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Questions without answers in Engineering Mathematics
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1
GATE ECE 2024 | Question: 1
The general form of the complementary function of a differential equation is given by $y(t)=(A t+B) e^{-2 t}$, where $A$ and $B$ ...
The general form of the complementary function of a differential equation is given by $y(t)=(A t+B) e^{-2 t}$, where $A$ and $B$ are real constants determined by the init...
admin
46.4k
points
558
views
admin
asked
Feb 16
Differential Equations
gateece-2024
differential-equations
second-order-differential-equation
engineering-mathematics
+
–
0
votes
0
answers
2
GATE ECE 2024 | Question: 2
In the context of Bode magnitude plots, $40 \mathrm{~dB} /$ decade is the same as $\_\_\_\_\_\_.$ $12 \mathrm{~dB} /$ octave $6 \mathrm{~dB} /$ octave $20 \mathrm{~dB} /$ octave $10 \mathrm{~dB} /$ octave
In the context of Bode magnitude plots, $40 \mathrm{~dB} /$ decade is the same as $\_\_\_\_\_\_.$$12 \mathrm{~dB} /$ octave$6 \mathrm{~dB} /$ octave$20 \ma...
admin
46.4k
points
579
views
admin
asked
Feb 16
Engineering Mathematics
gateece-2024
engineering-mathematics
bode-and-root-locus-plots
+
–
0
votes
0
answers
3
GATE ECE 2024 | Question: 6
Let $\hat{i}$ and $\hat{j}$ be the unit vectors along $x$ and $y$ axes, respectively and let $A$ be a positive constant. Which one of the following statements is true for the vector fields $\overrightarrow{F_{1}}=A(\hat{i} y+\hat{j} x)$ ... Only $\vec{F}_{2}$ is an electrostatic field. Neither $\vec{F}_{1}$ nor $\vec{F}_{2}$ is an electrostatic field. .
Let $\hat{i}$ and $\hat{j}$ be the unit vectors along $x$ and $y$ axes, respectively and let $A$ be a positive constant. Which one of the following statements is...
admin
46.4k
points
403
views
admin
asked
Feb 16
Vector Analysis
gateece-2024
vector-analysis
electrostatics
maxwell'
s-equations
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–
0
votes
0
answers
4
GATE ECE 2024 | Question: 20
Let $\mathbb{R}$ and $\mathbb{R}^{3}$ denote the set of real numbers and the three dimensional vector space over it, respectively. The value of $\alpha$ for which the set of vectors \[ \left\{\left[\begin{array}{lll} 2 & -3 & \alpha \end{array}\right], \quad\left ... ; 7 \end{array}\right]\right\} \] does not form a basis of $\mathbb{R}^{3}$ is $\_\_\_\_\_\_\_$.
Let $\mathbb{R}$ and $\mathbb{R}^{3}$ denote the set of real numbers and the three dimensional vector space over it, respectively. The value of $\alpha$ for which the set...
admin
46.4k
points
552
views
admin
asked
Feb 16
Linear Algebra
gateece-2024
numerical-answers
linear-algebra
vector-analysis
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–
0
votes
0
answers
5
GATE ECE 2024 | Question: 23
Suppose $\text{X}$ and $\text{Y}$ are independent and identically distributed random variables that are distributed uniformly in the interval $[0,1]$. The probability that $\text{X} \geq \text{Y}$ is $\_\_\_\_\_\_$.
Suppose $\text{X}$ and $\text{Y}$ are independent and identically distributed random variables that are distributed uniformly in the interval $[0,1]$. The probability tha...
