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1 IN-GATE-03 PAPE Q. No. 5 Carry One Mark Each 0. The dimension of the null space of the matrix 0 is 0 (A) 0 (B) (C) (D) 3 0 A = 0 ank A = 0 Dimension of the null space of A=3-=. 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 h ( t ) and h t are connected in cascade. Then the overall impulse response of the cascaded system is given by (A) a product of h ( t ) and h ( t ) (B) sum of h ( t ) and h ( t ) (C) convolution of h ( t ) and h ( t ) (D) subtraction of h ( t ) from h ( t ) x(t) h (t) h (t) y(t) x(t) h(t) *h(t) y(t) 4. The complex function tanh (s) is analytic over a region of the imaginary axis of the complex s-plane if the following is TUE everywhere in the region for all integers n (A) e ( s) 0 nπ 3 Answer: (D) s s e e Tanhs = s s isanalytic e e = (B) Im ( s) (C) Im ( s) (D) Im ( s) s s if e e 0 (n ) (n ) I m(s) π s e s i π nπ ( n ) π GATEFOUM- India s No. institute for GATE training

2 IN-GATE-03 PAPE 5. For a vector E, which one of the following statements is NOT TUE? (A) If E=0, E is called solenoidal Answer: (B) If (C) If (D) If E=0, E is called conservative E=0, E is called irrotational E=0, E is called irrotational (D).E = 0, Eis called irrotational, which is not true π 6. For a periodic signal v ( t) = 30sin00t 0cos300t 6sin 500t 4 fundamental frequency in rad/s is (A) 00 (B) 300 (C) 500 (D) 500 Answer: (A) ω o = 00 rad / sec fundamental 3ω = 300 rad / sec third harmonic o 5ω = 500 rad / sec fifth harmonic o 7. In the transistor circuit as shown below, the value of resistance E in kω is approximately, 0 5kΩ.5kΩ Ic = ma 0.µ F 6kΩ CE = 5 0.µ F v out E (A).0 (B).5 (C).0 (D).5 Answer (A) 6k B = 0 = k E = =.. E E = = = kω IE ma 8. A source vs ( t) = cos00π t has an internal impedance of ( 4 j3) Ω. If a purely resistive load connected to this source has to extract the maximum power out of the source, its value in Ω should be (A) 3 (B) 4 (C) 5 (D) 7 GATEFOUM- India s No. institute for GATE training

3 IN-GATE-03 PAPE For maximum power Transfer L = Z s = 4 3 = 5Ω 9. Which of the following statements is NOT TUE for a continuous time causal and stable LTI system? (A) All the poles of the system must lie on the left side of the jω-axis (B) Zeroes of the system can lie anywhere in the s-plane (C) All the poles must lie within s = (D) All the roots of the characteristic equation must be located on the left side of the jω-axis. 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 jω axis [efer Laplace transform]. 0. The operational amplifier shown in the circuit below has a slew rate of 0.8 / µ s. The input signal is 0.5sin ω t. The maximum frequency of input in khz for which there is no distortion in the output is 470kΩ 0.5sin ωt kω o (A) 3.84 (B) 5.0 (C) 50.0 (D) Answer: (A) S f = = ;where = = 5.34 sin ωt π π Max 6 0 i opk = 3.84kHz. Assuming zero initial condition, the response y ( t ) of the system given below to a unit step input u ( t ) is U( s) s Y ( s) (A) u ( t ) (B) t u ( t ) (C) u ( t) t t (D) e u ( t) GATEFOUM- India s No. institute for GATE training 3

