EC-GATE-2015 PAPER-01 General Aptitude Q. No. 1 5 Carry One Mark Each

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1 EC-GATE-5 PAPE- General Aptitude Q. No. 5 Carry One Mark Each. Choose the word most similar in meaning to the given word: Educe (A) Exert (B) Educate (C) Extract (D) Extend (C). If log x (5/7) = -/, then the value of x is (A) 4/5 (B) 5/4 (C) -5/49 (D) -49/5 (A) x x x a b a b. Operators, and aredefined by : a b ;a b ;a b ab. a b a b Find the value (A) - (B) - (C) (D) (C) (666)(66 6) Choose the most appropriate word from the options given below to complete the following sentence. The principal presented the chief guest with a, as token of appreciation. (A) momento (B) memento (C) momentum (D) moment (B) 5. Choose the appropriate word/phrase, out of the four options given below, to complete the following sentence: Frogs. (A) Croak (B) oar (C) Hiss (D) Patter (A) Frogs make croak sound. written permission.

2 EC-GATE-5 PAPE- Q. No. 6 Carry Two Marks Each 6. A cube of side units is formed using a set of smaller cubes of side unit. Find the proportion of the number of faces of the smaller cubes visible to those which are NOT visible. (A) : 4 (B) : (C) : (D) : (C) Number of faces per cube = 6 Total number of cubes = 9 = 7 Total number of faces = 7 6 = 6 Total number of non visible faces = 6-54 = 8 Numberof visible faces 54 Numberof non visible faces 8 7. Fill in the missing value Middle number is the average of the numbers on both sides. Average of 6 and 4 is 5 Average of (7+4) and (+) is 7 written permission.

3 EC-GATE-5 PAPE- Average of (+9+) and (++) is 8 Average of (4+) and (+) is 5 Therefore, Average of () and () is 8. Humpty Dumpty sits on a wall every day while having lunch. The wall sometimes breaks. A person sitting on the wall falls if the wall breaks. Which one of the statements below is logically valid and can be inferred from the above sentences? (A) Humpty Dumpty always falls while having lunch (B) Humpty Dumpty does not fall sometimes while having lunch (C) Humpty Dumpty never falls during dinner (D) When Humpty Dumpty does not sit on the wall, the wall does not break (B) 9. The following question presents a sentence, part of which is underlined. Beneath the sentence you find four ways of phrasing the underline part. Following the requirements of the standard written English, select the answer that produces the most effective sentence. Tuberculosis, together with its effects, ranks one of the leading causes of death in India. (A) ranks as one of the leading causes of death (B) rank as one of the leading causes of death (C) has the rank of one of the leading causes of death (D) are one of the leading causes of death (A). ead the following paragraph and choose the correct statement. Climate change has reduced human security and threatened human well being. An ignored reality of human progress is that human security largely depends upon environmental security. But on the contrary, human progress seems contradictory to environmental security. To keep up both at the required level is a challenge to be addressed by one and all. One of the ways to curb the climate change may be suitable scientific innovations, while the other may be the Gandhian perspective on small scale progress with focus on sustainability. (A) Human progress and security are positively associated with environmental security. (B) Human progress is contradictory to environmental security. (C) Human security is contradictory to environmental security. (D) Human progress depends upon environmental security. (B) written permission.

4 EC-GATE-5 PAPE- Electronics and Communication Engineering Q. No. 5 Carry One Mark Each. A region of negative differential resistance is observed in the current voltage characteristics of a silicon PN junction if (A) Both the P-region and the N-region are heavily doped (B) The N-region is heavily doped compared to the P-region (C) The P-region is heavily doped compared to the N-region (D) An intrinsic silicon region is inserted between the P-region and the N-region (A). A silicon sample is uniformly doped with donor type impurities with a concentration of 6 / cm. The electron and hole mobilities in the sample are cm / V s and 4cm / V s The charge of an electron is..5 P N q N D n cm respectively. Assume complete ionization of impurities. The resistivity of the sample in cm 9.6 C.. A unity negative feedback system has the open-loop transfer function Gs K s s s The value of the gain K (>) at which the root locus crosses the imaginary axis is. k C.Eis s s s s 4s s k By using outh Table, s row should be zero. For poles to be on imaginary axis k ; k should be Suppose A and B are two independent events with probabilities P A and P B. Let A and B be their complements. Which one of the following statements is FALSE? (A) PA B PAPB (B) PA \ B PA (C) PA B PA PB (D) PA B PAPB (C) We know that A and B are independent then P(A B) = P(A) P(B) written permission. 4 is.

