GATE EC 2006

Question 1
The rank of the matrix \begin{bmatrix} 1& 1& 1\\ 1& -1&0 \\ 1& 1& 1 \end{bmatrix} is
A
0
B
1
C
2
D
3
Engineering Mathematics   Linear Algebra
Question 1 Explanation: 
\begin{array}{l} R_{3} \rightarrow R_{1}-R_{3} \\ {\left[\begin{array}{lll} 1 & 1 & 1 \\ 1 & -1 & 0 \\ 0 & 0 & 0 \end{array}\right]}\\ \therefore \text{Rank}=2 \end{array}
Question 2
\bigtriangledown \times \bigtriangledown \times P, where P is a vector, is equal to
A
P \times \bigtriangledown \times P-\bigtriangledown ^{2}P
B
\bigtriangledown^{2} P+\bigtriangledown (\bigtriangledown \cdot P)
C
\bigtriangledown^{2} P+\bigtriangledown \times P
D
\bigtriangledown (\bigtriangledown \cdot P)-\bigtriangledown ^{2}P
Engineering Mathematics   Calculus
Question 2 Explanation: 
From vector triple product.
\begin{aligned} A \times(B \times C) &=B(A . C)-C(A . B) \\ A &=\nabla . B=\nabla, C=P \\ \nabla \times \nabla \times P &=(\nabla P)-P(\nabla \nabla)=\nabla(\nabla P)-\nabla^{2} P \end{aligned}
Question 3
\int \int (\bigtriangledown \times P)\cdot ds, where P is a vector, is equal to
A
\oint P\cdot dl
B
\oint \bigtriangledown \times \bigtriangledown \times P\cdot dl
C
\oint \bigtriangledown \times P\cdot dl
D
\int \int \int \bigtriangledown \cdot Pdv
Engineering Mathematics   Calculus
Question 3 Explanation: 
\iint(\Delta x P) d s=\oint P .d l (strokes Theorem)
Question 4
A probability density function is of the form
p(x)=Ke^{-\alpha |x|},x\in (-\infty ,\infty )
The value of K is
A
0.5
B
1
C
0.5\alpha
D
\alpha
Engineering Mathematics   Probability and Statistics
Question 4 Explanation: 
\begin{aligned} \int_{-\infty}^{\infty} p(x) d x&=1 \\ \int_{-\infty}^{\infty} K e^{-\alpha|x|} d x&=1 \\ \int_{-\infty}^{0} K e^{\mu x} d x+\int_{0}^{\infty} K e^{-u x}&=1\\ \Rightarrow \quad\frac{K}{\alpha}\left[e^{\alpha x}\right]_{-\infty}^{0}+\frac{K}{-\alpha}\left[e^{-\alpha x}\right]_{0}^{\infty}&=1 \\ \Rightarrow \quad \frac{K}{\alpha}+\frac{K}{\alpha}&=1\\ 2 K&=\alpha \\ \Rightarrow \quad K&=0.5 \alpha \end{aligned}
Question 5
A solution for the differential equation
\dot{x}(t)+2x(t)=\delta (t)
with initial condition x(0^-)=0 is
A
e^{-2t}u(t)
B
e^{2t}u(t)
C
e^{-t}u(t)
D
e^{t}u(t)
Engineering Mathematics   Differential Equations
Question 5 Explanation: 
\dot{x}(t)+2 x(t)=\delta(t)
Taking L.T. on both sides
\begin{aligned} s X(s)-x(0)+2 X(s)&=1 \\ X(s)[s+2]&=1 \\ X(s)&=\frac{1}{s+2} \\ x(t) &=e^{-2 t} u(t) \end{aligned}
Question 6
A low-pass filter having a frequency response H(j\omega )=A(\omega )e^{j\phi(\omega ) } does not produce any phase distortions if
A
A(\omega )=C\omega ^{2},\phi (\omega )=k\omega ^{3}
B
A(\omega )=C\omega ^{2},\phi (\omega )=k\omega
C
A(\omega )=C\omega ,\phi (\omega )=k\omega ^{2}
D
A(\omega )=C,\phi (\omega )=k\omega ^{-1}
Signals and Systems   Z-Transform
Question 6 Explanation: 
For distortionless transmission.
\frac{d \phi(\omega)}{d \omega}=\text { constant }
Phase response should be linear
\phi(\omega)=k \omega
Question 7
The values of voltage (V_{D}) across a tunnel-diode corresponding to peak and valley currents are V_{p} and V_{v} respectively. The range of tunnel-diode voltage V_{D} for which the slope of its I-V_{D} characteristics is negative would be
A
V_{D} \lt 0
B
0 \leq V_{D} \lt V_{p}
C
V_{p} \leq V_{D} \lt V_{v}
D
V_{D}\geq V_{v}
Electronic Devices   PN-Junction Diodes and Special Diodes
Question 7 Explanation: 


Question 8
The concentration of minority carriers in an extrinsic semiconductor under equilibrium is
A
Directly proportional to doping concentration
B
Inversely proportional to the doping concentration
C
Directly proportional to the intrinsic concentration
D
Inversely proportional to the intrinsic concentration
Electronic Devices   Basic Semiconductor Physics
Question 8 Explanation: 
n p=n_{i}^{2}
n_{i}= constant
For n -type p is minority carrier concentration
\begin{aligned} p&=\frac{n_{i}^{2}}{n} \\ p &\propto \frac{1}{n} \end{aligned}
Question 9
Under low level injection assumption, the injected minority carrier current for an extrinsic semiconductor is essentially the
A
Diffusion current
B
Drift current
C
Recombination current
D
Induced current
Electronic Devices   Basic Semiconductor Physics
Question 10
The phenomenon known as "Early Effect" in a bipolar transistor refers to a reduction of the effective base-width caused by
A
Electron - hole recombination at the base
B
The reverse biasing of the base - collector junction
C
The forward biasing of emitter-base junction
D
The early removal of stored base charge during saturation-to-cut off switching
Electronic Devices   BJT and FET Basics
There are 10 questions to complete.
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