4 IN-GATE-03 PAPE Integration of unit step function is ramp output u(s) s y(s) Writing in time domain u(t) u(t) y(t) y(t) = u(t)*u(t) = tu(t). The transfer function s s of the circuit shown below is 00 µ F ( s) 0kΩ 00 µ F ( s) (A) 0.5s (B) 3s 6 s s Answer: (D) Z (s) (C) s s (D) s s (s) Z(s) (s) s z (s) =, z 4 (s) = 4 0 s 0 s Z (s) q(s) s = = (s) Z (s) Z (s) s q f 3. The type of partial differential equation = t (A) Parabolic (B) Elliptic (C) Hyperbolic (D) Nonlinear Answer: (A) f t f = Here B 4AC = 0 x The equation is parabolic 4. The discrete-time transfer function z 0.5z (A) Non-minimum phase and unstable (B) Minimum phase and unstable (C) Minimum phase and stable (D) Non-minimum phase and stable f x is is GATEFOUM- India s No. institute for GATE training 4

5 IN-GATE-03 PAPE Answer: (D) z H(z) = 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, pole is at 0.5 This system is stable but non-minimum phase 5. Match the following biomedical instrumentation techniques with their application. P. Otoscopy U. espiratory volume measurement Q. Ultrasound Technique. Ear diagnostics. Spirometry W. Echo-cardiography S. Thermodilution Technique X. Heart-volume measurement (A) P-U;Q-;-X;S-W (B) P-;Q-U;-X;S-W (C) P-;Q-W;-U;S-X (D) P-;Q-W;-X;S-U 6. A continuous random variable X has a probability density function f x e, 0<x< P X > is x =, then (A) (B) 0.5 (C) 0.63 (D).0 Answer: (A) x x p(x > ) = e dx e e = = = 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 Answer: (A) (A) 5 (B) (C) 5 (D) 0 Given: f m = 5kHz According to sampling frequency fs f m; fs 0 khz So, only in option (a) it is less than 0KHz ie., (5KHz) 8. A differential pressure transmitter of a flow meter using venture tube reads Answer: Pa for a flow rate of a differential pressure of m / s. The approximate flow rate in Pa is (A) 0.30 (B) 0.8 (C) 0.83 (D) 0.60 (A) Q Qα p = Q p Q = 0.30 p 3 m / s for GATEFOUM- India s No. institute for GATE training 5

6 IN-GATE-03 PAPE 9. A bulb in a staircase has two switches, one switch being at the ground floor and the other one at the first floor. The bulb can be turned ON and also can be turned OFF by any one of the switches irrespective of the state of the other switch. The logic of switching of the bulb resembles (A) an AND gate (B) an O gate (C) an XO gate (D) a NAND gate Let Switches = p, p p, p Z(o / p) OFF OFF OFF OFF ON ON ON OFF ON ON ON OFF From Truth Table, it can be verified that Ex-O logic is implemented. 0. The impulse response of a system is h ( t) = tu ( t). For an input u ( t ) output is t (A) u ( t ) (B) ( ) t t u t U(t) t u (t) t u(t) LTI system t u t, the t u t ( ) (C) ( ) (D) ( ) For LTI system, if input gets delayed by one unit, output will also get delayed by one unit. (t ) u(t ) u(t ). Consider a delta connection of resistors and its equivalent star connection as shown. If all elements of the delta connection are scaled by a factor k, k>0, the elements of the corresponding star equivalent will be scaled by a factor of a C B b c A (A) C k (B) k (C) /k (D) k a b = a b c a C B b c A GATEFOUM- India s No. institute for GATE training 6

7 IN-GATE-03 PAPE B A a c = a b c bc = a b c Above expression shown that if a, b & c is scaled by k, A, B & C is scaled by k only.. An accelerometer has input range of 0-0g, natural frequency 30Hz and mass 0.00kg. The range of the secondary displacement transducer in mm required to cover the input range is Answer: (A) 0 to.76 (B) 0 to 9.8 (C) 0 to.0 (D) 0 to 5.0 a w x (A) = x = a ( πf ) for a = 0 log x 0 to In the circuit shown below what is the output voltage ( out ) in olts if a silicon transistor Q and an ideal op-amp are used? kω 5 Q 5 5 out (A) -5 (B) -0.7 (C) 0.7 (D) 5 5 kω v Q out out = 0.7v GATEFOUM- India s No. institute for GATE training 7

