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Hot questions in Engineering Mathematics
1
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41
TIFR ECE 2013 | Question: 10
Consider the following series of square matrices: \[ \begin{array}{l} H_{1}=[1], \\ H_{2}=\left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right], \end{array} \] and for $k=2,3, \ldots$, the $2^{k} \times 2^{k}$ matrix $H_{2^{k}}$ is recursively defined as \[ H_{2^{k}}=\ ... is $H_{2^{k}} H_{2^{k}}^{T}?)$ $0$ $2^{k}$ $2^{k / 2}$ $2^{k 2^{k-1}}$ $2^{k 2^{k}}$
Consider the following series of square matrices:\[\begin{array}{l}H_{1}= , \\H_{2}=\left[\begin{array}{cc}1 & 1 \\1 & -1\end{array}\right],\end{array}\]and for $k=2,3, \...
admin
46.4k
points
42
views
admin
asked
Dec 12, 2022
Linear Algebra
tifr2013
linear-algebra
determinant
+
–
1
votes
0
answers
42
TIFR ECE 2013 | Question: 12
Let $A$ be a Hermitian matrix and let $I$ be the Identity matrix with same dimensions as $A$. Then for a scalar $\alpha>0, A+\alpha I$ has the same eigenvalues as of $A$ but different eigenvectors the same eigenvalues and eigenvectors as of ... those of $A$ and same eigenvectors as of $A$ eigenvalues and eigenvectors with no relation to those of $A$ None of the above
Let $A$ be a Hermitian matrix and let $I$ be the Identity matrix with same dimensions as $A$. Then for a scalar $\alpha>0, A+\alpha I$ hasthe same eigenvalues as of $A$ b...
admin
46.4k
points
42
views
admin
asked
Dec 12, 2022
Linear Algebra
tifr2013
linear-algebra
eigen-values
+
–
1
votes
0
answers
43
TIFR ECE 2013 | Question: 13
Let $A$ be a square matrix and $x$ be a vector whose dimensions match $A$. Let $B^{\dagger}$ be the conjugate transpose of $B$. Then which of the following is not true: $x^{\dagger} A^{2} x$ is always non-negative $x^{\dagger} A x$ ... $A=A^{\dagger}$ then $x^{\dagger} A y$ is complex for some vector $y$ with same dimensions as $x$
Let $A$ be a square matrix and $x$ be a vector whose dimensions match $A$. Let $B^{\dagger}$ be the conjugate transpose of $B$. Then which of the following is not true:$x...
admin
46.4k
points
42
views
admin
asked
Dec 12, 2022
Linear Algebra
tifr2013
linear-algebra
matrices
+
–
1
votes
0
answers
44
TIFR ECE 2013 | Question: 17
Consider four coins, three of which are fair, that is they have heads on one side and tails on the other and both are equally likely to occur in a toss. The fourth coin has heads on both sides. Given that one coin amongst the four is picked at random and is tossed, and the ... is the probability that its other side is tails? $1 / 2$ $3 / 8$ $3 / 5$ $3 / 4$ $5 / 7$
Consider four coins, three of which are fair, that is they have heads on one side and tails on the other and both are equally likely to occur in a toss. The fourth coin h...
admin
46.4k
points
41
views
admin
asked
Dec 12, 2022
Probability and Statistics
tifr2013
probability-and-statistics
probability
random-variable
+
–
1
votes
0
answers
45
TIFR ECE 2013 | Question: 19
Which of the following is true for polynomials defined over real numbers $\mathbb{R}$. Every odd degree polynomial has a real root. Every odd degree polynomial has at least one complex root. Every even degree polynomial has at least one complex root. Every even degree polynomial has a real root. None of the above
Which of the following is true for polynomials defined over real numbers $\mathbb{R}$.Every odd degree polynomial has a real root.Every odd degree polynomial has at least...
admin
46.4k
points
40
views
admin
asked
Dec 12, 2022
Calculus
tifr2013
calculus
polynomials
+
–
1
votes
0
answers
46
TIFR ECE 2012 | Question: 15
Consider a string of length $1 \mathrm{~m}$. Two points are chosen independently and uniformly random on it thereby dividing the string into three parts. What is the probability that the three parts can form the sides of a triangle? $1 / 4$ $1 / 3$ $1 / 2$ $2 / 3$ $3 / 4$
Consider a string of length $1 \mathrm{~m}$. Two points are chosen independently and uniformly random on it thereby dividing the string into three parts. What is the prob...
admin
46.4k
points
166
views
admin
asked
Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
uniform-distribution
+
–
1
votes
0
answers
47
TIFR ECE 2012 | Question: 1
The minimum value of $f(x)=\ln \left(1+\exp \left(x^{2}-3 x+2\right)\right)$ for $x \geq 0$, where $\ln (\cdot)$ denotes the natural logarithm, is $\ln \left(1+e^{-1 / 4}\right)$ $\ln (5 / 3)$ $0$ $\ln \left(1+e^{2}\right)$ None of the above
The minimum value of $f(x)=\ln \left(1+\exp \left(x^{2}-3 x+2\right)\right)$ for $x \geq 0$, where $\ln (\cdot)$ denotes the natural logarithm, is$\ln \left(1+e^{-1 / 4}\...
