Among the following options, the antonym of the word 'nocturnal' is ______.

Correct answer: diurnal. Nocturnal means active at night. Diurnal means active during the day, so it is the antonym.

The statements (S1), (S2), and (S3) pertain to the scores obtained by students in an exam. The maximum possible marks in the exam is 150. (S1) The highest score is 100. (S2) The fourth highest score is 76. (S3) There are at least four students whose scores are within 25 of each other. Which one of the following options is necessarily correct?

Correct answer: (S1) and (S2) together imply (S3). If the highest score is 100 and the fourth highest score is 76, then the top four scores all lie from 76 to 100. Hence at least four students are within 24 marks of each other, so (S3) follows.

The figure below has exactly three intersecting line segments with a rectangular portion WXYZ missing. Which one of the following options P, Q, R, and S is the missing portion WXYZ?

Correct answer: S. The missing rectangle must continue the two diagonal segments and the nearly horizontal segment so that exactly three line segments intersect. Option S preserves those three intersecting segments.

Real numbers y, p, and n, all greater than 1, satisfy (log base p^(1/n) of y)(log base y^(1/n) of p) = 16, where the logarithms are taken to the bases p^(1/n) and y^(1/n). The value of n is ______.

Correct answer: 4. Let log_p y = a. Then log base p^(1/n) of y = n a, and log base y^(1/n) of p = n/a. Their product is n^2 = 16, so n = 4.

The following observation is made about the scores obtained by 100 students in an exam: 'For each student, there exists another student in the class such that their scores are at most 10 marks away.' If the above statement is false, which one of the following statements is necessarily true?

Correct answer: There exists at least one student in the class for whom the scores of all the other students are more than 10 marks away.. Negating 'for each student, there exists another student within 10 marks' gives 'there exists at least one student such that every other student's score is more than 10 marks away.'

Each one of the following clues contains a keyword that is partially filled. Clue 1: Synonym of recognize (8 letters): _D_NT_FY. Clue 2: A story long enough to fill a book (5 letters): ___EI_. Clue 3: Two of something (6 letters): ___PLE. Clue 4: A fraction of something, split equally into two parts (4 letters): ___F. The first letter of each of the keywords can be rearranged to form a four-letter word. Which one of the options below is a possible choice for the four-letter word?

Correct answer: ITCH. The clues point to words such as identify, tale/novel-type story word, couple, and half. Their initial letters can be arranged to form ITCH.

Three children P, Q, R and two grown-ups X, Y play a badminton doubles tournament. X and Y are parents to two of the children playing. The child of X is not the same as the child of Y. Exactly one of the children does not have a parent playing in the tournament. The following rules are followed: (i) A parent and his/her child cannot be on the same team. (ii) A match can feature at most one parent and his/her child, that is, a maximum of one parent-child pair can play in a match. Matches played: Match 1: Team 1 P and X, Team 2 Q and R. Match 2: Team 1 P and R, Team 2 X and Y. Match 3: Team 1 R and X, Team 2 Q and Y. Which one of the following options is correct?

The source image text appears inconsistent around the parent-child conditions, so the answer key is left pending until the official wording is confirmed.

Let Pk represent the perimeter of a square with sides of length k. The value of the expression P1 + P2 + P3 + ... + P10 is ______.

Correct answer: 220. A square of side k has perimeter Pk = 4k. Therefore P1 + P2 + ... + P10 = 4(1 + 2 + ... + 10) = 4 x 55 = 220.

The city of Atlantis was crafted by the God of the seas, Poseidon. It was made of alternating concentric circular rings of land (shaded) and water (not shaded), as represented in the figure. The radius of Inner Island was 2.5 stades, where a stade is an ancient Greek unit of length. The water surrounding Inner Island was one stade wide. This was surrounded by two pairs of alternating rings of land and water. The first pair of land and water was two stades wide each, and the outer pair was three stades wide each. The ratio of the surface area of the land to that of the water in the city of Atlantis is ______, rounded off to two decimal places.