8 IN-GATE-03 PAPE 4. In the feedback network shown below, if the feedback factor k is increased, then the v in v A 0 v f = kv out k Answer: (A) (A) input impedance increases and output impedance decreases (B) input impedance increases and output impedance also increases (C) input impedance decreases and output impedance also decreases (D) input impedance decreases and output impedance increases In voltage-voltage feedback A 0 in AMP AMPo out k = ( A K) in AMP 0 out as K in AMPo = A K 0, out 5. The Bode plot of a transfer function G(s) is shown in the figure below. Gain ( db) ω rad/s The gain ( 0 log G ( s ) ) is 3dB and -8dB at rad/s and 0 rad/s respectively. The phase is negative for all ω. Then G ( s ) is GATEFOUM- India s No. institute for GATE training 8

9 IN-GATE-03 PAPE (A) 39.8 s 39.8 (B) s (C) 3 s 3 (D) s Any two paints on same line segment of Bode plot satisfies the equation of straight line. 0log G ( jω ) w ( rad ) i.e, G G log ω log ω For the initial straight line G G = log ω log ω ω 0 3 = 40log N = slope of the line segment. 40 db dec ω = = k ;Where N is type of system here initial slope is = k k = k = 39.8 G s 39.8 = s Hence w db 40 Hence N = dec Q. No Carry One Mark Each dy xy 0, y 0 dx = = using Euler's predictor corrector (improved Euler Cauchy) method with a step size of 0., the value of y after the first step is 6. While numerically solving the differential equation Answer: (A).00 (B).03 (C) 0.97 (D) 0.96 (D) dy dx = = = = xy,x0 0, y0, h 0. GATEFOUM- India s No. institute for GATE training 9

10 IN-GATE-03 PAPE f(x, y) ; y = y h.f(x, y ) = (0.)f(0,) = p c p and y = yo f(x 0,y 0) f(x,y ) = (0.) f 0, f(0.,) = 0.96, is the value of y after first step, using Euler s predictor corrector method. 7. One pair of eigen vectors corresponding to the two eigen values of the matrix 0 is 0 (A) j, j (B) 0, 0 Answer: (D) λ = j, j are eigen values j x 0 (A ji)x = 0 = j y 0 j Eigen vector for j is j x 0 and (A ji)x = 0 = j y 0 Eigen vector for j is j (C), 0 j (D), j j 8. The digital circuit shown below uses two negative edge triggered D flip flops. Assuming initial condition of Q and Q0 as zero, the output QQ0 of this circuit is D Q D0 Q0 D Flip.Flop D Flip.Flop Q Q0 (A) 00,0,0,,00... clock (B) 00,0,,0,00... (C) 00,,0,0,00... (D) 00,,,,00... Q Q0 Qnext Q0next The counter is 00, 0,, 0, 00,. D D = Q 0 = Q 0 GATEFOUM- India s No. institute for GATE training 0

11 IN-GATE-03 PAPE 9. Considering the transformer to be ideal, the transmission parameter 'A' of the port network shown in the figure below is Ω : Ω I 5Ω 5Ω I ' ' Answer: (A) (A).3 (B).4 (C) 0.5 (D) The following arrangement consists of an ideal transformer and an attenuator, which attenuates by a factor of 0.8. An ac voltage WX = 00 is applied across WX to get an open circuit voltage YZ across YZ. Next, an ac voltage YZ is applied across YZ to get an open circuit voltage YZ / WX, WX / YZ are respectively = 00 WX across WX. Then, W :.5 Y (A) 5 / 00 and 80 / 00 (B) 00 /00 and 80 / 00 (C) 00 /00 and 00 / 00 (D) 80 / 00 and 80 /00 W For a transform X = N N :.5 N :N =.5 X Y Z The potentiometer gives an attenuation factor of 0.8 over υ Z Hence yz yz = υ = υ yz = wx YZ YZ = WX = WX GATEFOUM- India s No. institute for GATE training

