-sense stationary random process, then which of the following is NOT true? (A) ( ) ( ) R R t t = - (B) ( ) ( ) 0 R R t = (C) ( ) ( ) R R t t = - - (D) The mean square value of the process is ( ) 0 R Input Output Input Output EC? GATE Paper 2007 18. If ( ) S f is the power spectral density of a real, wide-sense stationary random process, then which of the following is ALWAYS true? (A) ( ) ( ) 0 S S f = (B) ( ) 0 S f = (C) ( ) ( ) S f S f - = - (D) ( ) 0 S f df 8 -8 = ? 19. A plane wave of wavelength ? is traveling in a direction making an angle 30° with positive x-axis and 90° with positive y-axis. The E u r field of the plane wave can be represented as ( ) is constant o E (A) µ 3 0 j t x z E yE e p p ? ? ? ? ? - - ? ? ? ? ? ? = u r (B) µ 3 0 j t x z E yE e p p ? ? ? ? ? - - ? ? ? ? ? ? = u r (C) µ 3 0 j t x z E yE e p p ? ? ? ? ? + + ? ? ? ? ? ? = u r (D) µ 3 0 j t x z E yE e p p ? ? ? ? ? - + ? ? ? ? ? ? = u r 20. If C is a closed curve enclosing a surface S, then the magnetic field intensity , H u r the current density J r and the electric flux density D u r are related by (A) . . s c D H ds J dl t ? ? ? = + ? ? ? ? ? ? ? ?? ? u r u r r r r Ñ (B) . . c s D H dl J d ds t ? ? ? = + ? ? ? ? ? ? ? ? ?? u r r u r r r Ò (C) . . s c D H ds J dl t ? ? ? = + ? ? ? ? ? ? ? ?? ? u r u r r r r Ò (D) . . c s D H dl J ds t ? ? ? = + ? ? ? ? ? ? ? ? ?? u r u r r r r Ñ Q.21 – Q.75 Carry Two Marks Each 21. It is given that 1 2 , ,.... M X x X are M non-zero, orthogonal vectors. The dimension of the vector space spanned by the 2M vectors 1 2 1 2 , ,... , , ,... M M X X X X X X - - - is: (A) 2M (B) M + 1 (C) M (D) dependent on the choice of 1 2 , ,... M X X X 22. Consider the function ( ) 2 2. f x x x = - - The maximum value of ( ) f x in the closed interval 4,4 - ? ? ? ? is: (A) 18 (B) 10 (C) -2.25 (D) indeterminate 23. An examination consists of two papers, Paper 1 and Paper 2. The probability of failing in Paper 1 is 0.3 and that in Paper 2 is 0.2. Given that a student has failed in Paper 2, the probability of failing in Paper 1 is 0.6. The probability of a student failing in both the papers is: (A) 0.5 (B) 0.18 (C) 0.12 (D) 0.06 EC? GATE Paper 2007 24. The solution of the differential equation 2 2 2 2 d y k y y dx = - under the boundary conditions (i) 1 y y = at 0 x = and (ii) 2 y y = at x = 8 , where , k 1 y and 2 y are constants, is (A) ( ) ( ) 2 1 2 2 exp y y y x k y = - - + (B) ( ) ( ) 2 1 1 exp y y y x k y = - - + (C) ( ) ( ) 1 2 1 sinh y y y x k y = - + (D) ( ) ( ) 1 2 2 exp y y y x k y = - - + 25. The equation 3 2 4 4 0 x x x - + - = is to be solved using the Newton-Raphson method. If 2 x = is taken as the initial approximation of the solution, then the next approximation using this method will be: (A) 2 3 (B) 4 3 (C) 1 (D) 3 2 26. Three functions ( ) ( ) ( ) 1 2 3 , and , f t f t f t which are zero outside the interval [0,T], are shown in the figure. Which of the following statements is correct? (A) ( ) ( ) 1 2 and f t f t are orthogonal (B) ( ) ( ) 1 3 and f t f t are orthogonal (C) ( ) ( ) 2 3 and f t f t are orthogonal (D) ( ) ( ) 1 2 and f t f t are orthonormal 27. If the semi-circular contour D of radius 2 is as shown in the figure, then the value of the integral ( ) 2 1 1 D ds s - ? Ñ is: (A) jp (B) -jp (C) -p (D) p t t t 2 -2 1 2 -1 1 -3 ( ) 1 f t ( ) 2 f t ( ) 3 f t 0 3 T 2 3 T T j? j2 D 2 0 0 s -j2Read More

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