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46.4k
points
117
views
admin
asked
Dec 8, 2022
Calculus
tifr2012
calculus
maxima-minima
+
–
1
votes
0
answers
48
TIFR ECE 2012 | Question: 19
$X$ and $Y$ are two $3$ by $3$ matrices. If \[ X Y=\left(\begin{array}{rrr} 1 & 3 & -2 \\ -4 & 2 & 5 \\ 2 & -8 & -1 \end{array}\right) \] then $X$ has rank $2$ at least one of $X, Y$ is not invertible $X$ can't be an invertible matrix $X$ and $Y$ could both be invertible. None of the above
$X$ and $Y$ are two $3$ by $3$ matrices. If\[X Y=\left(\begin{array}{rrr}1 & 3 & -2 \\-4 & 2 & 5 \\2 & -8 & -1\end{array}\right)\]then$X$ has rank $2$at least one of $X, ...
admin
46.4k
points
114
views
admin
asked
Dec 8, 2022
Linear Algebra
tifr2012
linear-algebra
determinant
+
–
1
votes
0
answers
49
TIFR ECE 2012 | Question: 12
In modeling the number of health insurance claims filed by an individual during a three year period, an analyst makes a simplifying assumption that for all non-negative integer up to $5$. \[ p_{n+1}=\frac{1}{2} p_{n} \] where $p_{n}$ denotes the probability that a ... files more than two claims in this period? $7 / 31$ $29 / 125$ $1 / 3$ $13 / 125$ None of the above
In modeling the number of health insurance claims filed by an individual during a three year period, an analyst makes a simplifying assumption that for all non-negative i...
admin
46.4k
points
95
views
admin
asked
Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
conditional-probability
+
–
1
votes
0
answers
50
TIFR ECE 2012 | Question: 13
Consider a single amoeba that at each time slot splits into two with probability $p$ or dies otherwise with probability $1-p$. This process is repeated independently infinitely at each time slot, i.e. if there are any amoebas left at time slot $t$, then they all split independently into ... $\min \left\{\frac{1 \pm \sqrt{1-4 p(1-p)}}{2(1-p)}\right\}$ None of the above
Consider a single amoeba that at each time slot splits into two with probability $p$ or dies otherwise with probability $1-p$. This process is repeated independently infi...
admin
46.4k
points
92
views
admin
asked
Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
independent-events
+
–
1
votes
0
answers
51
TIFR ECE 2012 | Question: 17
Let $A=U \Lambda U^{\dagger}$ be a $n \times n$ matrix, where $U U^{\dagger}=I$. Which of the following statements is TRUE. The matrix $I+A$ has non-negative eigen values The matrix $I+A$ is symmetic $\operatorname{det}(I+A)=\operatorname{det}(I+\Lambda)$ $(a)$ and $(c)$ $(b)$ and $(c)$ $(a), (b)$ and $(c)$
Let $A=U \Lambda U^{\dagger}$ be a $n \times n$ matrix, where $U U^{\dagger}=I$. Which of the following statements is TRUE.The matrix $I+A$ has non-negative eigen valuesT...
admin
46.4k
points
92
views
admin
asked
Dec 8, 2022
Linear Algebra
tifr2012
linear-algebra
eigen-values
+
–
1
votes
0
answers
52
TIFR ECE 2012 | Question: 2
Let $\alpha_{1}, \alpha_{2}, \cdots, \alpha_{k}$ be complex numbers. Then \[ \lim _{n \rightarrow \infty}\left|\sum_{i=1}^{k} \alpha_{i}^{n}\right|^{1 / n} \] is $0$ $\infty$ $\alpha_{k}$ $\alpha_{1}$ $\max _{j}|\alpha_{j}|$
Let $\alpha_{1}, \alpha_{2}, \cdots, \alpha_{k}$ be complex numbers. Then\[\lim _{n \rightarrow \infty}\left|\sum_{i=1}^{k} \alpha_{i}^{n}\right|^{1 / n}\]is$0$$\infty$$\...
