Page 1 |IN-GATE-2013 PAPER| 1 Q. No. 1 – 25 Carry One Mark Each 1. The dimension of the null space of the matrix 0 1 1 1 1 0 1 0 1 ? ? ? ? - ? ? ? ? - - ? ? is (A) 0 (B) 1 (C) 2 (D) 3 Answer: (B) Exp: 0 1 1 A 1 1 0 Rank A 2 1 0 1 ? ? ? ? = - ? = ? ? ? ? - - ? ? Dimension of the null space of A=3-2=1 ? 2. If the A- matrix of the state space model of a SISO linear time invariant system is rank deficient, the transfer function of the system must have (A) a pole with positive real part (B) a pole with negative real part (C) a pole with positive imaginary part (D) a pole at the origin Answer: (D) 3. Two systems with impulse responses ( ) 1 h t and ( ) 2 h t are connected in cascade. Then the overall impulse response of the cascaded system is given by (A) a product of ( ) 1 h t and ( ) 2 h t (B) sum of ( ) 1 h t and ( ) 2 h t (C) convolution of ( ) 1 h t and ( ) 2 h t (D) subtraction of ( ) 2 h t from ( ) 1 h t Answer: (C) Exp: 4. The complex function tanh (s) is analytic over a region of the imaginary axis of the complex s-plane if the following is TRUE everywhere in the region for all integers n (A) ( ) Re s 0 = (B) ( ) Im s n ? p (C) ( ) n Im s 3 p ? (D) ( ) ( ) 2n 1 Im s 2 + p ? Answer: (D) Exp: Tanhs s s s s e e isanalytic e e - - - = + s s 2s m if e e 0 (2n 1) e 1 s i 2 (2n 1) I (s) 2 - + ? + ? ? - ? ? p + ? ? p x(t) y(t) 1 2 h(t) *h(t) x(t) y(t) 1 h (t) 2 h (t) Page 2 |IN-GATE-2013 PAPER| 1 Q. No. 1 – 25 Carry One Mark Each 1. The dimension of the null space of the matrix 0 1 1 1 1 0 1 0 1 ? ? ? ? - ? ? ? ? - - ? ? is (A) 0 (B) 1 (C) 2 (D) 3 Answer: (B) Exp: 0 1 1 A 1 1 0 Rank A 2 1 0 1 ? ? ? ? = - ? = ? ? ? ? - - ? ? Dimension of the null space of A=3-2=1 ? 2. If the A- matrix of the state space model of a SISO linear time invariant system is rank deficient, the transfer function of the system must have (A) a pole with positive real part (B) a pole with negative real part (C) a pole with positive imaginary part (D) a pole at the origin Answer: (D) 3. Two systems with impulse responses ( ) 1 h t and ( ) 2 h t are connected in cascade. Then the overall impulse response of the cascaded system is given by (A) a product of ( ) 1 h t and ( ) 2 h t (B) sum of ( ) 1 h t and ( ) 2 h t (C) convolution of ( ) 1 h t and ( ) 2 h t (D) subtraction of ( ) 2 h t from ( ) 1 h t Answer: (C) Exp: 4. The complex function tanh (s) is analytic over a region of the imaginary axis of the complex s-plane if the following is TRUE everywhere in the region for all integers n (A) ( ) Re s 0 = (B) ( ) Im s n ? p (C) ( ) n Im s 3 p ? (D) ( ) ( ) 2n 1 Im s 2 + p ? Answer: (D) Exp: Tanhs s s s s e e isanalytic e e - - - = + s s 2s m if e e 0 (2n 1) e 1 s i 2 (2n 1) I (s) 2 - + ? + ? ? - ? ? p + ? ? p x(t) y(t) 1 2 h(t) *h(t) x(t) y(t) 1 h (t) 2 h (t) |IN-GATE-2013 PAPER| 2 5. For a vector E, which one of the following statements is NOT TRUE? (A) If E=0, E is called solenoidal ?· (B) If E=0, E is called conservative ?× (C) If E=0, E is called irrotational ?× (D) If E=0, E is called irrotational ?