admin
46.4k
points
657
views
admin
asked
Feb 16
Probability and Statistics
gateece-2024
probability
random-variable
probability-and-statistics
numerical-answers
+
–
0
votes
0
answers
6
GATE ECE 2024 | Question: 26
Consider the Earth to be a perfect sphere of radius $\text{R}$. Then the surface area of the region, enclosed by the $60^{\circ} \mathrm{N}$ latitude circle, that contains the north pole in its interior is $\_\_\_\_\_\_$. $(2-\sqrt{3}) \pi R^{2}$ $\frac{(\sqrt{2}-1) \pi R^{2}}{2}$ $\frac{2 \pi R^{2}}{3}$ $\frac{(2+\sqrt{3}) \pi R^{2}}{8 \sqrt{2}}$
Consider the Earth to be a perfect sphere of radius $\text{R}$. Then the surface area of the region, enclosed by the $60^{\circ} \mathrm{N}$ latitude circl...
admin
46.4k
points
549
views
admin
asked
Feb 16
Engineering Mathematics
gateece-2024
engineering-mathematics
geometry
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–
0
votes
0
answers
7
GATE ECE 2024 | Question: 33
Let $z$ be a complex variable. If $f(z)=\frac{\sin (\pi z)}{z^{2}(z-2)}$ and $C$ is the circle in the complex plane with $|z|=3$ then $\oint_{C} f(z) d z$ is $\_\_\_\_\_\_$. $\pi^{2} j$ $j \pi\left(\frac{1}{2}-\pi\right)$ $j \pi\left(\frac{1}{2}+\pi\right)$ $-\pi^{2} j$
Let $z$ be a complex variable. If $f(z)=\frac{\sin (\pi z)}{z^{2}(z-2)}$ and $C$ is the circle in the complex plane with $|z|=3$ then $\oint_{C} f(z) d z$ ...
admin
46.4k
points
202
views
admin
asked
Feb 16
Complex Analysis
gateece-2024
complex-analysis
+
–
0
votes
0
answers
8
GATE ECE 2024 | Question: 44
Let $F_{1}, F_{2}$, and $F_{3}$ be functions of $(x, y, z)$. Suppose that for every given pair of points $A$ and $B$ in space, the line integral $\int_{C}\left(F_{1} \mathrm{~d} x+F_{2} \mathrm{~d} y+F_{3} \mathrm{~d} z\right)$ evaluates to ...
Let $F_{1}, F_{2}$, and $F_{3}$ be functions of $(x, y, z)$. Suppose that for every given pair of points $A$ and $B$ in space, the line integral $\int_{C}\left(F_{1} \mat...
admin
46.4k
points
308
views
admin
asked
Feb 16
Vector Analysis
gateece-2024
vector-analysis
scalar-function
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–
0
votes
0
answers
9
TIFR ECE 2023 | Question: 1
Consider a fair coin with probability of heads and tails equal to $1 / 2$. Moreover consider two dice, first $\mathrm{D}_{1}$ that has three faces numbered $1,3,5$ and second $\mathrm{D}_{2}$ that has three faces numbered $2,4,6$ ... dice in the experiment. What is $\mathbb{E}[X]$ ? $\frac{7}{2}$ $4$ $3$ $\frac{9}{2}$ None of the above
Consider a fair coin with probability of heads and tails equal to $1 / 2$. Moreover consider two dice, first $\mathrm{D}_{1}$ that has three faces numbered $1,3,5$ and se...
admin
46.4k
points
314
views
admin
asked
Mar 14, 2023
Probability and Statistics
tifrece2023
probability
+
–
0
votes
0
answers
10
TIFR ECE 2023 | Question: 2
$\begin{array}{rlr}a^*=\max_{x, y} & x^2+y^2-8 x+7 \\ \text { s.t. } & \qquad x^2+y^2 \leq 1 \\ & \qquad \qquad y \geq 0\end{array}$ Then $a^{\star}$ is $16$ $14$ $12$ $10$ None of the above
$\begin{array}{rlr}a^*=\max_{x, y} & x^2+y^2-8 x+7 \\ \text { s.t. } & \qquad x^2+y^2 \leq 1 \\ & \qquad \qquad y \geq 0\end{array}$Then $a^{\star}$ is$16$$14$$12$$10$Non...
admin
46.4k
points
151
views
admin
asked
Mar 14, 2023
Linear Algebra
tifrece2023
engineering-mathematics
linear-algebra
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–
0
votes
0
answers
11
TIFR ECE 2023 | Question: 7
Let $f(x)$ be a positive continuous function on the real line that is the density of a random variable $X$. The differential entropy of $X$ is defined to be $-\int_{-\infty}^{\infty} f(x) \ln f(x) d x$. In which case does $X$ have the least differential entropy? You may use these facts: The ... $f(x):=(1 / 4) e^{-|x| / 2}$. $f(x):=e^{-2|x|}$.
