The state equation and the output equation of a control system are given below:

$$\mathop x\limits^. = \left[ {\matrix{ { - 4} & { - 1.5} \cr 4 & 0 \cr } } \right]x + \left[ {\matrix{ 2 \cr 0 \cr } } \right]u,$$

$$y = \left[ {\matrix{ {1.5} & {0.625} \cr } } \right]x.$$

The transfer function representation of the system is

The figure below shows the Bode magnitude and phase plots of a stable transfer function

$$G(s) = {{{n_0}} \over {{s^3} + {d_2}{s^2} + {d_1}s + {d_0}}}$$.

Consider the negative unity feedback configuration with gain k in the feedforward path. The closed loop is stable for k < k₀. The maximum value of k₀ is ______.

The Nyquist stability criterion and the Routh criterion both are powerful analysis tools for determining the stability of feedback controllers. Identify which of the following statements is FALSE.

Consider p(s) = s³ + $${a_2}$$s² + $${a_1}$$s + $${a_0}$$ with all real coefficients. It is known that its derivative p'(s) has no real roots. The number of real roots of p(s) is

For a unity feedback control system with the forward path transfer function

$$G(s) = {K \over {s\left( {s + 2} \right)}}$$

The peak resonant magnitude M_r of the closed-loop frequency response is 2. The corresponding value of the gain K (correct to two decimal places) is _________.

The logic gates shown in the digital circuit below use strong pull-down nMOS transistors for LOW logic level at the outputs. When the pull-downs are off, high-value resistors set the output logic levels to HIGH (i.e. the pull-ups are weak). Note that some nodes are intentionally shorted to implement “wired logic”. Such shorted nodes will be HIGH only if the outputs of all the gates whose outputs are shorted are HIGH.


The number of distinct values of X₃X₂X₁X₀ (out of the 16 possible values) that give 𝑌 = 1 is _______.

A traffic signal cycles from GREEN to YELLOW, YELLOW to RED and RED to GREEN. In each cycle, GREEN is turned on for 70 seconds, YELLOW is turned on for 5 seconds and the RED is turned on for 75 seconds. This traffic light has to be implemented using a finite state machine (FSM). The only input to this FSM is a clock of 5 second period. The minimum number of flip-flops required to implement this FSM is _______.

In the circuit shown below, a positive edge-triggered D Flip-Flop is used for sampling input data D_in using clock CK. The XOR gate outputs 3.3 volts for logic HIGH and 0 volts for logic LOW levels. The data bit and clock periods are equal and the value of $${{\Delta T} \over {{T_{CK}}}}$$ = 0.15, where the parameters $$\Delta T$$ and T_CK are shown in the figure. Assume that the Flip-Flop and the XOR gate are ideal.

If the probability of input data bit (D_in) transition in each clock period is 0.3, the average value (in volts, accurate to two decimal places) of the voltage at node X, is _______.

A 2 $$ \times $$ 2 ROM array is built with the help of diodes as shown in the circuit below. Here W₀ and W₁ are signals that select the word lines and B₀ and B₁ are signals that are output of the sense amps based on the stored data corresponding to the bit lines during the read operation.
During the read operation, the selected word line goes high and the other word line is in a high impedance state. As per the implementation shown in the circuit diagram above, what are the bits corresponding to D_ij (where i = 0 or 1 and j = 0 or 1) stored in the ROM?

A four-variable Boolean function is realized using 4 $$ \times $$ 1 multiplexers as shown in the figure.
The minimized expression for F(U, V, W, X) is

A function F(A, B, C) defined by three Boolean variables A, B and C when expressed as sum of products is given by

F = $$\overline A .\overline B .\overline C + \overline A .B.\overline C + A.\overline B .\overline C $$

where, $$\overline A $$, $$\overline B $$, and $$\overline C $$ are the complements of the respective variables. The product of sums (POS) form of the function F is

Consider matrix $$A = \left[ {\matrix{ k & {2k} \cr {{k^2} - k} & {{k^2}} \cr } } \right]$$ and

vector $$X = \left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right]$$.

The number of distinct real values of k for which the equation AX = 0 has infinitely many solutions is _______.

Let M be a real 4 $$ \times $$ 4 matrix. Consider the following statements :

S1: M has 4 linearly independent eigenvectors.

S2: M has 4 distinct eigenvalues.

S3: M is non-singular (invertible).

Which one among the following is TRUE?

The position of a particle y(t) is described by the differential equation :

$${{{d^2}y} \over {d{t^2}}} = - {{dy} \over {dt}} - {{5y} \over 4}$$.

The initial conditions are y(0) = 1 and $${\left. {{{dy} \over {dt}}} \right|_{t = 0}}$$ = 0.

The position (accurate to two decimal places) of the particle at t = $$\pi $$ is _______.

Let r = x² + y - z and z³ - xy + yz + y³ = 1. Assume that x and y are independent variables. At (x, y, z) = (2, -1, 1), the value (correct to two decimal places) of $${{\partial r} \over {\partial x}}$$ is ________________.

A curve passes through the point ($$x$$ = 1, $$y$$ = 0) and satisfies the differential equation

$${{dy} \over {dx}} = {{{x^2} + {y^2}} \over {2y}} + {y \over x}$$. The equation that describes the curve is

Let $$f\left( {x,y} \right) = {{a{x^2} + b{y^2}} \over {xy}}$$, where $$a$$ and $$b$$ are constants. If $${{\partial f} \over {\partial x}} = {{\partial f} \over {\partial y}}$$ at x = 1 and y = 2, then the relation between $$a$$ and $$b$$ is

The contour C given below is on the complex plane $$z = x + jy$$, where $$j = \sqrt { - 1} $$. The value of the integral $${1 \over {\pi j}}\oint\limits_C {{{dz} \over {{z^2} - 1}}} $$ is ________________.