· Answer: (D) Exp: .E 0, E ? = is called irrotational, which is not true 6. For a periodic signal ( ) v t 30sin100t 10cos300t 6sin 500t 4 p ? ? = + + + ? ? ? ? , the fundamental frequency in rad/s is (A) 100 (B) 300 (C) 500 (D) 1500 Answer: (A) Exp: o 100 rad / sec fundamental ? = o o 3 300 rad / sec third harmonic 5 500 rad / sec fifth harmonic ? = ? = 7. In the transistor circuit as shown below, the value of resistance E R in kO is approximately, (A) 1.0 (B) 1.5 (C) 2.0 (D) 2.5 Answer (A) Exp: ( ) B 6k V 10 2.8V 6 15 k = × = + E E E E V 2.8 0.7 2.1V V 2.1 R 1k I 2mA ? = - = ? = = = O 8. A source ( ) s v t V cos100 t = p has an internal impedance of ( ) 4 j3 + O . If a purely resistive load connected to this source has to extract the maximum power out of the source, its value in O should be (A) 3 (B) 4 (C) 5 (D) 7 0.1 F µ 15kO 1.5kO 10V c I 2mA = 0.1 F µ CE V 5V = 6kO E R out v Page 3 |IN-GATE-2013 PAPER| 1 Q. No. 1 – 25 Carry One Mark Each 1. The dimension of the null space of the matrix 0 1 1 1 1 0 1 0 1 ? ? ? ? - ? ? ? ? - - ? ? is (A) 0 (B) 1 (C) 2 (D) 3 Answer: (B) Exp: 0 1 1 A 1 1 0 Rank A 2 1 0 1 ? ? ? ? = - ? = ? ? ? ? - - ? ? Dimension of the null space of A=3-2=1 ? 2. If the A- matrix of the state space model of a SISO linear time invariant system is rank deficient, the transfer function of the system must have (A) a pole with positive real part (B) a pole with negative real part (C) a pole with positive imaginary part (D) a pole at the origin Answer: (D) 3. Two systems with impulse responses ( ) 1 h t and ( ) 2 h t are connected in cascade. Then the overall impulse response of the cascaded system is given by (A) a product of ( ) 1 h t and ( ) 2 h t (B) sum of ( ) 1 h t and ( ) 2 h t (C) convolution of ( ) 1 h t and ( ) 2 h t (D) subtraction of ( ) 2 h t from ( ) 1 h t Answer: (C) Exp: 4. The complex function tanh (s) is analytic over a region of the imaginary axis of the complex s-plane if the following is TRUE everywhere in the region for all integers n (A) ( ) Re s 0 = (B) ( ) Im s n ? p (C) ( ) n Im s 3 p ? (D) ( ) ( ) 2n 1 Im s 2 + p ? Answer: (D) Exp: Tanhs s s s s e e isanalytic e e - - - = + s s 2s m if e e 0 (2n 1) e 1 s i 2 (2n 1) I (s) 2 - + ? + ? ? - ? ? p + ? ? p x(t) y(t) 1 2 h(t) *h(t) x(t) y(t) 1 h (t) 2 h (t) |IN-GATE-2013 PAPER| 2 5. For a vector E, which one of the following statements is NOT TRUE? (A) If E=0, E is called solenoidal ?· (B) If E=0, E is called conservative ?× (C) If E=0, E is called irrotational ?× (D) If E=0, E is called irrotational ?· Answer: (D) Exp: .E 0, E ? = is called irrotational, which is not true 6. For a periodic signal ( ) v t 30sin100t 10cos300t 6sin 500t 4 p ? ? = + + + ? ? ? ? , the fundamental frequency in rad/s is (A) 100 (B) 300 (C) 500 (D) 1500 Answer: (A) Exp: o 100 rad / sec fundamental ? = o o 3 300 rad / sec third harmonic 5 500 rad / sec fifth harmonic ? = ? = 7. In the transistor circuit as shown below, the value of resistance E R in kO is approximately, (A) 1.0 (B) 1.5 (C) 2.0 (D) 2.5 Answer (A) Exp: ( ) B 6k V 10 2.8V 6 15 k = × = + E E E E V 2.8 0.7 2.1V V 2.1 R 1k I 2mA ? = - = ? = = = O 8. A source ( ) s v t V cos100 t = p has an internal impedance of ( ) 4 j3 + O . If a purely resistive load connected to this source has to extract the maximum power out of the source, its value in O should be (A) 3 (B) 4 (C) 5 (D) 7 0.1 F µ 15kO 1.5kO 10V c I 2mA = 0.1 F µ CE V 5V = 6kO E R out v |IN-GATE-2013 PAPER| 3 Answer: (C) Exp: For maximum power Transfer L s 2 2 R Z 4 3 5 = = + = O 9. Which of the following statements is NOT TRUE for a continuous time causal and stable LTI system? (A) All the poles of the system must lie on the left side of the j?