Let $f(x)$ be a positive continuous function on the real line that is the density of a random variable $X$. The differential entropy of $X$ is defined to be $-\int_{-\inf...
admin
46.4k
points
126
views
admin
asked
Mar 14, 2023
Probability and Statistics
tifrece2023
engineering-mathematics
probability-and-statistics
+
–
0
votes
0
answers
12
TIFR ECE 2023 | Question: 8
Suppose a bag contains $5$ red balls, $3$ blue balls, and $2$ black balls. Balls are drawn without replacement until the bag is empty. Let $X_{i}$ be a random variable which takes value $1$ if the $i$-th ball drawn is red, value $2$ if that ball is blue, and $3$ if it is ... $\text{(i), (ii),}$ and $\text{(iii)}$ None of $\text{(i), (ii),}$ or $\text{(iii)}$
Suppose a bag contains $5$ red balls, $3$ blue balls, and $2$ black balls. Balls are drawn without replacement until the bag is empty. Let $X_{i}$ be a random variable wh...
admin
46.4k
points
142
views
admin
asked
Mar 14, 2023
Probability and Statistics
tifrece2023
engineering-mathematics
probability
+
–
0
votes
0
answers
13
TIFR ECE 2023 | Question: 9
Consider an $n \times n$ matrix $A$ with the property that each element of $A$ is non-negative and the sum of elements of each row is $1$. Consider the following statements. $1$ is an eigenvalue of $A$ The magnitude of any eigenvalue of $A$ is at ... statements $1$ and $3$ are correct Only statements $2$ and $3$ are correct All statements $1,2$ , and $3$ are correct
Consider an $n \times n$ matrix $A$ with the property that each element of $A$ is non-negative and the sum of elements of each row is $1$.Consider the following statement...
admin
46.4k
points
132
views
admin
asked
Mar 14, 2023
Linear Algebra
tifrece2023
engineering-mathematics
linear-algebra
+
–
0
votes
0
answers
14
TIFR ECE 2023 | Question: 10
Convolution between two functions $f(t)$ and $g(t)$ is defined as follows: $f(t) * g(t)=\int_{-\infty}^{\infty} f(\tau) g(t-\tau) d \tau$ Let $u(t)$ be the unit-step function, i.e., $u(t)=1$ for $t \geq 0$ and $u(t)=0$ for $t<0$. What is $f(t) * g(t)$ ... $\frac{1}{2}(\exp (-t)+\sin (t)-2 \cos (t)) u(t)$ $\frac{1}{2}(\exp (-t)-\sin (t)+2 \cos (t)) u(t)$
Convolution between two functions $f(t)$ and $g(t)$ is defined as follows:$$f(t) * g(t)=\int_{-\infty}^{\infty} f(\tau) g(t-\tau) d \tau$$Let $u(t)$ be the unit-step func...
admin
46.4k
points
140
views
admin
asked
Mar 14, 2023
Calculus
tifrece2023
engineering-mathematics
calculus
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–
0
votes
0
answers
15
TIFR ECE 2023 | Question: 11
Consider the function $f(x)=x e^{|x|}+4 x^{2}$ for values of $x$ which lie in the interval $[-1,1]$. In this domain, suppose the function attains the minimum value at $x^{*}$. Which of the following is true? $-1 \leq x^{*}<-0.5$ $-0.5 \leq x^{*}<0$ $x^{*}=0$ $0<x^* \leq 0.5$ $0.5<x^* \leq 1$
Consider the function$$f(x)=x e^{|x|}+4 x^{2}$$for values of $x$ which lie in the interval $[-1,1]$. In this domain, suppose the function attains the minimum value at $x^...