12 IN-GATE-03 PAPE wx 00 Since potentiometer and transformer are bilateral elements. Hence = The open loop transfer function of a dc motor is given as a yz ω s 0 =. When s 0s connected in feedback as shown below, the approximate value of K a that will reduce the time constant of the closed loop system by one hundred times as compared to that of the open loop system is ( s ) ± K a a s 0 0s ω ( s) (A) (B) 5 (C) 0 (D) 00 τ openloop= τ closedloop= = Ka = k 0k 0Ka CLTF = = 0K 0 a a s ( 0k a) 0s 0k a = 0 3. Two magnetically uncoupled inductive coils have Q factor q and q at the chosen operating frequency. Their respective resistances are and. When "connected in series, the effective Q factor of the series combination at the same operating frequency is (A) q q (B) ( / q ) ( / q ) (C) ( q q ) / ( ) (D) ( q q ) / ( ) Q Factor of a inductive coil. wl wl wl Q = Q = & Q = L When such two coils are connected in series individual inductances and resistances are added. Hence, Leq = L L GATEFOUM- India s No. institute for GATE training

13 IN-GATE-03 PAPE Hence eq = Q eq ωl ωl ωl ω ( L L ) = = = eq ( ) Q Q Q Q = = eq 33. For the circuit shown below, the knee current of the ideal Zener diode is 0 ma. To maintain 5 across L, the minimum value of the load resistor L in Ω and the minimum power rating of the Zener diode in mw, respectively, are 00Ω 0 I Load z = 5 L (A) 5 and 5 (B) 5 and 50 (C) 50 and 5 (D) 50 and 50 Exp Lmin 5 = I L max I00 = = = 50 ma I = I I = 40 ma LMax 00 knee 5 Lmin = 000 = 5Ω 40 Minimum power rating of Zener should = 50 ma x 5 = 50 mw 34. The impulse response of a continuous time system is given by h ( t) = δ ( t ) δ ( t 3). The value of the step response at t = is (A) 0 (B) (C) (D) 3 GATEFOUM- India s No. institute for GATE training 3

14 IN-GATE-03 PAPE h ( t) = δ ( t ) δ ( t ) u ( t ) h ( t ) y ( t ) = ( ) ( ) = ( ) = y t u t u t 3 y u u 35. Signals from fifteen thermocouples are multiplexed and each one is sampled once per second with a6 bit ADC. The digital samples are converted by a parallel to serial converter to generate a serial PCM signal. This PCM signal is frequency modulated with FSK modulator with 00 Hz as and 960 Hz as 0. The minimum band allocation required for faithful reproduction of the signal by the FSK receiver without considering noise is (A) 840Hz to 30Hz (C) 080Hz to 30Hz (B) 960Hz to 00Hz (D) 70Hz to 440Hz Answer: (A) Data rate from ADC is 6 x 5 bits/second=40 bits/second The bandwidth required for transmitting 40 bits/second = 0 Hz. (Half of bit rate). Now when each pulse representing s and 0 s get FSK modulator with 960 Hz and 00 Base band signal before modulation The spectrum of signal after modulation Thus all frequency from 840 to 30 is required for allocation 36. Three capacitors C,C and C 3 whose values are 0 µ F,5µ F and µ F respectively, have breakdown voltages of 0, 5, and respectively. For the interconnection shown below, the maximum safe voltage in olts that can be applied across the combination, and the corresponding total charge in µ C stored in the effective capacitance across the terminals are, respectively (A).8 and 36 (B) 7 and 9 (C).8 and 3 (D) 7 and 80 C C 3 C GATEFOUM- India s No. institute for GATE training 4