-axis (B) Zeroes of the system can lie anywhere in the s-plane (C) All the poles must lie within s 1 = (D) All the roots of the characteristic equation must be located on the left side of the j?-axis. Answer: (C) Exp: For an LTI system to be stable and causal all poles or roots of characteristic equation must lie on LHS of s-plane i.e., left hand side of j axis ?- [Refer Laplace transform]. 10. The operational amplifier shown in the circuit below has a slew rate of 0.8V / s µ . The input signal is 0.25sin t ? . The maximum frequency of input in kHz for which there is no distortion in the output is (A) 23.84 (B) 25.0 (C) 50.0 (D) 46.60 Answer: (A) Exp: Max 0 i 6 opk SR 0.8 470 f ;where V V 5.34 sin t 2 V 22 2 5.34 10 23.84kHz - = = = = - ? p p× × = 11. Assuming zero initial condition, the response ( ) y t of the system given below to a unit step input ( ) u t is (A) ( ) u t (B) ( ) t u t (C) ( ) 2 t u t 2 (D) ( ) t e u t - ( ) Y s 1 s ( ) U s + - 0.25sin t ? 22kO 470kO o V Page 4 |IN-GATE-2013 PAPER| 1 Q. No. 1 – 25 Carry One Mark Each 1. The dimension of the null space of the matrix 0 1 1 1 1 0 1 0 1 ? ? ? ? - ? ? ? ? - - ? ? is (A) 0 (B) 1 (C) 2 (D) 3 Answer: (B) Exp: 0 1 1 A 1 1 0 Rank A 2 1 0 1 ? ? ? ? = - ? = ? ? ? ? - - ? ? Dimension of the null space of A=3-2=1 ? 2. If the A- matrix of the state space model of a SISO linear time invariant system is rank deficient, the transfer function of the system must have (A) a pole with positive real part (B) a pole with negative real part (C) a pole with positive imaginary part (D) a pole at the origin Answer: (D) 3. Two systems with impulse responses ( ) 1 h t and ( ) 2 h t are connected in cascade. Then the overall impulse response of the cascaded system is given by (A) a product of ( ) 1 h t and ( ) 2 h t (B) sum of ( ) 1 h t and ( ) 2 h t (C) convolution of ( ) 1 h t and ( ) 2 h t (D) subtraction of ( ) 2 h t from ( ) 1 h t Answer: (C) Exp: 4. The complex function tanh (s) is analytic over a region of the imaginary axis of the complex s-plane if the following is TRUE everywhere in the region for all integers n (A) ( ) Re s 0 = (B) ( ) Im s n ? p (C) ( ) n Im s 3 p ? (D) ( ) ( ) 2n 1 Im s 2 + p ? Answer: (D) Exp: Tanhs s s s s e e isanalytic e e - - - = + s s 2s m if e e 0 (2n 1) e 1 s i 2 (2n 1) I (s) 2 - + ? + ? ? - ? ? p + ? ? p x(t) y(t) 1 2 h(t) *h(t) x(t) y(t) 1 h (t) 2 h (t) |IN-GATE-2013 PAPER| 2 5. For a vector E, which one of the following statements is NOT TRUE? (A) If E=0, E is called solenoidal ?· (B) If E=0, E is called conservative ?× (C) If E=0, E is called irrotational ?× (D) If E=0, E is called irrotational ?