5 EC-GATE-5 PAPE- A and B are independent then P(A/B) = P(A) and P(B/A) = P(B) Also if A and B are independent then AandB are also independent i.e., PA B PAPB (A), (B), (D) are correct (C) is false Since P(A B) = P(A)+P(B)-P(A B) = P(A)+P(B)-P(A)P(B) 5. The waveform of a periodic signal x(t) is shown in the figure. xt 4 4 t A signal g(t) is defined by. g t t x. The average power of g(t) is x(t) lim t dt Power of T 9t 9. Watts 6. In the network shown in the figure, all resistors are identical with. The resistance ab in of the network is. a ab b written permission. 5

6 EC-GATE-5 PAPE- eq By bridge condition eq eq 7. Consider a system of linear equations: x y z x y 4z and x 4y 6z k. The value of k for which the system has infinitely many solutions is. x-y+z = - x-y+4z = -x+4y-6z = k Augmented matrix A B K, K The system will have infinitely many solutions if p(a/b) = p(a) = r < number of variables k- = k = written permission. 6

7 EC-GATE-5 PAPE The polar plot of the transfer function (A) first quadrant (C) third quadrant Gs (A) s s Put s j j G j j, M, M s G s for s (B) second quadrant (D) fourth quadrant will be in the j So, zero is nearer to imaginary axis. Hence plot will move clockwise direction. It is first quadrant. 9. Let z x iy be a complex variable. Consider that contour integration is performed along the unit circle in anticlockwise direction. Which one of the following statements is NOT TUE? z (A) The residue of at z is / z (B) (C) z dz C dz i C z (D) z (complex conjugate of z) is an analytical function f z (D) z x iy u x,v y u and v x x written permission. 7

8 EC-GATE-5 PAPE- u and v y x y y u v i.e., C equations not satisfied z is not analytic (A) z = is a simple pole z z z esidue at z is limz. lim z z z z z (B) Since z is analytic everywhere Using Cauchy s integral theorem, C z dz y. Consider the signal st mtcos f t mˆ t f t where mˆ t denotes the Hilbert c transform of m(t) and the bandwidth of m(t) is very small compared to f c. The signal s(t) is a (A) high-pass signal (B) low-pass signal (C) band-pass signal (D) double sideband suppressed carrier signal (C) Given s(t) is the equation for single side band modulation (lower side band). Thus it is a Band pass signal.. For the circuit with ideal diodes shown in the figure, the shape of the output v out for the in given sine wave input v will be c.5t T in out (A).5T T (B).5T T (C).5T T (D).5T T (C) written permission. 8

9 EC-GATE-5 PAPE- The circuit can be re drawn as a d D V o b D D c During positive pulse, both diodes are forward biased. So output pulse of +ve polarity is produced. As polarity at d and c is given opposite to input terminals, hence +ve pulse is inverted. During negative pulse, both diodes are reverse biased. So, V = V. The result of the convolution xtt t is (A) xt t (B) xt t (C) xt t (D) xt t (D) x( t)* t t x( t)* (t t ) x t t t x t t. The value of p such that the vector. is an eigenvector of the matrix 4 p 4 4 is 7 4 AX X P 4 4 P 7 6 P 7 and 6 i.e., Equation gives P 7 4 P 7 written permission. 9

10 EC-GATE-5 PAPE In the given circuit, the values of V and V respectively are 4 V I I 5A 4 4 V (A) 5V, 5V (B) V, V (C) 5V, 5V (D) V, V (A) By nodal analysis 5 I I I 4I 5 5 I A 4 V 4I 5volts V 4(5) V V 5 volts 5. In an 885 microprocessor, the shift registers which store the result of an addition and the overflow bit are, respectively (A) B and F (B) A and F (C) H and F (D) A and C (B) In an 885 microprocessor, after performing the addition, result is stored in accumulator and if any carry (overflow bit) is generated it updates flags. 6. Negative feedback in a closed-loop control system DOES NOT (A) reduce the overall gain (B) reduce bandwidth (C) improve disturbance rejection (D) reduce sensitivity to parameter variation (B) Negative feedback in a closed loop (i) Increases bandwidth (ii) educes gain (iii) Improve distance rejection written permission.