admin
46.4k
points
137
views
admin
asked
Mar 14, 2023
Linear Algebra
tifrece2023
engineering-mathematics
+
–
0
votes
0
answers
16
TIFR ECE 2023 | Question: 13
Let $X$ be a random variable which takes values $1$ and $-1$ with probability $1 / 2$ each. Suppose $Y=X+N$, where $N$ is a random variable independent of $X$ ... $0$ $1 / 8$ $1 / 4$ $1 / 2$ None of the above
Let $X$ be a random variable which takes values $1$ and $-1$ with probability $1 / 2$ each. Suppose $Y=X+N$, where $N$ is a random variable independent of $X$ with the fo...
admin
46.4k
points
138
views
admin
asked
Mar 14, 2023
Probability and Statistics
tifrece2023
engineering-mathematics
probability
+
–
0
votes
0
answers
17
TIFR ECE 2023 | Question: 14
Suppose that $Z \sim \mathcal{N}(0,1)$ is a Gaussian random variable with mean zero and variance $1$. Let $F(z) \equiv \mathbb{P}(Z \leq z)$ be the cumulative distribution function $\operatorname{(CDF)}$ of $Z$. Define a new random variable $Y$ as $Y=F(Z)$. This means that the ... of $\mathbb{E}[Y]$ is: $F(1)$ $1$ $\frac{1}{2}$ $\frac{1}{\sqrt{2 \pi}}$ $\frac{\pi}{4}$
Suppose that $Z \sim \mathcal{N}(0,1)$ is a Gaussian random variable with mean zero and variance $1$. Let $F(z) \equiv \mathbb{P}(Z \leq z)$ be the cumulative distributio...
admin
46.4k
points
150
views
admin
asked
Mar 14, 2023
Vector Analysis
tifrece2023
engineering-mathematics
gausss-theorem
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–
1
votes
0
answers
18
TIFR ECE 2015 | Question: 2
Let $x[n]=a^{\lfloor n \mid}$, ( $a$ is real, $0<a<1$ ) and the discrete time Fourier transform $\text{(DTFT)}$ of $x[n]$ is given by $X(\omega)=\sum_{n=-\infty}^{\infty} x[n] e^{-j \omega n}$. Then the $\text{DTFT}$ ... zero only at one value of $\omega \in[-\pi, \pi]$ Its maximum value is larger than $1$ Its minimum value is less than $-1$ None of the above
Let $x[n]=a^{\lfloor n \mid}$, ( $a$ is real, $0<a<1$ ) and the discrete time Fourier transform $\text{(DTFT)}$ of $x[n]$ is given by $X(\omega)=\sum_{n=-\infty}^{\infty}...
admin
46.4k
points
107
views
admin
asked
Dec 15, 2022
Calculus
tifr2015
calculus
discrete-fourier-transform
+
–
1
votes
0
answers
19
TIFR ECE 2015 | Question: 6
$\textbf{A}$ is an $n \times n$ square matrix of reals such that $\mathbf{A y}=\mathbf{A}^{T} \mathbf{y}$, for all real vectors $\mathbf{y}$. Which of the following can we conclude? $\mathbf{A}$ is invertible $\mathbf{A}^{T}=\mathbf{A}$ $\mathbf{A}^{2}=\mathbf{A}$ Only (i) Only (ii) Only (iii) Only (i) and (ii) None of the above
$\textbf{A}$ is an $n \times n$ square matrix of reals such that $\mathbf{A y}=\mathbf{A}^{T} \mathbf{y}$, for all real vectors $\mathbf{y}$. Which of the following can w...
