GATE EC 2014 SET-3

Question 1
The maximum value of the function f(x)=ln(1+x)-x(\; where \; x>-1) occurs at x = ____.
A
0
B
1
C
0.5
D
-1
Engineering Mathematics   Calculus
Question 1 Explanation: 
\begin{aligned} f^{\prime}(x) &=\frac{1}{1+x}-1=0 \\ \frac{1-1-x}{1+x} &=0 \\ \frac{x}{1+x} &=0 \\ x &=0\\ f^{\prime \prime}(x)&=\frac{-1}{(1+x)^{2}} \\ f^{\prime}(0)&=-1 \lt 0 \end{aligned}
f(x) have maximum value at x=0
\begin{aligned} f(0)&=\ln (1+0)-0=0 \\ t_{\max }&=0 \end{aligned}
Question 2
Which ONE of the following is a linear non-homogeneous differential equation, where x and y are the independent and dependent variables respectively ?
A
\frac{dy}{dx}+xy=e^{-x}
B
\frac{dy}{dx}+xy=0
C
\frac{dy}{dx}+xy=e^{-y}
D
\frac{dy}{dx}+e^{-y}=0
Engineering Mathematics   Differential Equations
Question 2 Explanation: 
General form of linear differential equation
\frac{d y}{d x}+p y=\theta when P and \theta can be function of x
Only option (A) is in this form.
Question 3
Match the application to appropriate numerical method.
A
P1-M3, P2-M2, P3-M4, P4-M1
B
P1-M3, P2-M1, P3-M4, P4-M2
C
P1-M4, P2-M1, P3-M3, P4-M2
D
P1-M2, P2-M1, P3-M3, P4-M4
Engineering Mathematics   Numerical Methods
Question 4
An unbiased coin is tossed an infinite number of times. The probability that the fourth head appears at the tenth toss is
A
0.067
B
0.073
C
0.082
D
0.091
Engineering Mathematics   Probability and Statistics
Question 4 Explanation: 
It means 3-head appears in 1^{\text {st }} 9 trials.
Probability of getting exactly 3 head in 1^{\text {st }} 9 trials
\begin{aligned} &=\text{ Coefficient } p^{3}\text{ in }(4+p) 9 \\ \text{When, }&[\overline{p+q=1}] \\ &={ }^{9} C_{3} q^{6} p^{3}\\ \text{when, }\quad \rho&= \text{ probability of occure of head}\\ q &=\text { probability of occure of tail } \\ &={ }^{9} C_{3} \times\left(\frac{1}{2}\right)^{6}\left(\frac{1}{2}\right)^{3} \\ &={ }^{9} C_{3} \times\left(\frac{1}{2}\right)^{9} \end{aligned}
and in 10^{\text {th }} trial head must appears.
So required probability
\begin{aligned} &={ }^{9} C_{3}\left(\frac{1}{2}\right)^{9} \times \frac{1}{2} \\ &=\frac{9 \times 8 \times 7}{3 !} \times\left(\frac{1}{2}\right)^{10}=\frac{84}{1024}=0.082 \end{aligned}
Question 5
If z = xy ln(xy), then
A
x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=0
B
y\frac{\partial z}{\partial x}=x\frac{\partial z}{\partial y}
C
x\frac{\partial z}{\partial x}=y\frac{\partial z}{\partial y}
D
y\frac{\partial z}{\partial x}+x\frac{\partial z}{\partial y}=0
Engineering Mathematics   Calculus
Question 5 Explanation: 
\begin{aligned} \frac{\partial z}{\partial x}&=yln(x y)+\frac{x y}{x y} y\\ \frac{\partial z}{\partial x}&=y[\ln (x y)+1] &\ldots(i)\\ \frac{\partial z}{\partial y}&=x \ln (x y)+\frac{x y}{x y} \times x\\ \frac{\partial z}{\partial y}&=x[\ln (x y)+1]\\ \text{Here}\quad x\frac{\partial z}{\partial x}&=y \frac{\partial z}{\partial y} \end{aligned}
Question 6
A series RC circuit is connected to a DC voltage source at time t = 0. The relation between the source voltage V_{S}, the resistance R, the capacitance C, and the current i(t) is given below :
V_{S}=Ri(t)+\frac{1}{C}\int_{0}^{t}i(u)du
Which one of the following represents the current i(t) ?
A
A
B
B
C
C
D
D
Network Theory   Sinusoidal Steady State Analysis
Question 6 Explanation: 
Given that:
V_{s}=R i(t)+\frac{1}{C} \int_{0}^{t} i(t) d t\qquad\ldots(i)
Using Laplace transform,
\begin{aligned} \text { WS) }&=R I(s)+\frac{1}{C s} I(s) &\ldots(ii)\\ \text { or } I(s)&=\frac{V(s)}{\left(R+\frac{1}{C S}\right)} &\ldots(iii)\\ \text { For, } V(s)&=\frac{1}{s}&\ldots(iv) \end{aligned}
From equation (iii) and (iv),
I(s)=\frac{C}{(R C s+1)}=\frac{1}{R\left(s+\frac{1}{R C}\right)}\qquad\ldots(v)
Using inverse Laplace transform in equation (v), we get,
i(t)=\frac{1}{R} e^{-t / R C}
Thus, option (A) is correct.
Question 7
In the figure shown, the value of the current I (in Amperes) is_____.
A
0
B
0.25
C
0.5
D
1
Network Theory   Network Theorems
Question 7 Explanation: 


Using super position theorem, when 5 V source acting alone, we get


I_{1}=\frac{V}{R_{\mathrm{eq}}}=\frac{5}{10+5+5}=\frac{1}{4} \mathrm{A}\quad \ldots(i)
When 1 A source acting alone, we get


I_{2}=\frac{1 \times 5}{5+10+5}=\frac{5}{20}=\frac{1}{4} \mathrm{A}\quad \ldots(ii)
Therefore,
I=I_{1}+I_{2}=\frac{1}{2} A=0.5 \mathrm{A}
Question 8
In MOSFET fabrication, the channel length is defined during the process of
A
isolation oxide growth
B
channel stop implantation
C
poly-silicon gate patterning
D
lithography step leading to the contact pads
Electronic Devices   IC Fabrication
Question 8 Explanation: 
Channel length is defined during the poly-silicon gate pattering.
Question 9
A thin P-type silicon sample is uniformly illuminated with light which generates excess carriers. The recombination rate is directly proportional to
A
the minority carrier mobility
B
the minority carrier recombination lifetime
C
the majority carrier concentration
D
the excess minority carrier concentration
Electronic Devices   Basic Semiconductor Physics
Question 10
At T = 300 K, the hole mobility of a semiconductor \mu _{P}=500 cm^{2}/V-s and \frac{KT}{q}=26mV. The hole diffusion constant D_{P} \; in \; cm^{2}/s is _____.
A
11
B
12
C
13
D
14
Electronic Devices   Basic Semiconductor Physics
Question 10 Explanation: 
\begin{array}{l} \frac{D_{p}}{\mu_{p}}=v_{T} \\ D_{p}=\mu_{p} V_{T}=500 \times 26 \times 10^{-3} \\ D_{p}=13 \mathrm{cm}^{2} / \mathrm{s} \end{array}
There are 10 questions to complete.
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