Correct answer: 0.75. The radii are 2.5, 3.5, 5.5, 7.5, 10.5, and 13.5 stades. Land area is pi[2.5^2 + (5.5^2 - 3.5^2) + (10.5^2 - 7.5^2)] = 78.25pi. Water area is pi[(3.5^2 - 2.5^2) + (7.5^2 - 5.5^2) + (13.5^2 - 10.5^2)] = 104pi. The ratio is 78.25/104 = 0.7524, so the answer is 0.75.

A reference figure transforms the text '12 34' into a shown output. Applying the same transformation to the text 'GATE - 2026 Aptitude Test', which one of the four shown options P, Q, R, and S is obtained?

The uploaded screenshot for this figure-based transformation question is not clear enough to confirm the official transformation and answer key. The answer is left pending until the saved figure or official key is available.

Consider the differential equation dw/dt = A w, with w(0) = [1, 1]^T. If w(t) = e^t xhat + e^(-2t) yhat is the solution to the equation where xhat and yhat are unit vectors along the positive x and y axes respectively, then which of the following options is the correct matrix representing A?

Correct answer: [[1, 0], [0, -2]]. Q.11 - GATE Shortcut Given: w(t) = e^t ux + e^(-2t) uy Observe the exponents: e^t gives lambda1 = 1 e^(-2t) gives lambda2 = -2 The corresponding eigenvectors are the standard basis vectors: ux = [1, 0]^T and uy = [0, 1]^T. Since the eigenvectors are the standard x-axis and y-axis unit vectors, the matrix is directly diagonal: A = [[1, 0], [0, -2]]. Quick Answer: A = [[1, 0], [0, -2]] Option (A).

A surface is given by z^2 = 2x^2 - y^2 and n and -n are unit normal vectors to the surface at the point (1, 0, sqrt(2)). Which of the following vectors can be n?

Correct answer: (sqrt(2) i - k) / sqrt(3). Q.12 - GATE Shortcut Given: z^2 = 2x^2 - y^2 Point: (1, 0, sqrt(2)) Convert to implicit form: F(x, y, z) = 2x^2 - y^2 - z^2 Compute gradient: grad F = (4x, -2y, -2z) At (1, 0, sqrt(2)): grad F = (4, 0, -2sqrt(2)) Normalize: |grad F| = sqrt(16 + 8) = 2sqrt(6) Unit normal: n = (4, 0, -2sqrt(2)) / (2sqrt(6)) = (sqrt(2)i - k) / sqrt(3) Quick Answer: n = (sqrt(2)i - k) / sqrt(3) Option (D). Key Formula: For a surface F(x, y, z) = 0, the unit normal is obtained from grad F / |grad F|.

The Laplace Transform of the signal x(t) = u(t - 2) * (t u(t)) is given by which of the following expressions? Here * represents the convolution operator.

Correct answer: e^(-2s) / s^3. L{u(t - 2)} = e^(-2s)/s and L{t u(t)} = 1/s^2. Convolution in time corresponds to multiplication in the Laplace domain, so X(s) = e^(-2s)/s^3.

Consider carrier transport in a Zener diode in the breakdown region. Which is the dominant transport mechanism for current flow in this case?

Correct answer: Tunneling. In the Zener breakdown region, the strong electric field across the heavily doped junction causes quantum mechanical tunneling of carriers.

Two analog signals x1(t) and x2(t), in seconds, are sampled at a rate fs = 40 Hz, where x1(t) = cos(20 pi t), t >= 0, and x2(t) = cos(100 pi t), t >= 0. The first ten samples, starting from t = 0, are considered for the analysis. Which of the following statements is TRUE?

Correct answer: All of the fourth to seventh samples of x1(t) are equal to the corresponding samples of x2(t).. With fs = 40 Hz, Ts = 1/40 s. The samples are x1[n] = cos(20pi n/40) = cos(pi n/2) and x2[n] = cos(100pi n/40) = cos(5pi n/2). Since cos(5pi n/2) = cos(pi n/2 + 2pi n), the sampled sequences are identical.

The response y[n] of a discrete-time system obeys the relation y[n] = (5/6)y[n - 1] - (1/6)y[n - 2] + x[n]. The input to the system is x[n] = delta[n] - (1/3)delta[n - 1]. Which of the following options is TRUE for y[n]?