15 IN-GATE-03 PAPE c () c c 7 3 c 5...() c c (3) From (), 7.5 olts From (),.8 olts From (3), 0 olts To operate Circuit safe, should be minimum of those = ceff = c (c C 3) = 0µ F µ F = µ F 7 7 Q = c.8 = 3µ c eff 37. The maximum value of the solution y(t) of the differential equation y t y t = 0 y 0 = and y 0 =, for t 0 is Answer: with initial conditions (A) (B) (C) π (D).. y(t) y(t) = 0 Taking Laplace on both sides, we get s s y(s) = s y(s) = y(t) = cos t sint s s = max imum value of y(t) is The Laplace Transform representation of the triangular pulse shown below is (A) (B) e s s s s e e s x ( t) (C) (D) s s e e s s s e e s Answer: (D) From the graph we have x(t) = t for 0 < t < = t for < t < st st s s 0 s L x(t) = e tdt e ( t)dt = e e t GATEFOUM- India s No. institute for GATE training 5

16 IN-GATE-03 PAPE 39. In the circuit shown below, if the source voltage s o = olts, then the Thevenin's equivalent voltage in olts as seen by the load resistance L is ~ S 3 Ω j4 Ω j6ω 5 Ω I L j40i 0 L I L = 0 Ω (A) TH L L o (B) = 0 o (C) 4 C L = = Tan j4 TH 3 j4 5 = = o (D) o A signal t ( t) = 0 0sin00π t 0sin4000π t 0sin00000π t is supplied to a filter circuit (shown below) made up of ideal op amps. The least attenuated frequency component in the output will be 0.µ F kω µ F kω 0.µ F 750Ω 0.µ F ( t) ( t) (A) 0 Hz (B) 50 Hz (C) khz (D) 50 khz Answer: (A) Least attenuated signal frequency is 0Hz 4. The signal flow graph for a system is given below. The transfer function this system is given as Y s U s for GATEFOUM- India s No. institute for GATE training 6

17 IN-GATE-03 PAPE U ( s ) s s Y ( s ) 4 s (A) 5s 6s s (B) s 6s Answer: (A) By using Mason s gain formula s (C) s 4s y(s) s s s s s = = = u(s) 5 6s 5 5s 6s s 4s s 4 0 (D) 5s 6s 4. A voltage 000 sin ωt olts is applied across YZ. Assuming ideal diodes, the voltage measured across WX in olts, is kω W X (A) sin t Answer: (D). ω kω (B) ( ω ω ) (C) ( sin ωt sin ω t ) / (D) 0 for all t yz Z sin t sin t / 0 ω t During the half cycle All Diodes OFF & Hence = = 0 During e half cycle = 0. GATEFOUM- India s No. institute for GATE training 7

18 IN-GATE-03 PAPE 43. In the circuit shown below the op amps are ideal. Then out in volts is kω kω 5 5 kω 5 5 out kω kω (A) 4 (B) 6 (C) 8 (D) 0 kω kω out =? kω out out = = 4 = = 8 kω kω 44. In the circuit shown below, Q has negligible collector to emitter saturation voltage and the diode drops negligible voltage across if under forward bias. If cc is 5, X and Y are digital signals with 0 as logic 0 and cc as logic, then the Boolean expression for Z is cc Z X Q Diode (A) XY (B) XY (C) XY (D) XY (B) Y GATEFOUM- India s No. institute for GATE training 8