admin
46.4k
points
104
views
admin
asked
Dec 15, 2022
Linear Algebra
tifr2015
linear-algebra
matrices
+
–
1
votes
0
answers
20
TIFR ECE 2015 | Question: 7
Let $A$ be an $8 \times 8$ matrix of the form \[ \left[\begin{array}{cccc} 2 & 1 & \ldots & 1 \\ 1 & 2 & \ldots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \ldots & 2 \end{array}\ ... $\operatorname{det}(A)=9$ $\operatorname{det}(A)=18$ $\operatorname{det}(A)=14$ $\operatorname{det}(A)=27$ None of the above
Let $A$ be an $8 \times 8$ matrix of the form\[\left[\begin{array}{cccc}2 & 1 & \ldots & 1 \\1 & 2 & \ldots & 1 \\\vdots & \vdots & \ddots & \vdots \\1 & 1 & \ldots & 2\e...
admin
46.4k
points
118
views
admin
asked
Dec 15, 2022
Linear Algebra
tifr2015
linear-algebra
determinant
+
–
1
votes
0
answers
21
TIFR ECE 2015 | Question: 8
Let $X$ and $Y$ be two independent and identically distributed random variables. Let $Z=\max (X, Y)$ and $W=\min (X, Y)$. Which of the following is true? $Z$ and $W$ are independent $E(X Z)=E(Y W)$ $E(X Y)=E(Z W)$ $(a), (b)$, and $(c)$ $(a)$ and $(b)$ only
Let $X$ and $Y$ be two independent and identically distributed random variables. Let $Z=\max (X, Y)$ and $W=\min (X, Y)$. Which of the following is true?$Z$ and $W$ are i...
admin
46.4k
points
101
views
admin
asked
Dec 15, 2022
Probability and Statistics
tifr2015
probability-and-statistics
probability
random-variable
+
–
1
votes
0
answers
22
TIFR ECE 2015 | Question: 9
Consider a random variable $X$ that takes integer values $1$ through $10$ each with equal probability. Now consider random variable \[ Y=\min (7, \max (X, 4)). \] What is the variance of $Y?$ $121 / 4$ $37 / 20 $ $9 / 5$ $99 / 12$ None of the above
Consider a random variable $X$ that takes integer values $1$ through $10$ each with equal probability. Now consider random variable\[Y=\min (7, \max (X, 4)).\]What is the...
admin
46.4k
points
100
views
admin
asked
Dec 15, 2022
Probability and Statistics
tifr2015
probability-and-statistics
probability
random-variable
+
–
1
votes
0
answers
23
TIFR ECE 2015 | Question: 10
Let $X$ be a uniform random variable between $[0,1]$. And let \[ M=\min _{m X \geq 1, m \in \mathbb{N}} m . \] Then which of the following is true? $E(M)=\infty$ $E(M) \in[5,10]$ $E(M)=\exp (1)$ $E(M)=\pi$ None of the above
Let $X$ be a uniform random variable between $[0,1]$. And let\[M=\min _{m X \geq 1, m \in \mathbb{N}} m .\]Then which of the following is true?$E(M)=\infty$$E(M) \in[5,10...
admin
46.4k
points
91
views
admin
asked
Dec 15, 2022
Probability and Statistics
tifr2015
probability-and-statistics
probability
random-variable
+
–
1
votes
0
answers
24
TIFR ECE 2015 | Question: 13
Let \[ A=\left(\begin{array}{ccc} 1 & 1+\varepsilon & 1 \\ 1+\varepsilon & 1 & 1+\varepsilon \\ 1 & 1+\varepsilon & 1 \end{array}\right) \] Then for $\varepsilon=10^{-6}, A$ has only negative eigenvalues only non-zero eigenvalues only positive eigenvalues one negative and one positive eigenvalue None of the above
Let\[A=\left(\begin{array}{ccc}1 & 1+\varepsilon & 1 \\1+\varepsilon & 1 & 1+\varepsilon \\1 & 1+\varepsilon & 1\end{array}\right)\]Then for $\varepsilon=10^{-6}, A$ haso...