Taylor series expansion of $$f\left( x \right) = \int\limits_0^x {{e^{ - \left( {{{{t^2}} \over 2}} \right)}}} dt$$ around 𝑥 = 0 has the form

f(x) = $${a_0} + {a_1}x + {a_2}{x^2} + ...$$

The coefficient $${a_2}$$ (correct to two decimal places) is equal to _______.

Let X₁ , X₂ , X₃ and X₄ be independent normal random variables with zero mean and unit variance. The probability that X₄ is the smallest among the four is _______.

Consider the network shown below with R₁ = 1 $$\Omega $$, R₂ = 2 $$\Omega $$ and R₃ = 3 $$\Omega $$. The network is connected to a constant voltage source of 11 V.

The magnitude of the current (in amperes, accurate to two decimal places) through the source is _______.

The ABCD matrix for a two-port network is defined by :

$$\left[ {\matrix{ {{V_1}} \cr {{I_1}} \cr } } \right] = \left[ {\matrix{ A & B \cr C & D \cr } } \right]\left[ {\matrix{ {{V_2}} \cr { - {I_2}} \cr } } \right]$$

The parameter B for the given two-port network (in ohms, correct to two decimal places) is _______.

For the circuit given in the figure, the magnitude of the loop current (in amperes, correct to three decimal places) 0.5 second after closing the switch is _______.

For the circuit given in the figure, the voltage V_C (in volts) across the capacitor is :

Let the input be u and the output be y of a system, and the other parameters are real constants. Identify which among the following systems is not a linear system:

Let 𝑥(𝑡) be a periodic function with period 𝑇 = 10. The Fourier series coefficients for this series are denoted by 𝑎_𝑘, that is

$$x\left( t \right) = \sum\limits_{k = - \infty }^\infty {{a_k}} {e^{jk{{2\pi } \over T}t}}$$

The same function 𝑥(𝑡) can also be considered as a periodic function with period T' = 40. Let b_k be the Fourier series coefficients when period is taken as T'. If $$\sum\limits_{k = - \infty }^\infty {\left| {{a_k}} \right|} = 16$$, then $$\sum\limits_{k = - \infty }^\infty {\left| {{b_k}} \right|} = 16$$ is equal to

Let X[k] = k + 1, 0 ≤ k ≤ 7 be 8-point DFT of a sequence x[n],

where X[k] = $$\sum\limits_{n = 0}^{N - 1} {x\left[ n \right]{e^{ - j2\pi nk/N}}} $$.

The value (correct to two decimal places) of $$\sum\limits_{n = 0}^3 {x\left[ {2n} \right]} $$ is ___________.

A discrete time all-pass system has two of its poles at 0.25$$\angle 0^\circ $$ and $$\angle 30^\circ $$. Which one of the following statements about the system is TRUE?

A cab was involved in a hit and run accident at night. You are given the following data about the cabs in the city and the accident.

(i) 85% of cabs in the city are green and the remaining cabs are blue.

(ii) A witness identified the cab involved in the accident as blue.

(iii) It is known that a witness can correctly identify the cab colour only 80% of the time.

Which of the following options is closest to the probability that the accident was caused by a blue cab?

If the number 715 ■ 423 is divisible by 3 (■ denotes the missing digit in the thousandths place), then the smallest whole number in the place of ■ is _______.

A 1.5 m tall person is standing at a distance of 3 m from a lamp post. The light from the lamp at the top of the post casts her shadow. The length of the shadow is twice her height. What is the height of the lamp post in meters?

"Even though there is a vast scope for its __________, tourism has remained a/an _________ area."

The words that best fill the blanks in the above sentence are

What is the value of $$1 + {1 \over 4} + {1 \over {16}} + {1 \over {64}} + {1 \over {256}} + ......$$?

"By giving him the last __________ of the cake, you will ensure lasting ________ in our house today."

The words that best fill the blanks in the above sentence are

A coastal region with unparalleled beauty is home to many species of animals. It is dotted with coral reefs and unspoilt white sandy beaches. It has remained inaccessible to tourists due to poor connectivity and lack of accommodation. A company has spotted the opportunity and is planning to develop a luxury resort with helicopter service to the nearest major city airport. Environmentalists are upset that this would lead to the region becoming crowded and polluted like any other major beach resorts.

Which one of the following statements can be logically inferred from the information given in the above paragraph?

Two alloys A and B contain gold and copper in the ratios of 2 : 3 and 3 : 7 by mass, respectively. Equal masses of alloys A and B are melted to make an alloy C. The ratio of gold to copper in alloy C is

The Cricket Board has long recognized John's potential as a leader of the team. However, his on-field temper has always been a matter of concern for them since his junior days. While this aggression has filled stadia with die-hard fans, it has taken a toll on his own batting. Until recently, it appeared that he found it difficult to convert his aggression into big scores. Over the past three seasons though, that picture of John has been replaced by a cerebral, calculative and successful batsman-captain. After many years, it appears that the team has finally found a complete captain.

Which of the following statements can be logically inferred from the above paragraph?

(i) Even as a junior cricketer. John was considered a good captain.

(ii) Finding a complete captain is a challenge.

(iii) Fans and the Cricket Board have differing views on what they want in a captain

(iv) Over the past three seasons John has accumulated big scores.

Leila aspires to buy a car worth Rs. 10,00,000 after 5 years. What is the minimum amount in Rupees that she should deposit now in a bank which offers 10% annual rate of interest, if the interest was compounded annually?