11 EC-GATE-5 PAPE A 6 Kb (=6,84 bit) memory array is designed as a square with an aspect ratio of one (number of rows is equal to the number of columns). The minimum number of address lines needed for the row decoder is. 7 Generally the structure of a memory chip = Number of ow Number of column = M N The number of address line required for row decoder is n where M = n or n = log M As per information given in question : M = N So M N = M M = M = 6k = 4 M = 4 M = 8 n = log 8 = 7 8. A function f (x) x x is defined in the closed interval [-, ]. The value of x, in the open interval (-, ) for which the mean value theorem is satisfied, is (A) -/ (B) -/ (C) / (D) / (B) By Lagrange s mean value theorem f f f ' x -x+x = x -x- = So x =, -/ x= -/ only lies in (-,) 9. The electric field component of a plane wave traveling in a lossless dielectric medium is 8 z given by Ez,t aˆ y cos t V/ m. The wavelength (in m) for the wave is z E z, t cos t a m y written permission.

12 EC-GATE-5 PAPE- In the circuit shown, the switch SW is thrown from position A to position B at time t =. The energy in J taken from the V source to charge the. F capacitor from V to V is V B SW A.F t (A). (B).45 (C).9 (D) (C) So the capacitor in initially uncharged i.e. V C () = The capacitor will be charged to supply voltage V when the switch is in position B for time. So we need to find capacitor voltage c Vc t Vc V c Vc e e t 6 C. sec i dvc C dt d. e dt.. e e 6 t t t t So instantaneous power of source = V(t)i(t)..9 P t.e e t t E Pd t.9 e t dt.9 t.9 e 6.9.9J written permission.

13 EC-GATE-5 PAPE- In the circuit shown below, the Zener diode is ideal and the Zener voltage is 6V. The output voltage V (in volts) is. k V k V 5 Zener is not Breakdown Hence Vo 5V. Consider a four bit D to A converter. The analog value corresponding to digital signals of values and are V and.65v respectively. The analog value (in Volts) corresponding to the digital signal is..975 Analog output = [esolution] [Decimal equivalent of Binary] Ana log output In the circuit shown, at resonance, the amplitude of the sinusoidal voltage (in Volts) across the capacitor is. 4.mH cost Volts F 5 VC QV 9 L L Q.5 C 4 V C = 5-9 V C = 5V written permission.

14 EC-GATE-5 PAPE- 4 A sinusoidal signal of khz frequency is applied to a delta modulator. The sampling rate and step-size of the delta modulator are, samples per second and.v, respectively. To prevent slope overload, the maximum amplitude of the sinusoidal signal (in Volts) is (A) (A) To avoid slope overload, d.t s x(t) dt m max x(t) E sin f t m (B) (C) (D) d x(t) dt max Em E. f., E.. m m m 5. Consider a straight, infinitely long, current carrying conductor lying on the z-axis. Which one of the following plots (in linear scale) qualitatively represents the dependence of H on r, where H is the magnitude of the azimuthal component of magnetic field outside the conductor and r is the radial distance from the conductor? A B H H r r C D H H r (C) H I r r is the distance from current element. H r r written permission. 4

15 EC-GATE-5 PAPE- Q. No carry Two Marks Each 6. The input X to the Binary Symmetric Channel (BSC) shown in the figure is with probability.8. The cross-over probability is /7. If the received bit Y =, the conditional probability that was transmitted is. P[X ]. X 67 Y 7 7 P[X ] X P x. / 7 67 Y P x.8 / 7 6 / 7 P y / x P x Px / y P{y } Py / x 7 P{x }.8 6 P{y } Px / y The transmitted signal in a GSM system is of khz bandwidth and 8 users share a common bandwidth using TDMA. If at a given time users are talking in a cell, the total bandwidth of the signal received by the base station of the cell will be at least (in khz). 4 written permission. 5

16 EC-GATE-5 PAPE- Since GSM requires KHz and only 8 users can use it using TDMA, 9 th user needs another KHz. 9 th, th, th, th user can use another KHz bandwidth on time share basis. Thus for user we need 4 KHz bandwidth. 8. For the discrete-time system shown in the figure, the poles of the system transfer function are located at Xn Yn Z Z (A), (B), (C) (C), (D), Y(z) z H(z) X(z) 5 z 5 z z z z z z So, poles are z, z. 9. The circuit shown in the figure has an ideal opamp. The oscillation frequency and the condition to sustain the oscillations, respectively, are out C C (A) and (B) and 4 C C (C) and (D) and 4 C C (D) written permission. 6