Correct answer: Stable and causal response. The transfer function is H(z) = 1/[(1 - (1/2)z^-1)(1 - (1/3)z^-1)]. Multiplying by X(z) = 1 - (1/3)z^-1 cancels the pole at 1/3, giving Y(z) = 1/(1 - (1/2)z^-1). With the causal right-sided response, the pole magnitude is less than one, so the response is stable and causal.

The ideal OP-AMP circuit shown in the figure produces output voltage Vo = x when the switch S is open. Which of the options represents the output voltage when S is closed?

The saved circuit image is included for this question. The answer key is left pending because the component connections in the screenshot are too small to verify safely.

Consider the circuit shown in the figure with VBB = 3 V and VCC = 12 V. Assume VBE = 0.7 V and beta_dc = 99 for the BJT. Which of the following options is the correct value of the current IE?

The saved BJT circuit image is included for this question. The answer key is left pending until the exact source label and current direction from the official figure are confirmed.

A control system is shown in the figure. Which option represents the correct transfer function of the system?

The saved block diagram is included for this question. The answer key is left pending because the summing-junction signs in the screenshot are not clear enough to confirm the official transfer function.

In the circuit shown in the figure, A and B are logic inputs and Y is the logic output. Which of the following logic operations is realized by the circuit?

Correct answer: NOR. The CMOS network has the pull-up path active only when both inputs are 0, while the pull-down path conducts when either input is 1. Therefore the output is high only for A = 0 and B = 0, which is the NOR operation.

Consider the Friis transmission equation Pr = (Pt Gt Gr lambda^2) / ((4 pi D)^2), where Pr and Pt are the received and transmitted powers, respectively. Gt and Gr are the gains of transmitting and receiving antennas, respectively, D is the distance between the transmitting and receiving antennas, and lambda is the wavelength in free space. Given Gt = Gr = 1.0, lambda = 0.30 m, and Pt = 10 dBm, choose the distance D, in km, from the following options at which the received power Pr = -90 dBm.

Correct answer: 15 / (2pi). The power ratio is Pr/Pt = 10^((-90 - 10)/10) = 10^-10. Hence lambda/(4 pi D) = 10^-5, so D = 0.30/(4 pi x 10^-5) m = 7.5/pi km = 15/(2pi) km.

Consider a discrete memoryless source with an alphabet of four source symbols. s(n) is a multi-level random sequence of source symbols from the alphabet {-1, 0, +1, +2}. The symbols are equally likely and generated at 10^4 symbols per second. Which of the following options is the correct value of equivalent Nyquist bandwidth of s(n)?

Correct answer: 10 kHz. Four equally likely source symbols carry log2(4) = 2 bits per symbol. At 10^4 symbols/s, the equivalent bit rate is 20 kbps. The ideal Nyquist bandwidth is Rb/2 = 10 kHz.

The relation between the input current i(t) and the output voltage V(t) of a circuit is governed by the equation C dV/dt = i(t) - m(t). The circuit is excited by i(t) = q delta(t), where q is a real valued constant. V at t = 0- is V0. Which of the following is an equivalent representation of the above case?

Correct answer: C dV/dt = -m(t), with V(0+) = V0 + q/C. Integrating the equation around t = 0 gives C[V(0+) - V(0-)] = integral q delta(t) dt = q. Thus V(0+) = V0 + q/C. For t > 0, the impulse input is absent, so C dV/dt = -m(t).

The electric field of a monochromatic plane wave travelling in a lossless isotropic and homogeneous medium is given by E(z,t) = E0[cos(omega t - kz) xhat + sin(omega t - kz) yhat] in a right-handed orthogonal coordinate system. Which of the following is the correct polarization of the electromagnetic wave?

The field has equal orthogonal components in phase quadrature, so it is circularly polarized. The handedness convention depends on the viewing convention, so the answer key is left pending until the official convention/key is confirmed.

Consider a p-n junction diode when it is forward biased with 2 V. Which of the following is/are the correct magnitude(s) of the energy difference between quasi Fermi-levels, EFn in the n-side and EFp in the p-side?