admin
46.4k
points
100
views
admin
asked
Dec 15, 2022
Linear Algebra
tifr2015
linear-algebra
eigen-values
+
–
1
votes
0
answers
25
TIFR ECE 2015 | Question: 14
Consider a frog that lives on two rocks $A$ and $B$ and moves from one rock to the other randomly. If it is at Rock $A$ at any time, irrespective of which rocks it occupied in the past, it jumps back to Rock $A$ with probability $2 / 3$ and instead jumps to Rock ... of $n$ jumps as $n \rightarrow \infty?$ $1 / 2 $ $2 / 3$ $1$ The limit does not exist None of the above
Consider a frog that lives on two rocks $A$ and $B$ and moves from one rock to the other randomly. If it is at Rock $A$ at any time, irrespective of which rocks it occupi...
admin
46.4k
points
95
views
admin
asked
Dec 15, 2022
Probability and Statistics
tifr2015
probability-and-statistics
probability
conditional-probability
limits
+
–
1
votes
0
answers
26
TIFR ECE 2015 | Question: 15
Let $x_{1}=-1$ and $x_{2}=1$ be two signals that are transmitted with equal probability. If signal $x_{i}, i \in$ $\{1,2\}$ is transmitted, the received signal is $y=x_{i}+n_{i}$, where $n_{i}$ ... $\theta^{\star}$ to minimize the probability of error is $\leq 0$ None of the above.
Let $x_{1}=-1$ and $x_{2}=1$ be two signals that are transmitted with equal probability. If signal $x_{i}, i \in$ $\{1,2\}$ is transmitted, the received signal is $y=x_{i...
admin
46.4k
points
89
views
admin
asked
Dec 15, 2022
Probability and Statistics
tifr2015
probability-and-statistics
probability
normal-distribution
+
–
1
votes
0
answers
27
TIFR ECE 2014 | Question: 1
Consider two independent and identically distributed random variables $X$ and $Y$ uniformly distributed in $[0,1]$. For $\alpha \in[0,1]$, the probability that $\alpha \max (X, Y)<\min (X, Y)$ is $1 /(2 \alpha)$. $\exp (1-\alpha)$ $1-\alpha$ $(1-\alpha)^{2}$ $1-\alpha^{2}$
Consider two independent and identically distributed random variables $X$ and $Y$ uniformly distributed in $[0,1]$. For $\alpha \in[0,1]$, the probability that $\alpha \m...
admin
46.4k
points
129
views
admin
asked
Dec 14, 2022
Probability and Statistics
tifr2014
probability-and-statistics
probability
uniform-distribution
+
–
1
votes
0
answers
28
TIFR ECE 2014 | Question: 2
Evaluate the limit \[ \lim _{n \rightarrow \infty}\left(2 n^{4}\right)^{\frac{1}{3 n}} . \] $e$ $1$ $2^{\frac{1}{3}}$ $0$ None of the above
Evaluate the limit\[\lim _{n \rightarrow \infty}\left(2 n^{4}\right)^{\frac{1}{3 n}} .\]$e$$1$$2^{\frac{1}{3}}$$0$None of the above
admin
46.4k
points
88
views
admin
asked
Dec 14, 2022
Calculus
tifr2014
calculus
limits
+
–
1
votes
0
answers
29
TIFR ECE 2014 | Question: 3
For a non-negative continuous random variable $X$, which of the following is TRUE? $E\{X\}=\int_{0}^{\infty} P(X>x) d x$. $E\{X\}=\int_{0}^{\infty} P(X \leq x) d x$. $P(X<x) \leq \frac{E\{X\}}{x}$. $(a)$ and $(c)$. None of the above.
For a non-negative continuous random variable $X$, which of the following is TRUE?$E\{X\}=\int_{0}^{\infty} P(X>x) d x$.$E\{X\}=\int_{0}^{\infty} P(X \leq x) d x$.$P(X<x)...
