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Recent questions in Engineering Mathematics
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1
GATE ECE 2021 | Question: 1
The vector function $F\left ( r \right )=-x\hat{i}+y\hat{j}$ is defined over a circular arc $C$ shown in the figure. The line integral of $\int _{C} F\left ( r \right ).dr$ is $\frac{1}{2}$ $\frac{1}{4}$ $\frac{1}{6}$ $\frac{1}{3}$
Arjun
asked
in
Vector Analysis
Feb 20, 2021
by
Arjun
6.0k
points
375
views
gateec-2021
vector-analysis
vector-in-planes
0
votes
0
answers
2
GATE ECE 2021 | Question: 2
Consider the differential equation given below. $\frac{dy}{dx}+\frac{x}{1-x^{2}}y=x\sqrt{y}$ The integrating factor of the differential equation is $\left ( 1-x^{2} \right )^{-3/4}$ $\left ( 1-x^{2} \right )^{-1/4}$ $\left ( 1-x^{2} \right )^{-3/2}$ $\left ( 1-x^{2} \right )^{-1/2}$
Arjun
asked
in
Differential Equations
Feb 20, 2021
by
Arjun
6.0k
points
153
views
gateec-2021
differential-equations
first-order-differential-equation
0
votes
0
answers
3
GATE ECE 2021 | Question: 3
Two continuous random variables $X$ and $Y$ are related as $Y=2X+3$ Let $\sigma ^{2}_{X}$ and $\sigma ^{2}_{Y}$denote the variances of $X$ and $Y$, respectively. The variances are related as $\sigma ^{2}_{Y}=2 \sigma ^{2}_{X}$ $\sigma ^{2}_{Y}=4 \sigma ^{2}_{X}$ $\sigma ^{2}_{Y}=5 \sigma ^{2}_{X}$ $\sigma ^{2}_{Y}=25 \sigma ^{2}_{X}$
Arjun
asked
in
Probability and Statistics
Feb 20, 2021
by
Arjun
6.0k
points
114
views
gateec-2021
probability-and-statistics
random-variable
variance
0
votes
0
answers
4
GATE ECE 2021 | Question: 16
If the vectors $(1.0,\:-1.0,\:2.0)$, $(7.0,\:3.0,\:x)$ and $(2.0,\:3.0,\:1.0)$ in $\mathbb{R}^{3}$ are linearly dependent, the value of $x$ is __________
Arjun
asked
in
Vector Analysis
Feb 20, 2021
by
Arjun
6.0k
points
151
views
gateec-2021
numerical-answers
vector-analysis
vector-in-planes
0
votes
0
answers
5
GATE ECE 2021 | Question: 17
Consider the vector field $F\:=\:a_{x}\left ( 4y-c_{1}z \right )+a_y\left ( 4x + 2z\right )+a_{z}\left ( 2y +z\right )$ in a rectangular coordinate system $(x,y,z)$ with unit vectors $a_{x},\:a_{y}$ and $a_{z}$. If the field $F$ is irrotational (conservative), then the constant $c_{1}$ (in integer) is _________________
Arjun
asked
in
Vector Analysis
Feb 20, 2021
by
Arjun
6.0k
points
128
views
gateec-2021
numerical-answers
vector-analysis
vector-in-planes
0
votes
0
answers
6
GATE ECE 2021 | Question: 26
Consider the integral $\oint _{c}\frac{sin\left ( x \right )}{x^{2}\left ( x^{2}+4 \right )}dx$ where $C$ is a counter-clockwise oriented circle defined as $\left | x-i \right |=2$. The value of the integral is $-\frac{\pi }{8}\sin\left ( 2i \right )$ $\frac{\pi }{8}\sin\left ( 2i \right )$ $-\frac{\pi }{4}\sin\left ( 2i \right )$ $\frac{\pi }{4}\sin\left ( 2i \right )$
Arjun
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in
Complex Analysis
Feb 20, 2021
by
Arjun
6.0k
points
215
views
gateec-2021
complex-analysis
0
votes
0
answers
7
GATE ECE 2021 | Question: 27
A box contains the following three coins. A fair coin with head on one face and tail on the other face. A coin with heads on both the faces. A coin with tails on both the faces. A coin is picked randomly from the box and tossed. Out of the two remaining coins in the box, one ... getting a head in the second toss is $\frac{2}{5}$ $\frac{1}{3}$ $\frac{1}{2}$ $\frac{2}{3}$