17 EC-GATE-5 PAPE- Frequency of Wein bridge oscillator is, but in the question time constant is C doubled so, frequency becomes half. C Z j jc Z j c j j j c Z Z Z A source emits bit with probability and bit with probability. The emitted bits are communicated to the receiver. The receiver decides for either or based on the received value. It is given that the conditional density functions of are as, x,, x 5, f (r) 4 and f (r) 6, otherwise,, otherwise. The minimum decision error probability is (A) (B) / (C) /9 (D) /6 (D). For a silicon diode with long P and N regions, the accepter and donor impurity concentrations 7 5 are cm and cm, respectively. The lifetimes of electrons in P region and holes in N region are both s. The electron and hole diffusion coefficients are 49cm / s and 6cm / s, respectively. Assume kt / q 6mV, the intrinsic carrier 9 concentration is cm and q.6 C. When a forward voltage of 8 mv is applied across the diode, the hole current density in na / cm injected from P region to N regions is written permission. 7

18 EC-GATE-5 PAPE- q.d p.p Hole current density L P n n N Given i d N p d cm n 5 sec p n qva kt e Va 8mV,D p 6, Dn 49 qn D i p qva Jp kt N e d p e na / cm. For the NMOSFET in the circuit shown, the threshold voltage is V th, where Vth. The source voltage V ss is varied from to V DD. Neglecting the channel length modulation, the drain current I D as a function V ss is represented by. V DD V SS A I D B I D V DD V th V SS V th V SS written permission. 8

19 EC-GATE-5 PAPE- C I D D I D V DD V th V SS V DD V th V SS (A) V GS = V DS Hence MOS Transistor is in saturation. In saturation, D GS r DD SS T I k V V k V V V As V I Not linearly because square factor SS D Hence option (A) correct. In the system shown in figure (a), m(t) is a low-pass signal with bandwidth W Hz. The frequency response of the band-pass filter H(f) is shown in figure (b). If it is desired that the output signal z(t) = x(t), the maximum value of W (in Hz) should be strictly less than. mtcos4t Hf yt x t x t zt x t Amplifier Hf a Band pass filter f Hz 5 x(t) m(t).cos 4 t Spectrum of x(t) written permission. 9

20 EC-GATE-5 PAPE- X(f ) w w w w f y(t) =x(t)+x (t) Let us draw the spectrum of positive frequencies of y(t). y(t) m(t)cos 4t m (t)cos (4t) cos 48t m(t)cos 4t m (t) m (t) m (t) y(t) m(t)cos 4t cos 48 t m (t) w w y(t) Y(f ) w w w w 4 w 4 w w 7 4 w 7 w 5 written permission.

21 EC-GATE-5 PAPE In the circuit shown, I 8mA and I 4mA. Transistors T and T are identical. Assume o o that the thermal voltage VT is 6mV at 7 C. At 5 C, the value of the voltage V V V (in mv) is. V s I V V V I T T 8.88 We know that I VBE VT Ise I VBE VT Ise I I e At 5 O C, V V VT V 8mV T V V ln 8mV V V 8.88 mv 5. The electric field intensity of a plane wave traveling in free space is give by the following expression E x,t a 4 cos t k x V / m y In this field, consider a square area cm x cm on a plane x + y =. The total timeaveraged power (in mw) passing through the square area is. 5. E(x,t) = 4π cos( t - k x)v/m written permission.

22 EC-GATE-5 PAPE- P avg x y E a x x s 4 a 7.5a Power P P.ds a x y a x avg unit vector normal to thesurface x P 7.5a x. dydz s a 7.5 P 5. W P 5.mW 6. The maximum area (in square units) of a rectangle whose vertices lie on the ellipse x 4y is. rectangle o ellipse Let x, y be the length, breadth respectively of the rectangle inscribed in the ellipse x 4y, then Area of the rectangle (x) (y) i.e., 4xy Consider, f Area 6x y x 4x x y 4 f ' x x x x y y 8 8 written permission.