Correct answer: 2 eV. Under an applied forward bias V, the separation between the electron and hole quasi-Fermi levels is qV. For V = 2 V, the energy separation is 2 eV.

Consider the matrix M = [[2, 1, 1], [1, 3, 0], [-1, a, b]]. Which of the following options is/are TRUE if det(M) is not equal to 0?

Correct answer: B and D. The determinant is a + 5b + 3. Options A and C make the determinant zero. Options B and D give nonzero determinant values, so they satisfy det(M) != 0.

A binary ripple counter is designed to count from (0)10 to (64)10. Which of the following is/are the number of flip-flops required to design the counter?

Correct answer: 7. Counting from 0 to 64 requires 65 distinct states. Since 2^6 = 64 is not enough and 2^7 = 128 is enough, the counter needs 7 flip-flops.

Which optical sequential representation describes the dielectric loss tangent of a substance?

The screenshot text for this dielectric-loss question is partially unclear, so the answer key is left pending until the official wording is confirmed.

The figure shows the output characteristics of two different bipolar junction transistors (BJT1, BJT2) with magnitudes of Early voltages |VA1| and |VA2|. Which of the following options is/are correct regarding the Early voltages?

The saved Early-voltage characteristic image is included for this question. The answer key is left pending because the curve labels/slopes in the screenshot are not clear enough to confirm safely.

The output voltage Vo, in volts, for the network shown in the figure is ______, rounded off to two decimal places.

Correct answer: 1.00. Let the bottom node be ground and the top node be Vo. The 1 mA source injects current into the top node. For the two middle nodes A and B, the supernode equation gives A + B = Vo. KCL at the top node gives (Vo - A)/1k + (Vo - B)/1k = 1 mA, hence 2Vo - (A + B) = 1 V. Therefore Vo = 1.00 V.

Consider the circuit shown in the figure, where the input vi(t) is in volts. The average power in mW dissipated in the load resistance of 1 kohm at the resonant frequency is ______, rounded off to two decimal places.

Correct answer: 72.00. The source is ideal and applies vi(t) = 12 sin(omega t) across the parallel RLC network. At resonance, the load resistor still has 12 V peak across it, so Vrms = 12/sqrt(2) V. The load power is Vrms^2/R = 72/1000 W = 72.00 mW.

A wireless digital transmission scheme is using 16-QAM over an additive white Gaussian noise channel and a maximum-likelihood receiver. Consider the information bit rate from source to be 4 x 10^7 bits per second. The minimum transmission bandwidth, in MHz, of the modulated signal necessary for optimum recovery of information at the receiver is ______, rounded off to two decimal places.

Correct answer: 10.00. For 16-QAM, each symbol carries log2(16) = 4 bits. The symbol rate is 4 x 10^7 / 4 = 10^7 symbols/s. For ideal Nyquist pulse shaping in passband QAM, the minimum RF transmission bandwidth equals the symbol rate, which is 10.00 MHz.

The cutoff frequency, in GHz, for the dominant TEmn mode of an air-filled rectangular waveguide of inner dimension 2.8 inch x 0.14 inch is ______, rounded off to two decimal places.

The screenshot wording for this waveguide-dimension question is not clear enough to confirm the intended dominant-mode calculation and official answer key.

For a lossless passive two-port network, S^H S and S S^H have maximum eigenvalue equal to 1. For a lossy passive two-port network, S^H S and S S^H have maximum eigenvalue equal to 0.8. The percentage of power dissipated in the lossy network at the interconnection frequency is ______, rounded off to two decimal places.

Correct answer: 20.00. For a passive network, the largest eigenvalue of S^H S represents the maximum fraction of incident power that can appear as outgoing scattered power. If this value is 0.8, the remaining fraction is dissipated: 1 - 0.8 = 0.2, or 20.00%.

The negative edge-triggered JK flip-flop in the figure has J and K inputs tied to logic High and a square wave of frequency 600 cycles/second is applied to its clock (C) input. The frequency of output Q, in cycles/second, is ______, rounded off to two decimal places.