Arjun
asked
in
Probability and Statistics
Feb 20, 2021
by
Arjun
6.0k
points
224
views
gateec-2021
probability-and-statistics
probability
conditional-probability
0
votes
0
answers
8
GATE ECE 2021 | Question: 36
A real $2\times2$ non-singular matrix $A$ with repeated eigenvalue is given as $A=\begin{bmatrix} x & -3.0\\ 3.0 & 4.0 \end{bmatrix}$ where $x$ is a real positive number. The value of $x$ (rounded off to one decimal place) is ________________
Arjun
asked
in
Linear Algebra
Feb 20, 2021
by
Arjun
6.0k
points
98
views
gateec-2021
numerical-answers
linear-algebra
eigen-values
0
votes
0
answers
9
GATE ECE 2021 | Question: 37
For a vector field $D=\rho\cos^{2}\:\varphi \:a_{\rho }+z^{2}\sin^{2}\:\varphi \:a_{\varphi }$ in a cylindrical coordinate system $\left ( \rho ,\varphi ,z \right )$ with unit vectors $a_{\rho },a_{\varphi }$ and $a_{z}$, the ... $\left ( \rho =3, 0\leq z\leq 2 \right )$ (rounded off to two decimal places) is ________________
Arjun
asked
in
Vector Analysis
Feb 20, 2021
by
Arjun
6.0k
points
58
views
gateec-2021
numerical-answers
vector-analysis
0
votes
0
answers
10
GATE ECE 2020 | Question: 1
If $v_{1},v_{2}, \dots ,v_{6}$ are six vectors in $\mathbb{R}^{4}$ , which one of the following statements is $\text{FALSE}$? It is not necessary that these vectors span $\mathbb{R}^{4}$. These vectors are not linearly independent. Any four of these vectors form a basis ... $\mathbb{R}^{4}$ , then it forms a basis for $\mathbb{R}^{4}$.
go_editor
asked
in
Vector Analysis
Feb 13, 2020
by
go_editor
1.9k
points
281
views
gate2020-ec
vector-analysis
1
vote
0
answers
11
GATE ECE 2020 | Question: 2
For a vector field $\overrightarrow{A}$, which one of the following is $\text{FALSE}$? $\overrightarrow{A}$ is solenoidal if $\triangledown \cdot \overrightarrow{A}=0.$ $\triangledown \times \overrightarrow{A}$ ...
go_editor
asked
in
Vector Analysis
Feb 13, 2020
by
go_editor
1.9k
points
281
views
gate2020-ec
vector-analysis
0
votes
0
answers
12
GATE ECE 2020 | Question: 3
The partial derivative of the function $f(x, y, z) = e^{1-x\cos y} + xze^{-1/(1+y^{2})}$ with respect to $x$ at the point $(1,0,e)$ is $-1$ $0$ $1 \\$ $\dfrac{1}{e}$
go_editor
asked
in
Calculus
Feb 13, 2020
by
go_editor
1.9k
points
167
views
gate2020-ec
calculus
derivatives
partial-derivatives
0
votes
0
answers
13
GATE ECE 2020 | Question: 4
The general solution of $\dfrac{\mathrm{d^{2}} y}{\mathrm{d} x^{2}}-6\dfrac{\mathrm{d} y}{\mathrm{d} x}+9y=0$ is $y=C_{1}e^{3x}+C_{2}e^{-3x}$ $y=(C_{1}+C_{2}x)e^{-3x}$ $y=(C_{1}+C_{2}x)e^{3x}$ $y=C_{1}e^{3x}$
go_editor
asked
in
Differential Equations
Feb 13, 2020
by
go_editor
1.9k
points
120
views
gate2020-ec
differential-equations
second-order-differential-equation
0
votes
0
answers
14
GATE ECE 2020 | Question: 24
The random variable $Y=\int_{-\infty }^{\infty }W\left ( t \right )\phi \left ( t \right )dt, \text{ where } \phi \left ( t \right )=\begin{cases} 1; & 5\leq t\leq 7 &\\ 0; & \text{otherwise} \end{cases}$ and $W(t)$ is ... noise process with two-sided power spectral density $S_{W}\left ( f \right )=3 W/Hz$, for all $f$. The variance of $Y$ is ________.