admin
46.4k
points
106
views
admin
asked
Dec 14, 2022
Probability and Statistics
tifr2014
probability-and-statistics
probability
random-variable
+
–
1
votes
0
answers
30
TIFR ECE 2014 | Question: 5
The matrix \[ A=\left(\begin{array}{ccc} 1 & a_{1} & a_{1}^{2} \\ 1 & a_{2} & a_{2}^{2} \\ 1 & a_{3} & a_{3}^{2} \end{array}\right) \] is invertible when $a_{1}>a_{2}>a_{3}$ $a_{1}<a_{2}<a_{3}$ $a_{1}=3, a_{2}=2, a_{3}=4$ All of the above None of the above
The matrix\[A=\left(\begin{array}{ccc}1 & a_{1} & a_{1}^{2} \\1 & a_{2} & a_{2}^{2} \\1 & a_{3} & a_{3}^{2}\end{array}\right)\]is invertible when$a_{1}>a_{2}>a_{3}$$a_{1}...
admin
46.4k
points
95
views
admin
asked
Dec 14, 2022
Linear Algebra
tifr2014
linear-algebra
matrices
+
–
1
votes
0
answers
31
TIFR ECE 2014 | Question: 6
Let $g:[0, \pi] \rightarrow \mathbb{R}$ be continuous and satisfy \[ \int_{0}^{\pi} g(x) \sin (n x) d x=0 \] for all integers $n \geq 2$. Then which of the following can you say about $g?$ $g$ must be identically zero. $g(\pi / 2)=1$. $g$ need not be identically zero. $g(\pi)=0$. None of the above.
Let $g:[0, \pi] \rightarrow \mathbb{R}$ be continuous and satisfy\[\int_{0}^{\pi} g(x) \sin (n x) d x=0\]for all integers $n \geq 2$. Then which of the following can you ...
admin
46.4k
points
103
views
admin
asked
Dec 14, 2022
Calculus
tifr2014
calculus
definite-integrals
+
–
1
votes
0
answers
32
TIFR ECE 2014 | Question: 7
Let $A$ be an $n \times n$ real matrix. It is known that there are two distinct $n$-dimensional real column vectors $v_{1}, v_{2}$ such that $A v_{1}=A v_{2}$. Which of the following can we conclude about $A?$ All eigenvalues of $A$ are non-negative. $A$ is not full rank. $A$ is not the zero matrix. $\operatorname{det}(A) \neq 0$. None of the above.
Let $A$ be an $n \times n$ real matrix. It is known that there are two distinct $n$-dimensional real column vectors $v_{1}, v_{2}$ such that $A v_{1}=A v_{2}$. Which of t...
admin
46.4k
points
143
views
admin
asked
Dec 14, 2022
Linear Algebra
tifr2014
linear-algebra
eigen-values
+
–
1
votes
0
answers
33
TIFR ECE 2014 | Question: 8
Consider a square pulse $g(t)$ of height $1$ and width $1$ centred at $1 / 2$. Define $f_{n}(t)=\frac{1}{n}\left(g(t) *^{n} g(t)\right),$ where $*^{n}$ stands for $n$-fold convolution. Let $f(t)=\lim _{n \rightarrow \infty} f_{n}(t)$. Then, which ... $\infty$. $f(t)$ has width $\infty$ and height $1$ . $f(t)$ has width $0$ and height $\infty$. None of the above.
Consider a square pulse $g(t)$ of height $1$ and width $1$ centred at $1 / 2$. Define $f_{n}(t)=\frac{1}{n}\left(g(t) *^{n} g(t)\right),$ where $*^{n}$ stands for $n$-fol...
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TIFR ECE 2014 | Question: 12
Assume that $Y, Z$ are independent, zero-mean, continuous random variables with variances $\sigma_{Y}^{2}$ and $\sigma_{Z}^{2},$ respectively. Let $X=Y+Z$. The optimal value of $\alpha$ which minimizes $\mathbb{E}\left[(X-\alpha Y)^{2}\right]$ ... $1$ $\frac{\sigma_{Y}^{2}}{\sigma_{Z}^{2}}$ None of the above.
Assume that $Y, Z$ are independent, zero-mean, continuous random variables with variances $\sigma_{Y}^{2}$ and $\sigma_{Z}^{2},$ respectively. Let $X=Y+Z$. The optimal va...