23 EC-GATE-5 PAPE- f '' x 8 48x when x f is maximum at x = Area is maximum and the maximum area is 4 8 i.e., 7. In the given circuit, the maximum power (in Watts) that can be transferred to the load L is. 4 V rms j L.66 4 j V 45 ; V 4 j Th(rms) Th(mq) Z j j Th Z L th Maximum power transfer to L is 45 Pmax I L.66W j 8. A lead compensator network includes a parallel combination of and C in the feed-forward s path. If the transfer function of the compensator is Gc s, the value of C is s 4..5 s Given G(s) s 4 Zero = = C Pole = 4 = C So, C =.5 written permission.

24 EC-GATE-5 PAPE A MOSFET in saturation has a drain current of ma for VDS.5V. If the channel length modulation coefficient is.5 V, the output resistance in k of the MOSFET is. Under channel length modulation I I V D Dsat DS did I dv r r DS Dsat IDsat.5 k 4. The built-in potential of an abrupt p-n junction is.75v. If its junction capacitance C J at a reverse bias V of.5v is 5pF, the value of C in pf when V.5 C j 7.5V is. V bi V Cj Vbi V C V V j bi Cj C j Cj.5 pf 8 So, answer is.5 J 4. In the circuit shown, assume that the opamp is ideal. The bridge output voltage V (in mv) for.5 is. 5 V V written permission. 4

25 EC-GATE-5 PAPE- 5 I5 A I Vo V 5mV A V o A 4. The damping ratio of a series LC circuit can be expressed as (A) C L (C) L (B) C ' ' (In series LC circuit) Q C L L C (C) C L (D) L C 4. In the circuit shown, switch SW is closed at t =. Assuming zero initial conditions, the value of v t (in Volts) at t= sec is. c t V SW 5 F 6 vct.58 Vc 4V V C ( - ) = V At t =, C is open circuit V c ( ) = 4V written permission. 5

26 EC-GATE-5 PAPE thc sec V t V V V e t C C C C C 4 4e t V 4 4e.58V 44. Consider a uniform plane wave with amplitude E of V / m and.ghz frequency travelling in air, and incident normally on a dielectric medium with complex relative permittivity and permeability as shown in the figure. r r Air Dielectric r j j r cm E? E V/m Freq.GHz. The magnitude of the transmitted electric field component (in V/m) after it has travelled a distance of cm inside the dielectric region is. Dielectric air j j r E V m r = So, E = E = V/m E Electric field in the dielectric after travelling cm E z E e r j j j j j r r j j j j written permission. 6

27 EC-GATE-5 PAPE- z cm E e e. 45. A vector P is given by P x ya x y a x yza. Which one of the following statements x y z is TUE? (A) P is solenoidal, but not irrotational (B) P is irrotational, but not solenoidal, (C) P is neither solenoidal nor irrotational (D) P is both solenoidal and irrotational (A) P x ya x y a x yza x y z.p x y x y x y It is solenoidal. a a a x y z P x y z x y x y x yz a x x y a y xyz a z xy x So P is solenoidal but not irrotational. 46. All the logic gates shown in the figure have a propagation delay of ns. Let A=C= and B= until time t=. At t=, all the inputs flip (i.e., A = C = and B = ) and remain in that state. For t >, output Z = for a duration (in ns) of A B C Z 4 A B C X Y Z As per information given on question the waveforms of A, B, C are as follows written permission. 7

28 EC-GATE-5 PAPE- A B C t The logic to solve this question is first obtain X, Y waveform and using this obtain Z. To obtain X, initially assume delay of NOT gate is and draw its waveform and finally shift it by nsec to obtain actual X. Similar procedure to be followed for obtaining Y and Z i.e., first draw waveform with delay and at the end shift by the amount of delay given in question. X B with delay B t B t B with delay t nsec Y B.A A t B with delay t Y AB without delay z AB Y C AB with delay t nsec t 4nsec written permission. 8

29 EC-GATE-5 PAPE- AB t 4 C t t t 4 Z without delay t 4 Z with delay 4 6 Clearly we can say that output is high during nsec to 6 nsec i.e. a duration of 4 nsec KI 47. A plant transfer function is given as Gs Kp s. When the plant operates in a ss unity feedback configuration, the condition for the stability of the closed loop system is KI (A) Kp (B) KI Kp (C) KI Kp (D) KI Kp (A) Gs sk s p K s I C.E s s s K K H table s Kp s KI s K K s p K I I P For stable system st column elements must be positive K I KP K K I KP I I written permission. 9