go_editor
asked
in
Vector Analysis
Feb 13, 2020
by
go_editor
1.9k
points
109
views
gate2020-ec
numerical-answers
vector-analysis
gausss-theorem
1
vote
1
answer
15
GATE ECE 2020 | Question: 25
The two sides of a fair coin are labelled as $0$ to $1$. The coin is tossed two times independently. Let $M$ and $N$ denote the labels corresponding to the outcomes of those tosses. For a random variable $X$, defined as $X = \text{min}(M, N)$, the expected value $E(X)$ (rounded off to two decimal places) is ___________.
go_editor
asked
in
Probability and Statistics
Feb 13, 2020
by
go_editor
1.9k
points
187
views
gate2020-ec
numerical-answers
probability-and-statistics
probability
independent-events
random-variable
expectation
0
votes
0
answers
16
GATE ECE 2020 | Question: 26
Consider the following system of linear equations. $\begin{array}{llll} x_{1}+2x_{2}=b_{1} ; & 2x_{1}+4x_{2}=b_{2}; & 3x_{1}+7x_{2}=b_{3} ; & 3x_{1}+9x_{2}=b_{4} \end{array}$ Which one of the following conditions ensures that a solution exists for the above system? ... $b_{2}=2b_{1}$ and $3b_{1}-6b_{3}+b_{4}=0$ $b_{3}=2b_{1}$ and $3b_{1}-6b_{3}+b_{4}=0$
go_editor
asked
in
Linear Algebra
Feb 13, 2020
by
go_editor
1.9k
points
67
views
gate2020-ec
linear-algebra
system-of-equations
0
votes
0
answers
17
GATE ECE 2020 | Question: 27
Which one of the following options contains two solutions of the differential equation $\dfrac{\mathrm{d} y}{\mathrm{d} x}=\left ( y-1 \right )x?$ $\ln\mid y-1 \mid=0.5x^{2}+C$ and $y=1$ $\ln\mid y-1 \mid=2x^{2}+C$ and $y=1$ $\ln\mid y-1 \mid=0.5x^{2}+C$ and $y=-1$ $\ln\mid y-1 \mid=2x^{2}+C$ and $y=-1$
go_editor
asked
in
Differential Equations
Feb 13, 2020
by
go_editor
1.9k
points
70
views
gate2020-ec
differential-equations
0
votes
0
answers
18
GATE ECE 2020 | Question: 51
For the solid $S$ shown below, the value of $\underset{S}{\iiint} xdxdydz$ (rounded off to two decimal places) is _______________.
go_editor
asked
in
Calculus
Feb 13, 2020
by
go_editor
1.9k
points
125
views
gate2020-ec
numerical-answers
calculus
definite-integrals
0
votes
0
answers
19
GATE ECE 2020 | Question: 54
$X$ is a random variable with uniform probability density function in the interval $[-2,\:10]$. For $Y=2X-6$, the conditional probability $P\left ( Y\leq 7\mid X\geq 5 \right )$ (rounded off to three decimal places) is __________.