Correct answer: 300.00. With J = K = 1, a JK flip-flop toggles once on every active clock edge. Therefore the output frequency is half the clock frequency: 600/2 = 300.00 cycles/second.

Consider the two series Sa and Sb, where Sa is as shown in the question and Sb = 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + ... . Which of the following statements is correct for the two given series?

The expression for Sa is not legible enough in the screenshot to confirm the official convergence result. Sb is a geometric series with ratio 1/2 and converges.

The continuous-time signal x(t) is real, periodic with period T and satisfies the Dirichlet conditions. The Fourier series representation of x(t) is x(t) = sum from k = -infinity to infinity of ak exp(jk(2pi/T)t) and satisfies x(t - T/2) = -x(t). For any integer n, which of the following options is correct?

Correct answer: a2n = 0. Using the Fourier shift property, x(t - T/2) has coefficients ak exp(-jkpi). Since x(t - T/2) = -x(t), ak[(-1)^k + 1] = 0. For even k, this gives 2ak = 0, so all even-index coefficients are zero.

Let X, Y, Z and N be random variables. The variables X and N are independent of each other. X is uniformly distributed between -1 and 1. N follows a normal distribution with zero mean and unit variance. Y and Z are defined as shown in the question. Which of the following pairs represents the values of correlation between X and Y and between X and Z?

The definitions of Y and Z are not legible enough in the screenshot to confirm the official correlation values.

In the given circuit, L = 1 microH and C = 1 microF. The phasor diagram for IC and IL is also shown. Assume that the phase condition shown in the question holds at a frequency of 159.15 kHz. Among the following options, what is the nearest integer value asked for in the question?

The saved phasor/circuit image is included for this question. The final quantity in the screenshot is not clear enough to safely confirm the official answer key.

For the control system shown in the figure, the transfer function of a plant G(s) = 1 / ((s + 1)(s + 2)) is connected in cascade with a compensator C(s) = K(s + alpha), where K and alpha are positive real valued constants. Which of the following pairs (K, alpha) represent the correct values for the closed-loop system to have poles at (-3 +/- j sqrt(5))?

Correct answer: 3, 4. For unity negative feedback, the characteristic equation is (s + 1)(s + 2) + K(s + alpha) = 0, or s^2 + (3 + K)s + (2 + K alpha) = 0. Desired poles -3 +/- j sqrt(5) give s^2 + 6s + 14. Hence K = 3 and alpha = 4.

The state and output equations for a control system are xdot = [[-4, -1.5], [4, 0]]x + [2, 0]^T u and y = [1.5, 0.625]x. Which of the following expressions correctly represents the transfer function Y(s)/U(s) of the system with zero initial conditions?

Correct answer: (3s + 5) / (s^2 + 4s + 6). The transfer function is C(sI - A)^(-1)B. Here det(sI - A) = s^2 + 4s + 6 and C adj(sI - A)B = 3s + 5. Therefore Y(s)/U(s) = (3s + 5)/(s^2 + 4s + 6).

The address of the first location of a 256 kilo byte (KB) memory is (2500)H. Choose the correct address of the last location of the memory.

Correct answer: (424FF)H. 256 KB = 256 x 1024 = 0x40000 bytes. The last address is start address + size - 1 = 0x2500 + 0x40000 - 1 = 0x424FF.

Consider a real signal x(t), -infinity = 1. Let E[x(t)] = integral from -infinity to infinity of |x(t)|^2 dt. Which of the following options correctly represents the ratio E[x(t)] / E[3x(-3t + 5)]?

Correct answer: 1/3. E[x(t)] = 4 over an interval of length 1, so it is 4. The signal 3x(-3t + 5) has amplitude 6 and time support of length 1/3, so its energy is 36 x 1/3 = 12. The ratio is 4/12 = 1/3.

A QPSK modulated signal from an additive white Gaussian noise (AWGN) channel is received with an Eb/N0 = 8.4 dB at the input of a coherent QPSK demodulator. A maximum-likelihood reception method is used in the demodulator. Assume the complementary error function erfc(u) is approximately [1/(u sqrt(pi))] exp(-u^2). Which is the nearest bit error rate (BER) at the output of the demodulator?

