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QUESTION: 1

**Q. 1 – Q. 5 carry one mark each.**

Statement: You can always give me a ring whenever you need.

Which one of the following is the best inference from the above statement?

Solution:

QUESTION: 2

Complete the sentence:

Dare ____________ mistakes.

Solution:

QUESTION: 3

Choose the grammatically CORRECT sentence:

Solution:

QUESTION: 4

They were requested not to quarrel with others.

Which one of the following options is the closest in meaning to the word quarrel?

Solution:

QUESTION: 5

In the summer of 2012, in New Delhi, the mean temperature of Monday to Wednesday was 41ºC and of Tuesday to Thursday was 43ºC. If the temperature on Thursday was 15% higher than that of Monday, then the temperature in ºC on Thursday was

Solution:

Let the temperature of Monday be T_{M}

Sum of temperatures of Tuesday and Wednesday = T and

Temperature of Thursday =T_{Th}

QUESTION: 6

**Q. No. 6 – 10 Carry Two Marks Each**

Q. Find the sum to n terms of the series 10 + 84 + 734 + …

Solution:

QUESTION: 7

The set of values of p for which the roots of the equation 3x^{2} + 2x + p(p − 1) = 0 are of opposite sign is

Solution:

QUESTION: 8

A car travels 8 km in the first quarter of an hour, 6 km in the second quarter and 16km in the third quarter. The average speed of the car in km per hour over the entire journey is

Solution:

QUESTION: 9

What is the chance that a leap year, selected at random, will contain 53 Sundays?

Solution:

QUESTION: 10

**Statement:** There were different streams of freedom movements in colonial India carried out by the moderates, liberals, radicals, socialists, and so on.

Which one of the following is the best inference from the above statement?

Solution:

QUESTION: 11

**Q. No. 11 – 35 Carry One Mark Each**

The dimension of the null space of the matrix is

Solution:

QUESTION: 12

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

Solution:

QUESTION: 13

Two systems with impulse responses h_{1}(t) and h_{2}(t) are connected in cascade. Then the overall impulse response of the cascaded system is given by

Solution:

QUESTION: 14

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

Solution:

QUESTION: 15

For a vector E, which one of the following statements is NOT TRUE?

Solution:

QUESTION: 16

For a periodic signal v(t) = 30sin100t + 10cos300t + , the fundamental frequency in rad/s is

Solution:

QUESTION: 17

In the transistor circuit as shown below, the value of resistance R_{E} in kW is approximately,

Solution:

QUESTION: 18

A source v_{s}(t) = V cos100πt has an internal impedance of (4 + j3) ? . If a purely resistive load connected to this source has to extract the maximum power out of the source, its value in ? should be

Solution:

QUESTION: 19

Which of the following statements is NOT TRUE for a continuous time causal and stable LTI system?

Solution:

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

[Refer Laplace transform].

QUESTION: 20

The operational amplifier shown in the circuit below has a slew rate of 0.8V /μs . The input signal is 0.25sinωt . The maximum frequency of input in kHz for which there is no distortion in the output is

Solution:

QUESTION: 21

Assuming zero initial condition, the response y(t) of the system given below to a unit step input u(t) is

Solution:

QUESTION: 22

The transfer function of the circuit shown below is

Solution:

QUESTION: 23

The type of partial differential equation

Solution:

QUESTION: 24

The discrete-time transfer function

Solution:

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

QUESTION: 25

Match the following biomedical instrumentation techniques with their application.

Solution:

QUESTION: 26

A continuous random variable X has a probability density function

Solution:

QUESTION: 27

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

Solution:

Given: f_{m} = 5kHz

According to sampling frequency

f_{s} __>__ 2f_{m} ; f_{s} __>__ 10 KHz

So, only in option (a) it is less than 10KHz ie., (5KHz)

QUESTION: 28

A differential pressure transmitter of a flow meter using venture tube reads 2.5 ×10^{5} Pa for a flow rate of 0.5 m^{3}/s. The approximate flow rate in m^{3}/s for a differential pressure of 0.9 ×10^{5} Pa is

Solution:

QUESTION: 29

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

Solution:

QUESTION: 30

The impulse response of a system is h(t) = tu(t) . For an input u(t − 1) , the output is

Solution:

For LTI system, if input gets delayed by one unit, output will also get delayed by one unit.

QUESTION: 31

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

Solution:

Above expression shown that if R_{a} , R_{b} & R_{c }is scaled by k, RA, RB & RC is scaled by k only.

QUESTION: 32

An accelerometer has input range of 0-10g, natural frequency 30Hz and mass 0.001kg. The range of the secondary displacement transducer in mm required to cover the input range is

Solution:

QUESTION: 33

In the circuit shown below what is the output voltage V(_{out}) in Volts if a silicon transistor Q and an ideal op-amp are used?

Solution:

QUESTION: 34

In the feedback network shown below, if the feedback factor k is increased, then the

Solution:

QUESTION: 35

The Bode plot of a transfer function G(s) is shown in the figure below.