go_editor
asked
in
Probability and Statistics
Feb 13, 2020
by
go_editor
1.9k
points
79
views
gate2020-ec
numerical-answers
probability-and-statistics
probability
probability-density-function
uniform-distribution
0
votes
0
answers
20
GATE2016 EC-3: 3
The probability of getting a “head” in a single toss of a biased coin is 0.3. The coin is tossed repeatedly till a “head” is obtained. If the tosses are independent, then the probability of getting “head” for the first time in the fifth toss is _________
KUSHAGRA गुप्ता
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in
Probability and Statistics
Nov 21, 2019
by
KUSHAGRA गुप्ता
270
points
62
views
gate2016-ec
probability
1
vote
0
answers
21
GATE2009 EC: 11
A fair coin is tossed 10 times. What is the probability that ONLY the first two tosses will yield heads. (A) $\left(\dfrac{1}{2}\right)^{2}$ (B) $^{10}C_2\left(\dfrac{1}{2}\right)^{2}$ (C) $\left(\dfrac{1}{2}\right)^{10}$ (D) $^{10}C_2\left(\dfrac{1}{2}\right)^{10}$
KUSHAGRA गुप्ता
asked
in
Probability and Statistics
Nov 21, 2019
by
KUSHAGRA गुप्ता
270
points
51
views
gate2009-ec
probability
2
votes
0
answers
22
GATE ECE 2019 | Question: 1
Which one of the following functions is analytic over the entire complex plane? $\ln(z)$ $e^{1/z}$ $\frac{1}{1-z}$ $\cos(z)$
Arjun
asked
in
Complex Analysis
Feb 12, 2019
by
Arjun
6.0k
points
270
views
gate2019-ec
complex-analysis
0
votes
0
answers
23
GATE ECE 2019 | Question: 2
The families of curves represented by the solution of the equation $\frac{dy}{dx}=\: – \left(\frac{x}{y} \right)^n$ for $n=-1$ and $n= +1,$ respectively, are Parabolas and Circles Circles and Hyperbolas Hyperbolas and Circles Hyperbolas and Parabolas
Arjun
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in
Differential Equations
Feb 12, 2019
by
Arjun
6.0k
points
109
views
gate2019-ec
differential-equations
0
votes
0
answers
24
GATE ECE 2019 | Question: 16
The value of the contour integral $\frac{1}{2\pi j} \oint\left(z+\frac{1}{z}\right)^{2}dz$ evaluated over the unit circle $\mid z \mid=1$ is_______.
Arjun
asked
in
Calculus
Feb 12, 2019
by
Arjun
6.0k
points
81
views
gate2019-ec
numerical-answers
calculus
integrals
0
votes
1
answer
25
GATE ECE 2019 | Question: 17
The number of distinct eigenvalues of the matrix $A=\begin{bmatrix} 2&2&3&3\\0&1&1&1\\0&0&3&3\\0&0&0&2 \end{bmatrix}$ is equal to ____________.
Arjun
asked
in
Linear Algebra
Feb 12, 2019
by
Arjun
6.0k
points
118
views
gate2019-ec
numerical-answers
linear-algebra
matrices
eigen-values
0
votes
0
answers
26
GATE ECE 2019 | Question: 18
If $X$ and $Y$ are random variables such that $E\left[2X+Y\right]=0$ and $E\left[X+2Y\right]=33$, then $E\left[X\right]+E\left[Y\right]=$___________.
Arjun
asked
in
Probability and Statistics
Feb 12, 2019
by
Arjun
6.0k
points
84
views
gate2019-ec
numerical-answers
probability-and-statistics
probability
random-variable
expectation
0
votes
0
answers
27
GATE ECE 2019 | Question: 19
The value of the integral $ \displaystyle{}\int_{0}^{\pi} \int_{y}^{\pi}\dfrac{\sin x}{x}dx\: dy ,$ is equal to __________.