30 EC-GATE-5 PAPE A -input majority gate is defined by the logic function Ma,b,c ab bc ca. Which one of the following gates is represented by the function M(M(a,b,c),M(a,b,c),c)? (A) -input NAND gate (C) -input NO gate (B) and (D) Ma,b,c ab bc ac m,5,6,7 M a,b,c m,,,4 X let say for simplicity M a,b,c ab bc ac m,4,6,7 Y let c m,,5,7 z let f M a,b,c,m a,b,c,c f x, y,z xy yz zx (B) -input XO gate (D) -input XNO gate m,,,4 m,4,6,7 m,4,6,7 m,,5,7 m,,5,7 m,,,4 m,4 m7 m AND operater is like intersection m,,4,7 O operator is like union A B C A B C standard result Both options D and B are correct 49. The longitudinal component of the magnetic field inside an air-filled rectangular waveguide made of a perfect electric conductor is given by the following expression 9 H x,y,z,t. cos 5x cos. y cos t z A / m z The cross-sectional dimensions of the waveguide are given as a =.8m and b=.m. The mode of propagation inside the waveguide is (A) TM (B) TM (C) TE (D) TE (C) mx 5 x m 5a a ny. y n.b b Given is H z means TE mode mode TE written permission.

31 EC-GATE-5 PAPE The open-loop transfer function of a plant in a unity feedback configuration is given as Ks 4 Gs. The value of the gain K(>) for which j lies on the root locus s 8 s 9 is. 5.5 By magnitude condition s j G s H s So, k j 7 j j 4 j 8 5 K 5.5 So K value is = Two sequences [a, b, c] and [A, B, C] are related as, A a j B W W b where W e. 4 C W c W If another squence p,q, r is derived as, p A / q W W W B /, 4 4 r W C / W W then the relationship between the sequences [p, q, r] and [a, b, c] is p,q,r b,a,c p,q,r b,c,a (A) (B) (C) p,q,r c,a,b (D) p,q,r c,b,a (C) Consider, written permission.

32 EC-GATE-5 PAPE- 4 w w 6 w w w w w w w w w w w w w ; w w w ; w w w w A w w a B w w b C w w w w w w c w w w w w w a w w w w b w c w w w w w j w e, w w w w w w p a c q b a r c b d y 5. The solution of the differential equation t (A) t e dt dy y with y() y' is dt t (B) te t (C) te (D) te t (B) Differential equation is (D +D+).y = D +D+ = (D+) = D = -,- t solution is y t c ct e C.F t t y' t c e c c t e y() = ; y(t) = (+t)e -t y' gives c = and c +c (-) = c = written permission.

33 EC-GATE-5 PAPE The pole-zero diagram of a causal and stable discrete-time system is shown in the figure. The zero at the origin has multiplicity 4. The impulse response of the system is h[n]. If h[] =, we can conclude Imz ez (A) h[n] is real for all n (B) h[n] is purely imaginary for all n (C) h[n] is real for only even n (D) h[n] is purely imaginary for only odd n (A) x 54. Which one of the following graphs describes the function f x e x x? A B f (x) f (x) x x C f (x) D f (x) x x (B) f() = e -x (x +x+) f() = f(.5) =.67 For positive values of x, function never goes negative. written permission.

34 EC-GATE-5 PAPE The Boolean expression FX,Y, Z XYZ XYZ XYZ XYZ converted into the canonical product of sum (POS) form is (A) X Y ZX Y ZX Y ZX Y Z (B) X Y ZX Y ZX Y ZX Y Z (C) X Y ZX Y ZX Y ZX Y Z (D) X Y ZX Y ZX Y ZX Y Z (A) Given minterms are: F X, Y, Z XYZ XYZ XYZ XYZ So,maxtermisF X,Y,Z m,,,5 POS X Y Z X Y Z X Y Z X Y Z written permission. 4

Q. 1 Q. 25 carry one mark each.

Q. 1 Q. 25 carry one mark each. Q. Q. 5 carry one mark each. Q. Consider a system of linear equations: x y 3z =, x 3y 4z =, and x 4y 6 z = k. The value of k for which the system has infinitely many solutions is. Q. A function 3 = is