19 IN-GATE-03 PAPE X Y Z The circuit below incorporates a permanent magnet moving coil milli ammeter of range ma having a series resistance of 0kΩ. Assuming constant diode forward resistance of 50 Ω a forward diode drop of 0.7 and infinite reverse diode resistance for each diode, the reading of the meter in ma is ma 0kΩ 5, 50Hz 0kΩ 0 (A) 0.45 (B) 0.5 (C) 0.7 (D) 0.9 Answer: (A) 46. Measurement of optical absorption of a solution is disturbed by the additional stray light falling at the photo-detector. For estimation of the error caused by stray light the following data could be obtained from controlled experiments. Photo-detector output without solution and without stray light is 500 µ W Photo-detector output without solution and with stray light is 600 µ W Photo-detector output with solution and with stray light is 00 µ W The percent error in computing absorption coefficient due to stray light is (A).50 (B) 3.66 (C) (D) Two ammeters A and A measure the same current and provide readings I and I respectively. The ammeter errors can be characterized as independent zero mean Gaussian random variables of standard deviations σ and σ respectively. The value of the current is computed as: I I I is Answer: = µ µ. The value of (A) σ σ σ (C) (B) σ σ σ which gives the lowest standard deviation of I (C) σ σ σ σ (D) σ σ GATEFOUM- India s No. institute for GATE training 9

20 IN-GATE-03 PAPE Common Data for Questions: 48 & 49 A tungsten wire used in a constant current hot wire anemometer has the following properties: esistance at 0ºC is 0 Ω, Surface area is of resistance of the tungsten wire is 4 0 m. Linear temperature coefficient /º C, Convective heat transfer coefficient is 5.W/ m /ºC, flowing air temperature is 30ºC, wire current is 00mA, mass specific heat product is J/º C 48. The thermal time constant of the hot wire under flowing air condition in ms is (A) 4.5 (B).5 (C) 6.5 (D) At steady state, the resistance of the wire in ohms is (A) (B) 0.44 (C).5 (D) 4.8 Common Data for Questions: 50 & 5 A piezo electric force sensor, connected by a cable to a voltage amplifier has the following parameters: Crystal properties: Stiffness 5 0 rad/s, Force-to-Charge sensitivity angle assumed negligible. Cable properties : capacitance 6 0 N / m, Damping ratio 0.0, natural frequency 9 0 F 9 0 C / N, Capacitance 9 0 F with its loss with its resistance assumed negligable Amplifier properties: Input impedance M Ω, Bandwidth MHz, Gain The maximum frequency of a force signal in Hz below the natural frequency within its useful midband range of measurement, for which the gain amplitude is less than.05 approximately is: Answer: (A) 35 (B) 350 (C) 3500 (D) 6 0 (D) 3 5. The minimum frequency of a force signal in Hz within its useful mid-band range of measurement for which the gain amplitude is more than 0.95 approximately is: Answer: (A) 6 (B) 60 (C) 600 (D) 6 0 (B) 3 Linked Answer Questions: Q.5 to Q.55 Carry Two Marks Each Statement for Linked Answer Questions: 5 & 53 Consider a plant with transfer function G ( s) = ( s ) 3. Let K u and Tu be the ultimate gain and ultimate period corresponding to the frequency response based closed loop Ziegler-Nichols cycling method respectively. The Ziegler-Nichols tuning rule for a P-controller is given as: K = 0.5K u GATEFOUM- India s No. institute for GATE training 0

21 IN-GATE-03 PAPE 5. The values of K u and T u respectively are: Answer: (A) and π (B) 8 and π (C) 8 and π / 3 (D) and π / 3 (C) 53. The gain of the transfer function between the plant output and an additive load π disturbance input of frequency in closed loop with a P-controller designed T according to the Ziegler Nichols tuning rule as given above is (A) -.0 (B) 0.5 (C).0 (D).0 u Statement for Linked Answer Questions: 54 & 55 A differential amplifier with signal terminals X, Y, Z is connected as shown in figure(a) below for CM measurement where the differential amplifier has an additional constant offset voltage in the output. The observations obtained are: =, = 3m and when = 3, = 4m when i o i o 5 X Y Z Strain Gage X Y Differential Amplifier Z Figure : a Figure :b 54. Assuming its differential gain to be 0 and the op-amp to be otherwise ideal, the CM is Answer: (A) 0 (B) (C) i 0 = CM CM = (0) = 0 m (C) 4 0 (D) The differential amplifier is connected as shown in the figure(b) above to a single strain gage bridge. Let the strain gage resistance vary around its no load resistance by ± %. Assume the input impedance of the amplifier to be high compared to the equivalent source resistance of the bridge, and the common mode characteristic to be as obtained above. The output voltage in m varies approximately from GATEFOUM- India s No. institute for GATE training