The gain is 32dB and -8dB at 1 rad/s and 10 rad/s respectively. The phase is negative for all ω. Then G(s) is

Solution:

Any two paints on same line segment of Bode plot satisfies the equation of straight line.

QUESTION: 36

**Q. No. 36 – 65 Carry Two Mark Each**

Q. While numerically solving the differential equation using Euler's predictor– corrector (improved Euler – Cauchy) method with a step size of 0.2, the value of y after the first step is

Solution:

= 0.96, is the value of y after first step, using Euler’s predictor – corrector method.

QUESTION: 37

One pair of eigen vectors corresponding to the two eigen values of the matrix is

Solution:

QUESTION: 38

The digital circuit shown below uses two negative edge–triggered D–flip–flops. Assuming initial condition of Q1 and Q0 as zero, the output Q1Q0 of this circuit is

Solution:

QUESTION: 39

Considering the transformer to be ideal, the transmission parameter 'A' of the 2–port network shown in the figure below is

Solution:

QUESTION: 40

The following arrangement consists of an ideal transformer and an attenuator, which attenuates by a factor of 0.8. An ac voltage V_{WX1} = 100V is applied across WX to get an open circuit voltage V_{YZ1} across YZ. Next, an ac voltage V_{YZ2} = 100V is applied across YZ to get an open circuit voltage V_{WX2} across WX. Then, V_{YZ1} / V_{WX1 , }V_{WX2} / _{ }V_{YZ2 }are respectively

Solution:

Since potentiometer and transformer are bilateral elements. Hence

QUESTION: 41

The open–loop transfer function of a dc motor is given as . When connected in feedback as shown below, the approximate value of a 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

Solution:

QUESTION: 42

Two magnetically uncoupled inductive coils have Q factor q_{1} and q_{2} at the chosen operating frequency. Their respective resistances are R_{1} and R_{2} . When "connected in series, the effective Q factor of the series combination at the same operating frequency is

Solution:

QUESTION: 43

For the circuit shown below, the knee current of the ideal Zener diode is 10 mA. To maintain 5 V across R_{L} , the minimum value of the load resistor R_{L} in ? and the minimum power rating of the Zener diode in mW, respectively, are

Solution:

QUESTION: 44

The impulse response of a continuous time system is given by h(t) = δ (t − 1) + δ (t − 3) . The value of the step response at t = 2 is

Solution:

QUESTION: 45

Signals from fifteen thermocouples are multiplexed and each one is sampled once per second with a16–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 1200 Hz as 1 and 960 Hz as 0. The minimum

band allocation required for faithful reproduction of the signal by the FSK receiver without considering noise is

Solution:

Data rate from ADC is 16 x 15 bits/second=240 bits/second

The bandwidth required for transmitting 240 bits/second = 120 Hz. (Half of bit rate).

Now when each pulse representing ‘1’s and ‘0’ s get FSK modulator with 960 Hz and 1200

Base band signal before modulation

The spectrum of signal after modulation

Thus all frequency from 840 to 1320 is required for allocation

QUESTION: 46

Three capacitors C_{1} ,C_{2} and C_{3} whose values are 10μF,5μF and 2μF respectively, have breakdown voltages of 10V, 5V, and 2V respectively. For the interconnection shown below, the maximum safe voltage in Volts that can be applied across the combination, and the corresponding total charge in μC stored in the effective capacitance across the terminals are, respectively

Solution:

QUESTION: 47

The maximum value of the solution y(t) of the differential equation

Solution:

QUESTION: 48

The Laplace Transform representation of the triangular pulse shown below is

Solution:

QUESTION: 49

In the circuit shown below, if the source voltage Volts, then the Thevenin's equivalent voltage in Volts as seen by the load resistance R_{L} is

Solution:

QUESTION: 50

A signal is supplied to a filter circuit (shown below) made up of ideal op–amps. The least attenuated frequency component in the output will be

Solution:

Least attenuated signal frequency is 0Hz

QUESTION: 51

The signal flow graph for a system is given below. The transfer function Y(s) / U(s) for this system is given as

Solution:

By using Mason’s gain formula

QUESTION: 52

A voltage 1000 sin ωt Volts is applied across YZ. Assuming ideal diodes, the voltage measured across WX in Volts, is

Solution:

QUESTION: 53

In the circuit shown below the op–amps are ideal. Then V_{out} in volts is

Solution:

QUESTION: 54

In the circuit shown below, Q_{1} has negligible collector–to–emitter saturation voltage and the diode drops negligible voltage across if under forward bias. If V_{cc} is + 5V , X and Y are digital signals with 0 V as logic 0 and V_{cc} as logic 1, then the Boolean expression for Z is

Solution:

(B)

QUESTION: 55

The circuit below incorporates a permanent magnet moving coil milli–ammeter of range 1 mA having a series resistance of 10k?. Assuming constant diode forward resistance of 50W a forward diode drop of 0.7 V and infinite reverse diode resistance for each diode, the reading of the meter in mA is