Correct answer: 10^-4. For coherent QPSK, BER = 0.5 erfc(sqrt(Eb/N0)). With Eb/N0 = 8.4 dB = 6.92, sqrt(Eb/N0) is about 2.63. Using the given approximation gives BER close to 1.0 x 10^-4.

What is the 10's complement of (47)10?

Correct answer: 53. For a two-digit decimal number, the 10's complement is 100 - N. Thus the 10's complement of 47 is 100 - 47 = 53.

Consider a real baseband signal x(t) = e^(-2t), for t in seconds and t >= 0. If 99% of the energy of x(t) lies within B Hz, then which of the following options is TRUE for the value of B?

Correct answer: 63/pi Hz < B < 64/pi Hz. For x(t) = e^(-2t)u(t), X(f) = 1/(2 + j2pi f). The fraction of energy in [-B, B] is (2/pi)tan^-1(pi B). Setting this to 0.99 gives pi B = tan(0.99pi/2), so B is about 20.26 Hz, which lies between 63/pi and 64/pi Hz.

Consider the discrete-time system S with input x[n] and output y[n] as shown in the figure. The two sub-systems represented by their impulse responses h1[n] and h2[n] are linear and time invariant. Which of the following statements is necessarily TRUE?

Correct answer: S is linear and time invariant.. The system is built only from LTI subsystems, unit delays, additions, and subtractions. These interconnections preserve linearity and time invariance. Causality is not necessarily guaranteed because h1[n] and h2[n] are not stated to be causal.

A Boolean function f(x,y,z), with x as MSB and z as LSB, is realized by a 4:1 multiplexer (MUX) with select lines S1 and S0, where S1 is MSB and S0 is LSB, and inputs I0, I1, I2, I3 as shown in the figure. Which of the following options is the correct expression of f(x,y,z)?

Correct answer: xy + zbar. The select inputs are x and z. From the figure, I0 = 1, I1 = 0, I2 = 1, and I3 = y. Therefore f = 1 for xz = 00, 0 for 01, 1 for 10, and y for 11, which simplifies to zbar + xy.

A shift-left Shift Register (SR) and a D flip-flop are connected to a synchronized clock as shown in the figure. Assume that the SR and D flip-flops are initially cleared and the XOR gate has no propagation delay. Which of the following options gives the correct binary representation (b7 b6 b5 b4 b3 b2 b1 b0) of the content of the shift register immediately after the 5th clock transition (positive edge)?

Correct answer: 00011111. Initially the shift register and D flip-flop output are 0. The XOR with logic 1 feeds the complement of Q into the serial input, so the first five shifts insert five 1s at the right end while shifting left. After five positive edges, the register content is 00011111.

A small signal source, Vi(t) = A cos(10^5 t) + B sin(10^7 t), is applied to a BJT circuit as shown in the figure. Assume zero source resistance, VBE = 0.7 V, beta_dc = 99, Early voltage = 100 V and thermal voltage = 25 mV. Effect of internal parasitic capacitances of the BJT may be neglected. Which expression is the best approximation of the output voltage Vo(t)?

Correct answer: -9.1[A cos(10^5 t) + B sin(10^7 t)]. The divider bias gives about 1 mA collector current, so gm is about 0.04 S and ro is about 100 kOhm. The 10 microF bypass capacitor shorts the 1 kOhm emitter resistor at both angular frequencies, leaving the 500 Ohm unbypassed emitter resistance. The midband common-emitter gain is approximately -gm(RC || ro)/(1 + gmRE), which is about -9.1 for both components.

Consider the two-port network as shown in the figure. Which of the following options provides the correct set of values of A, B, C and D parameters?

Correct answer: A = 2, B = 3 x 10^3 ohm, C = 10^-3 ohm^-1, D = 2. The network is a cascade of a 1 kOhm series arm, a 1 kOhm shunt arm, and another 1 kOhm series arm. Multiplying [1 Z; 0 1][1 0; Y 1][1 Z; 0 1] with Z = 1000 and Y = 0.001 gives A = D = 2, B = 3000 ohm, and C = 0.001 ohm^-1.

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