22 IN-GATE-03 PAPE (A) 8 to -8 (B) 8 to - (C) to - (D) 99 to -0 Q. No Carry One Mark Each 56. Statement: You can always give me a ring whenever you need. Which one of the following is the best inference from the above statement? (A) Because I have a nice caller tune (B) Because I have a better telephone facility (C) Because a friend in need in a friend indeed (D) Because you need not pay towards the telephone bills when you give me a ring 57. Complete the sentence: Dare mistakes. (A) commit (B) to commit (C) committed (D) committing 58. Choose the grammatically COECT sentence: (A) Two and two add four (B) Two and two become four (C) Two and two are four (D) Two and two make four Answer: (D) 59. They were requested not to quarrel with others. Which one of the following options is the closest in meaning to the word quarrel? (A) make out (B) call out (C) dig out (D) fall out Answer: (D) 60. In the summer of 0, in New Delhi, the mean temperature of Monday to Wednesday was 4ºC and of Tuesday to Thursday was 43ºC. If the temperature on Thursday was 5% higher than that of Monday, then the temperature in ºC on Thursday was (A) 40 (B) 43 (C) 46 (D) 49 - Let the temperature of Monday be T M Sum of temperatures of Tuesday and Wednesday = T and Temperature of Thursday =T Th GATEFOUM- India s No. institute for GATE training

23 IN-GATE-03 PAPE Now, T T = 4 3 = 3 th m & T T = 43 3 = 9 T T = 6, Also T =.5T Th m Th m 0.5T = 6 T = 40 m m = = O Temperature of thursday C Q. No Carry Two Marks Each 6. Find the sum to n terms of the series (A) Answer: (D) n ( ) (B) n ( ) 9 9 -Using the answer options, substitute n =. The sum should add up to 94 Alternative Solution: 8 n ( ) 9 9 (C) 8 The given series is n terms n ( ) 9 9 n (D) ( 9 ) ( 9 3) ( 9 5) ( 9 7 )...nterms 3 ( nterms ) ( nterms) n n 9 ( 9 a ) ( r ) = = = n s n = (r > ) and 9 r Sum of first n oddnaturalnumbers is n 6. The set of values of p for which the roots of the equation are of opposite sign is (A) (,0) (B) ( 0, ) (C) (, ) (D) ( 0, ) n 3x x p p = 0 Since the roots are of opposite sign, the product of roots will be negative. ( ) p p < 0 p ( p ) < 0 ( p 0) ( p ) < 0 0 < p < 3 Thus therequiredset of valuesis 0, 63. A car travels 8 km in the first quarter of an hour, 6 km in the second quarter and 6km in the third quarter. The average speed of the car in km per hour over the entire journey is Exp (A) 30 (B) 36 (C) 40 (D) 4 :-Average speed = Total dis tance Total time = = 40km / hr GATEFOUM- India s No. institute for GATE training 3

24 IN-GATE-03 PAPE 64. What is the chance that a leap year, selected at random, will contain 53 Sundays? Answer: (A) (A) /7 (B) 3/7 (C) /7 (D) 5/7 - There are 5 complete weeks in a calendar year Number of days in a leap year = = 364days Probability of 53 Saturdays = Statement: There were different streams of freedom movements in colonial India carried out by the moderates, liberals, radicals, socialists, and so on. Answer: (D) Which one of the following is the best inference from the above statement? (A) The emergence of nationalism in colonial India led to our Independence (B) Nationalism in India emerged in the context of colonialism (C) Nationalism in India is homogeneous (D) Nationalism in India is heterogeneous GATEFOUM- India s No. institute for GATE training 4