Solution:

QUESTION: 56

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 μW

Photo-detector output without solution and with stray light is 600 μW

Photo-detector output with solution and with stray light is 200 μW

The percent error in computing absorption coefficient due to stray light is

Solution:

QUESTION: 57

Two ammeters 1 A and 2 A measure the same current and provide readings I_{1 }and I_{2} respectively. The ammeter errors can be characterized as independent zero mean Gaussian random variables of standard deviations σ_{1} and σ_{2} respectively. The value of the current is computed as:

. The value of V_{1} which gives the lowest standard deviation of I is

Solution:

QUESTION: 58

**Common Data for Questions: 58 & 59**

A tungsten wire used in a constant current hot wire anemometer has the following properties:

Resistance at 0ºC is 10?, Surface area is 10^{-4}m^{2} . Linear temperature coefficient of resistance of the tungsten wire is 4.8 ×10^{−3}/ºC , Convective heat transfer coefficient is 25.2W/m^{2} /ºC , flowing air temperature is 30ºC, wire current is 100mA, mass specific heat product is 2.5 x 10^{5} J/ºC

**Q. The thermal time constant of the hot wire under flowing air condition in ms is**

Solution:

QUESTION: 59

**Common Data for Questions: 58 & 59**

A tungsten wire used in a constant current hot wire anemometer has the following properties:

Resistance at 0ºC is 10?, Surface area is 10^{-4}m^{2} . Linear temperature coefficient of resistance of the tungsten wire is 4.8 ×10^{−3}/ºC , Convective heat transfer coefficient is 25.2W/m^{2} /ºC , flowing air temperature is 30ºC, wire current is 100mA, mass specific heat product is 2.5 x 10^{5} J/ºC

**Q. At steady state, the resistance of the wire in ohms is**

Solution:

QUESTION: 60

**Common Data for Questions: 60 & 61**

A piezo electric force sensor, connected by a cable to a voltage amplifier has the following parameters:

Crystal properties: Stiffness 10^{6} N /m, Damping ratio 0.01, natural frequency 10^{5} rad/s , Force-to-Charge sensitivity 10^{-9} C / N, Capacitance 10^{-9} F with its loss angle assumed negligible.

Cable properties : capacitance 2×10^{-9} F with its resistance assumed negligable Amplifier properties: Input impedance 1M?, Bandwidth 1MHz, Gain 3

**Q. 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 1.05 approximately is:**

Solution:

QUESTION: 61

**Common Data for Questions: 60 & 61**

A piezo electric force sensor, connected by a cable to a voltage amplifier has the following parameters:

Crystal properties: Stiffness 10^{6} N /m, Damping ratio 0.01, natural frequency 10^{5} rad/s , Force-to-Charge sensitivity 10^{-9} C / N, Capacitance 10^{-9} F with its loss angle assumed negligible.

Cable properties : capacitance 2×10^{-9} F with its resistance assumed negligable Amplifier properties: Input impedance 1M?, Bandwidth 1MHz, Gain 3

Q. 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:

Solution:

QUESTION: 62

**Linked Answer Questions: Q.62 to Q.65 Carry Two Marks Each**

**Statement for Linked Answer Questions: 62 & 63**

Consider a plant with transfer function . Let K_{u} and T_{u} 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}

**Q. The values of K _{u} and T_{u} respectively are:**

Solution:

QUESTION: 63

**Linked Answer Questions: Q.62 to Q.65 **

**Carry Two Marks Each Statement for Linked Answer Questions: 62 & 63 **

Consider a plant with transfer function G(s) = 1 / (s + 1)^{3}. Let K_{u} and T_{u} 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 = 0.5Ku

**Q. The gain of the transfer function between the plant output and an additive load disturbance input of frequency 2π / T _{u }in closed loop with a P-controller designed according to the Ziegler Nichols tuning rule as given above is**

Solution:

QUESTION: 64

**Statement for Linked Answer Questions: 64 & 65**

A differential amplifier with signal terminals X, Y, Z is connected as shown in figure(a) below for CMRR measurement where the differential amplifier has an additional constant offset voltage in the output. The observations obtained are:

when V_{i} = 2V, V_{o} = 3mV and when V_{i} = 3V, V_{o} = 4mV

**Q. Assuming its differential gain to be 10 and the op-amp to be otherwise ideal, the CMRR is**

Solution:

QUESTION: 65

**Statement for Linked Answer Questions: 64 & 65**

A differential amplifier with signal terminals X, Y, Z is connected as shown in figure(a) below for CMRR measurement where the differential amplifier has an additional constant offset voltage in the output. The observations obtained are:

when V_{i} = 2V, V_{o} = 3mV and when V_{i} = 3V, V_{o} = 4mV

**Q. **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 R by ±1%. 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 mV varies approximately from

Solution:

### Instrumentation (IN) 2013 GATE Paper with solution

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