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TIFR ECE 2014 | Question: 13
Let function $f: \mathbf{R} \rightarrow \mathbf{R}$ be convex, i.e., for $x, y \in \mathbf{R}, \alpha \in[0,1], f(\alpha x+(1-\alpha) y) \leq$ $\alpha f(x)+(1-\alpha) f(y)$. Then which of the following is $\text{TRUE?}$ $f(x) \leq f(y)$ whenever ... $f$ and $g$ are both convex, then $\min \{f, g\}$ is also convex. For a random variable $X, E(f(X)) \geq f(E(X))$.
Let function $f: \mathbf{R} \rightarrow \mathbf{R}$ be convex, i.e., for $x, y \in \mathbf{R}, \alpha \in[0,1], f(\alpha x+(1-\alpha) y) \leq$ $\alpha f(x)+(1-\alpha) f(y...
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TIFR ECE 2014 | Question: 14
Suppose that a random variable $X$ has a probability density function \[ \begin{aligned} f(x) & =c(x-4) \quad \text { for } 4 \leq x \leq 6 \\ & =0 \quad \text { for all other } x \end{aligned} \] for some constant $c$. What is the expected value of $X$ given that $X \geq 5?$ $5 \frac{5}{9}$ $5 \frac{1}{2}$ $5 \frac{3}{4}$ $5 \frac{1}{4}$ $5 \frac{5}{8}$
Suppose that a random variable $X$ has a probability density function\[\begin{aligned}f(x) & =c(x-4) \quad \text { for } 4 \leq x \leq 6 \\& =0 \quad \text { for all othe...
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TIFR ECE 2014 | Question: 16
A fair dice (with faces numbered $1, \ldots, 6$ ) is independently rolled twice. Let $X$ denote the maximum of the two outcomes. The expected value of $X$ is $4 \frac{1}{2}$ $3 \frac{1}{2}$ $5$ $4 \frac{17}{36} $ $4 \frac{3}{4}$
A fair dice (with faces numbered $1, \ldots, 6$ ) is independently rolled twice. Let $X$ denote the maximum of the two outcomes. The expected value of $X$ is$4 \frac{1}{2...
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TIFR ECE 2014 | Question: 17
Let $X$ be a Gaussian random variable with mean $\mu_{1}$ and variance $\sigma_{1}^{2}$. Now, suppose that $\mu_{1}$ itself is a random variable, which is also Gaussian distributed with mean $\mu_{2}$ and variance $\sigma_{2}^{2}$. Then the distribution ... variable with mean $\mu_{2}$ and variance $\sigma_{1}^{2}+\sigma_{2}^{2}$. Has no known form. None of the above.
Let $X$ be a Gaussian random variable with mean $\mu_{1}$ and variance $\sigma_{1}^{2}$. Now, suppose that $\mu_{1}$ itself is a random variable, which is also Gaussian d...
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TIFR ECE 2014 | Question: 18
A non-negative loss in a car accident is distributed with the following probability density function \[ f(x)=\frac{1}{10} \exp (-x / 10) \] for $x \geq 0$. Suppose that first $5$ units of loss is incurred by the insured and the remaining loss if any is covered by the ... $5+10 \exp \left(-\frac{1}{2}\right)$ $15 \exp \left(-\frac{1}{2}\right)$
A non-negative loss in a car accident is distributed with the following probability density function\[f(x)=\frac{1}{10} \exp (-x / 10)\]for $x \geq 0$. Suppose that first...
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TIFR ECE 2014 | Question: 20
What is \[ \lim _{n \rightarrow \infty} \cos \frac{\pi}{2^{2}} \cos \frac{\pi}{2^{3}} \cdots \cos \frac{\pi}{2^{n}} ? \] $0$ $\pi / 2$ $1 / \sqrt{2}$ $2 / \pi$ None of the above.
What is\[\lim _{n \rightarrow \infty} \cos \frac{\pi}{2^{2}} \cos \frac{\pi}{2^{3}} \cdots \cos \frac{\pi}{2^{n}} ?\]$0$$\pi / 2$$1 / \sqrt{2}$$2 / \pi$None of the above....
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