# 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.