· Answer: (D) Exp: .E 0, E ? = is called irrotational, which is not true 6. For a periodic signal ( ) v t 30sin100t 10cos300t 6sin 500t 4 p ? ? = + + + ? ? ? ? , the fundamental frequency in rad/s is (A) 100 (B) 300 (C) 500 (D) 1500 Answer: (A) Exp: o 100 rad / sec fundamental ? = o o 3 300 rad / sec third harmonic 5 500 rad / sec fifth harmonic ? = ? = 7. In the transistor circuit as shown below, the value of resistance E R in kO is approximately, (A) 1.0 (B) 1.5 (C) 2.0 (D) 2.5 Answer (A) Exp: ( ) B 6k V 10 2.8V 6 15 k = × = + E E E E V 2.8 0.7 2.1V V 2.1 R 1k I 2mA ? = - = ? = = = O 8. A source ( ) s v t V cos100 t = p has an internal impedance of ( ) 4 j3 + O . If a purely resistive load connected to this source has to extract the maximum power out of the source, its value in O should be (A) 3 (B) 4 (C) 5 (D) 7 0.1 F µ 15kO 1.5kO 10V c I 2mA = 0.1 F µ CE V 5V = 6kO E R out v |IN-GATE-2013 PAPER| 3 Answer: (C) Exp: For maximum power Transfer L s 2 2 R Z 4 3 5 = = + = O 9. Which of the following statements is NOT TRUE for a continuous time causal and stable LTI system? (A) All the poles of the system must lie on the left side of the j?-axis (B) Zeroes of the system can lie anywhere in the s-plane (C) All the poles must lie within s 1 = (D) All the roots of the characteristic equation must be located on the left side of the j?-axis. Answer: (C) Exp: For an LTI system to be stable and causal all poles or roots of characteristic equation must lie on LHS of s-plane i.e., left hand side of j axis ?- [Refer Laplace transform]. 10. The operational amplifier shown in the circuit below has a slew rate of 0.8V / s µ . The input signal is 0.25sin t ? . The maximum frequency of input in kHz for which there is no distortion in the output is (A) 23.84 (B) 25.0 (C) 50.0 (D) 46.60 Answer: (A) Exp: Max 0 i 6 opk SR 0.8 470 f ;where V V 5.34 sin t 2 V 22 2 5.34 10 23.84kHz - = = = = - ? p p× × = 11. Assuming zero initial condition, the response ( ) y t of the system given below to a unit step input ( ) u t is (A) ( ) u t (B) ( ) t u t (C) ( ) 2 t u t 2 (D) ( ) t e u t - ( ) Y s 1 s ( ) U s + - 0.25sin t ? 22kO 470kO o V |IN-GATE-2013 PAPER| 4 Answer: (B) Exp: Integration of unit step function is ramp output Writing in time domain y(t) u(t)*u(t) tu(t) = = 12. The transfer function ( ) ( ) 2 1 V s V s of the circuit shown below is (A) 0.5s 1 s 1 + + (B) 3s 6 s 2 + + (C) s 2 s 1 + + (D) s 1 s 2 + + Answer: (D) Exp: 1 2 4 4 q 2 1 1 q 1 s 1 z (s) , z (s) 10 s 10 s Z (s) V (s) s 1 V (s) Z (s) Z (s) s 2 - - + = = + = = + + 13. The type of partial differential equation 2 2 f f t x ? ? = ? ? is (A) Parabolic (B) Elliptic (C) Hyperbolic (D) Nonlinear Answer: (A) Exp: 2 2 2 f f HereB 4AC 0 t x The equation is parabolic ? ? = - = ? ? ? 14. The discrete-time transfer function 1 1 1 2z 1 0.5z - - - - is (A) Non-minimum phase and unstable (B) Minimum phase and unstable (C) Minimum phase and stable (D) Non-minimum phase and stable + - + - ( ) 2 V s ( ) 1 V s 10kO 100 F µ 100 F µ u(s) y(s) 1 s u(t) y(t) u(t) 1 V (s) + + - - 1 Z (s) 2 Z(s) 2 V (s) Page 5 |IN-GATE-2013 PAPER| 1 Q. No. 1 – 25 Carry One Mark Each 1. The dimension of the null space of the matrix 0 1 1 1 1 0 1 0 1 ? ? ? ? - ? ? ? ? - - ? ? is (A) 0 (B) 1 (C) 2 (D) 3 Answer: (B) Exp: 0 1 1 A 1 1 0 Rank A 2 1 0 1 ? ? ? ? = - ? = ? ? ? ? - - ? ? Dimension of the null space of A=3-2=1 ? 