Arjun
asked
in
Calculus
Feb 12, 2019
by
Arjun
6.0k
points
116
views
gate2019-ec
numerical-answers
calculus
definite-integrals
0
votes
0
answers
28
GATE ECE 2019 | Question: 20
Let $Z$ be an exponential random variable with mean $1$. That is, the cumulative distribution function of $Z$ is given by $F_{Z}(x)= \left\{\begin{matrix} 1-e^{-x}& \text{if}\: x \geq 0 \\ 0& \text{if}\: x< 0 \end{matrix}\right.$ Then $Pr\left(Z>2 \mid Z>1\right),$ rounded off to two decimal places, is equal to ___________.
Arjun
asked
in
Probability and Statistics
Feb 12, 2019
by
Arjun
6.0k
points
170
views
gate2019-ec
numerical-answers
probability-and-statistics
probability
random-variable
0
votes
0
answers
29
GATE ECE 2019 | Question: 26
Consider a differentiable function $f(x)$ on the set of real numbers, such that $f(-1)=0$ and $ \mid f’(x) \mid \leq 2.$ Given these conditions, which one of the following inequalities is necessarily true for all $x \in[-2,2]?$ $f(x)\leq \frac{1}{2} \mid x+1 \mid$ $f(x)\leq 2 \mid x+1 \mid $ $f(x)\leq \frac{1}{2} \mid x \mid$ $f(x)\leq 2 \mid x \mid$
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Calculus
Feb 12, 2019
by
Arjun
6.0k
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121
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gate2019-ec
calculus
maxima-minima
0
votes
0
answers
30
GATE ECE 2019 | Question: 27
Consider the line integral $\int_{c} (xdy-ydx)$ the integral being taken in a counterclockwise direction over the closed curve $C$ that forms the boundary of the region $R$ shown in the figure below. The region $R$ is the area enclosed by the union of a $2 \times 3$ ... circle of radius $1$. The line integral evaluates to $6+ \dfrac{\pi}{2}$ $8+\pi$ $12+\pi$ $16+2\pi$
Arjun
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in
Calculus
Feb 12, 2019
by
Arjun
6.0k
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231
views
gate2019-ec
integrals
calculus
0
votes
0
answers
31
GATE ECE 2019 | Question: 43
Consider the homogenous ordinary differential equation $x^{2}\frac{d^{2}y}{dx^{2}}-3x\frac{dy}{dx}+3y=0, \quad x>0$ with $y(x)$ as a general solution. Given that $y(1)=1 \quad \text{and} \quad y(2)=14$ the value of $y(1.5),$ rounded off to two decimal places, is________.
Arjun
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Differential Equations
Feb 12, 2019
by
Arjun
6.0k
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97
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gate2019-ec
numerical-answers
differential-equations
engineering-mathematics
0
votes
0
answers
32
GATE ECE 2019 | Question: 47
A random variable $X$ takes values $-1$ and $+1$ with probabilities $0.2$ and $0.8$, respectively. It is transmitted across a channel which adds noise $N,$ so that the random variable at the channel output is $Y=X+N$. The noise $N$ is ... probability of error $Pr[ \hat{X} \neq X].$ The minimum probability of error, rounded off to $1$ decimal place, is _________.