2. If the A- matrix of the state space model of a SISO linear time invariant system is rank deficient, the transfer function of the system must have (A) a pole with positive real part (B) a pole with negative real part (C) a pole with positive imaginary part (D) a pole at the origin Answer: (D) 3. Two systems with impulse responses ( ) 1 h t and ( ) 2 h t are connected in cascade. Then the overall impulse response of the cascaded system is given by (A) a product of ( ) 1 h t and ( ) 2 h t (B) sum of ( ) 1 h t and ( ) 2 h t (C) convolution of ( ) 1 h t and ( ) 2 h t (D) subtraction of ( ) 2 h t from ( ) 1 h t Answer: (C) Exp: 4. The complex function tanh (s) is analytic over a region of the imaginary axis of the complex s-plane if the following is TRUE everywhere in the region for all integers n (A) ( ) Re s 0 = (B) ( ) Im s n ? p (C) ( ) n Im s 3 p ? (D) ( ) ( ) 2n 1 Im s 2 + p ? Answer: (D) Exp: Tanhs s s s s e e isanalytic e e - - - = + s s 2s m if e e 0 (2n 1) e 1 s i 2 (2n 1) I (s) 2 - + ? + ? ? - ? ? p + ? ? p x(t) y(t) 1 2 h(t) *h(t) x(t) y(t) 1 h (t) 2 h (t) |IN-GATE-2013 PAPER| 2 5. For a vector E, which one of the following statements is NOT TRUE? (A) If E=0, E is called solenoidal ?· (B) If E=0, E is called conservative ?× (C) If E=0, E is called irrotational ?× (D) If E=0, E is called irrotational ?· Answer: (D) Exp: .E 0, E ? = is called irrotational, which is not true 6. For a periodic signal ( ) v t 30sin100t 10cos300t 6sin 500t 4 p ? ? = + + + ? ? ? ? , the fundamental frequency in rad/s is (A) 100 (B) 300 (C) 500 (D) 1500 Answer: (A) Exp: o 100 rad / sec fundamental ? = o o 3 300 rad / sec third harmonic 5 500 rad / sec fifth harmonic ? = ? = 7. In the transistor circuit as shown below, the value of resistance E R in kO is approximately, (A) 1.0 (B) 1.5 (C) 2.0 (D) 2.5 Answer (A) Exp: ( ) B 6k V 10 2.8V 6 15 k = × = + E E E E V 2.8 0.7 2.1V V 2.1 R 1k I 2mA ? = - = ? = = = O 8. A source ( ) s v t V cos100 t = p has an internal impedance of ( ) 4 j3 + O . If a purely resistive load connected to this source has to extract the maximum power out of the source, its value in O should be (A) 3 (B) 4 (C) 5 (D) 7 0.1 F µ 15kO 1.5kO 10V c I 2mA = 0.1 F µ CE V 5V = 6kO E R out v |IN-GATE-2013 PAPER| 3 Answer: (C) Exp: For maximum power Transfer L s 2 2 R Z 4 3 5 = = + = O 9. Which of the following statements is NOT TRUE for a continuous time causal and stable LTI system? (A) All the poles of the system must lie on the left side of the j?-axis (B) Zeroes of the system can lie anywhere in the s-plane (C) All the poles must lie within s 1 = (D) All the roots of the characteristic equation must be located on the left side of the j?-axis. Answer: (C) Exp: For an LTI system to be stable and causal all poles or roots of characteristic equation must lie on LHS of s-plane i.e., left hand side of j axis ?- [Refer Laplace transform]. 