Arjun
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Probability and Statistics
Feb 12, 2019
by
Arjun
6.0k
points
124
views
gate2019-ec
numerical-answers
probability-and-statistics
probability
random-variable
uniform-distribution
0
votes
0
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33
GATE ECE 2016 Set 3 | Question: 1
Consider a $2\times2$ sqaure matrix $\textbf{A}= \begin{bmatrix} \sigma &x\\ \omega &\sigma \end{bmatrix},$ where $x$ is unknown. If the eigen values of the matrix $\textbf{A}$ are $(\sigma + j\omega)$ and $(\sigma - j\omega)$, then $x$ is equal to $+j\omega$ $-j\omega$ $+\omega$ $-\omega$
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Linear Algebra
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Milicevic3306
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60
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gate2016-ec-3
linear-algebra
matrices
0
votes
0
answers
34
GATE ECE 2016 Set 3 | Question: 2
For $f(z)= \large\frac{\sin(z)}{z^2}$, the residue of the pole at $z = 0$ is _______
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Complex Analysis
Mar 28, 2018
by
Milicevic3306
15.8k
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60
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gate2016-ec-3
numerical-answers
complex-analysis
0
votes
0
answers
35
GATE ECE 2016 Set 3 | Question: 3
The probability of getting a “head” in a single toss of a biased coin is $0.3$. The coin is tossed repeatedly till a head is obtained. If the tosses are independent, then the probability of getting “head” for the first time in the fifth toss is _______
Milicevic3306
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Probability and Statistics
Mar 28, 2018
by
Milicevic3306
15.8k
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112
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gate2016-ec-3
probability-and-statistics
probability
independent-events
0
votes
0
answers
36
GATE ECE 2016 Set 3 | Question: 4
The integral $\int\limits_{0}^{1}\large\frac{dx}{\sqrt{(1-x)}}$ is equal to _______
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in
Calculus
Mar 28, 2018
by
Milicevic3306
15.8k
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83
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gate2016-ec-3
numerical-answers
calculus
definite-integrals
0
votes
0
answers
37
GATE ECE 2016 Set 3 | Question: 5
Consider the first order initial value problem $y’= y+2x-x^2 ,\ y(0)=1,\ (0 \leq x < \infty)$ with exact solution $y(x) = x^2 + e^x$. For $x = 0.1$, the percentage difference between the exact solution and the solution obtained using a single iteration of the second-order Runga-Kutta method with step-size $h=0.1$ is _______
Milicevic3306
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Numerical Methods
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Milicevic3306
15.8k
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68
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gate2016-ec-3
numerical-answers
numerical-methods
0
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0
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38
GATE ECE 2016 Set 3 | Question: 26
The particular solution of the initial value problem given below is $\frac{d^2y}{dx^2}+12\frac{dy}{dx}+36y=0\hspace{0.3cm} \text{ with } \hspace{0.3cm}y(0)=3\hspace{0.3cm} \text{ and }\hspace{0.3cm} \frac{dy}{dx} \bigg| _{x=0} =-36$ $(3-18x)e^{-6x}$ $(3+25x)e^{-6x}$ $(3+20x)e^{-6x}$ $(3-12x)e^{-6x}$
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Differential Equations
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Milicevic3306
15.8k
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63
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gate2016-ec-3
differential-equations
1
vote
1
answer
39
GATE ECE 2016 Set 3 | Question: 27
If the vectors $e_1=(1,0,2)$, $e_2=(0,1,0)$ and $e_3=(-2,0,1)$ form an orthogonal basis of the three-dimensional real space $\mathbb{R}^3$, then the vector $\textbf{u}=(4,3,-3)\in \mathbb{R}^3 $ can be expressed as $\textbf{u}=-$ ... \frac{2}{5}$e_1+3e_2+$\large\frac{11}{5}$e_3 \\$ $\textbf{u}=-$\large\frac{2}{5}$e_1+3e_2-$\large\frac{11}{5}$e_3$
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Vector Analysis
Mar 28, 2018
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Milicevic3306
15.8k
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199
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gate2016-ec-3
vector-analysis
0
votes
0
answers
40
GATE ECE 2016 Set 3 | Question: 28
A triangle in the $xy$-plane is bounded by the straight lines $2x=3y, \: y=0$ and $x=3$. The volume above the triangle and under the plane $x+y+z=6$ is _______
Milicevic3306
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Vector Analysis
Mar 28, 2018
by
Milicevic3306
15.8k
points
82
views
gate2016-ec-3
numerical-answers
vector-analysis
vector-in-planes
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