10. The operational amplifier shown in the circuit below has a slew rate of 0.8V / s µ . The input signal is 0.25sin t ? . The maximum frequency of input in kHz for which there is no distortion in the output is (A) 23.84 (B) 25.0 (C) 50.0 (D) 46.60 Answer: (A) Exp: Max 0 i 6 opk SR 0.8 470 f ;where V V 5.34 sin t 2 V 22 2 5.34 10 23.84kHz - = = = = - ? p p× × = 11. Assuming zero initial condition, the response ( ) y t of the system given below to a unit step input ( ) u t is (A) ( ) u t (B) ( ) t u t (C) ( ) 2 t u t 2 (D) ( ) t e u t - ( ) Y s 1 s ( ) U s + - 0.25sin t ? 22kO 470kO o V |IN-GATE-2013 PAPER| 4 Answer: (B) Exp: Integration of unit step function is ramp output Writing in time domain y(t) u(t)*u(t) tu(t) = = 12. The transfer function ( ) ( ) 2 1 V s V s of the circuit shown below is (A) 0.5s 1 s 1 + + (B) 3s 6 s 2 + + (C) s 2 s 1 + + (D) s 1 s 2 + + Answer: (D) Exp: 1 2 4 4 q 2 1 1 q 1 s 1 z (s) , z (s) 10 s 10 s Z (s) V (s) s 1 V (s) Z (s) Z (s) s 2 - - + = = + = = + + 13. The type of partial differential equation 2 2 f f t x ? ? = ? ? is (A) Parabolic (B) Elliptic (C) Hyperbolic (D) Nonlinear Answer: (A) Exp: 2 2 2 f f HereB 4AC 0 t x The equation is parabolic ? ? = - = ? ? ? 14. The discrete-time transfer function 1 1 1 2z 1 0.5z - - - - is (A) Non-minimum phase and unstable (B) Minimum phase and unstable (C) Minimum phase and stable (D) Non-minimum phase and stable + - + - ( ) 2 V s ( ) 1 V s 10kO 100 F µ 100 F µ u(s) y(s) 1 s u(t) y(t) u(t) 1 V (s) + + - - 1 Z (s) 2 Z(s) 2 V (s) |IN-GATE-2013 PAPER| 5 Answer: (D) Exp: 1 1 1 2z H(z) 1 0.5z - - - = - For minimum phase system, all poles and zeros must lie inside the unit circle. For stable system, all poles must be inside the unit circle For the given system, Zero is at 2, pole is at 0.5 This system is stable but non-minimum phase 15. Match the following biomedical instrumentation techniques with their application. P. Otoscopy U. Respiratory volume measurement Q. Ultrasound Technique V. Ear diagnostics R. Spirometry W. Echo-cardiography S. Thermodilution Technique X. Heart-volume measurement (A) P-U;Q-V;R-X;S-W (B) P-V;Q-U;R-X;S-W (C) P-V;Q-W;R-U;S-X (D) P-V;Q-W;R-X;S-U Answer: (C) 16. A continuous random variable X has a probability density function ( ) x f x e , 0<x< - = 8 , then ( ) P X 1 > is (A) 0.368 (B) 0.5 (C) 0.632 (D) 1.0 Answer: (A) Exp: x x 1 1 1 p(x 1) e dx e e 0.368 8 8 - - - ? ? > = = - = = ? ? ? 17. A band limited signal with a maximum frequency of 5 kHz is to be sampled. According to the sampling theorem, the sampling frequency in kHz which is not valid is (A) 5 (B) 12 (C) 15 (D) 20 Answer: (A) Exp: Given: m f 5kHz = According to sampling frequency s m s f 2f ; f 10 kHz = = So, only in option (a) it is less than 10KHz ie., (5KHz) 18. A differential pressure transmitter of a flow meter using venture tube reads 5 2.5 10 Pa × for a flow rate of 3 0.5 m / s . The approximate flow rate in 3 m / s for a differential pressure of 5 0.9 10 Pa × is (A) 0.30 (B) 0.18 (C) 0.83 (D) 0.60 Answer: (A) Exp: 2 2 1 1 2 Q p Q p Q p Q 0.30 a ? = =Read More

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