admin
46.4k
points
90
views
admin
asked
Dec 8, 2022
Calculus
tifr2012
calculus
limits
+
–
1
votes
0
answers
53
TIFR ECE 2012 | Question: 14
Let $X$ and $Y$ be indepedent, identically distributed standard normal random variables, i.e., the probability density function of $X$ is \[f_{X}(x)=\frac{1}{\sqrt{2 \pi}} \exp \left(-\frac{x^{2}}{2}\right),-\infty<x<\infty. \] The random variable $Z$ is defined ... none of the above
Let $X$ and $Y$ be indepedent, identically distributed standard normal random variables, i.e., the probability density function of $X$ is\[f_{X}(x)=\frac{1}{\sqrt{2 \pi}}...
admin
46.4k
points
88
views
admin
asked
Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
normal-distribution
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–
1
votes
0
answers
54
TIFR ECE 2012 | Question: 20
Let $A$ be a $2 \times 2$ matrix with all entries equal to $1.$ Define $B=\sum_{n=0}^{\infty} A^{n} / n !$. Then $B=e^{2} A / 2$ $B=\left(\begin{array}{cc}1+e & e \\e & 1+e\end{array}\right)$ ... $B=\left(\begin{array}{cc}1+e^{2} & e^{2} \\e^{2} & 1+e^{2}\end{array}\right)$ None of the above
Let $A$ be a $2 \times 2$ matrix with all entries equal to $1.$ Define $B=\sum_{n=0}^{\infty} A^{n} / n !$. Then$B=e^{2} A / 2$$B=\left(\begin{array}{cc}1+e & e \\e & 1+e...
admin
46.4k
points
88
views
admin
asked
Dec 8, 2022
Linear Algebra
tifr2012
linear-algebra
matrices
+
–
1
votes
0
answers
55
TIFR ECE 2012 | Question: 10
Suppose three dice are rolled independently. Each dice can take values $1$ to $6$ with equal probability. Find the probability that the second highest outcome equals the average of the other two outcomes. Here, the ties may be resolved arbitrarily. $1 / 6$ $1 / 9$ $39 / 216$ $7 / 36$ $43 / 216$
Suppose three dice are rolled independently. Each dice can take values $1$ to $6$ with equal probability. Find the probability that the second highest outcome equals the ...
admin
46.4k
points
87
views
admin
asked
Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
independent-events
+
–
1
votes
0
answers
56
TIFR ECE 2012 | Question: 16
Let $P$ be a $n \times n$ matrix such that $P^{k}=\mathbf{0}$, for some $k \in \mathbb{N}$ and where $\mathbf{0}$ is an all zeros matrix. Then at least how many eigenvalues of $P$ are zero $1$ $n-1$ $n$ $0$ None of the above
Let $P$ be a $n \times n$ matrix such that $P^{k}=\mathbf{0}$, for some $k \in \mathbb{N}$ and where $\mathbf{0}$ is an all zeros matrix. Then at least how many eigenvalu...
admin
46.4k
points
87
views
admin
asked
Dec 8, 2022
Linear Algebra
tifr2012
linear-algebra
eigen-values
+
–
1
votes
0
answers
57
TIFR ECE 2012 | Question: 18
Under a certain coordinate transformation from $(x, y)$ to $(u, v)$ the circle $x^{2}+y^{2}=1$ shown below on the left side was transformed into the ellipse shown on the right side. If the transformation is of the form \[ \left[\begin{array}{l} u \\ v \end{array}\right]=\mathbf{A}\ ... \] $A_{1}$ only $A_{2}$ only $A_{1}$ or $A_{2}$ $A_{1}$ or $A_{3}$ $A_{2}$ or $A_{3}$
Under a certain coordinate transformation from $(x, y)$ to $(u, v)$ the circle $x^{2}+y^{2}=1$ shown below on the left side was transformed into the ellipse shown on the ...
admin
46.4k
points
81
views
admin
asked
Dec 8, 2022
Linear Algebra
tifr2012
linear-algebra
matrices
+
–
1
votes
0
answers
58
TIFR ECE 2012 | Question: 11
A Poisson random variable $X$ is given by $\operatorname{Pr}\{X=k\}=\mathrm{e}^{-\lambda} \lambda^{k} / k !, k=0,1,2, \ldots$ for $\lambda>0$. The variance of $X$ scales as $\lambda$ $\lambda^{2}$ $\lambda^{3}$ $\sqrt{\lambda}$ None of the above
A Poisson random variable $X$ is given by $\operatorname{Pr}\{X=k\}=\mathrm{e}^{-\lambda} \lambda^{k} / k !, k=0,1,2, \ldots$ for $\lambda>0$. The variance of $X$ scales ...
admin
46.4k
points
75
views
admin
asked
Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
poisson-distribution
+
–
1
votes
0
answers
59
TIFR ECE 2011 | Question: 13
If $a_k$ is an increasing function of $k$, i.e. $a_1<a_2<\ldots<a_k \ldots$. Then which of the following is $\text{TRUE.}$ $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{a_{k}}=\infty$ ... . Either $(a)$ or $(b)$. $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{a_{k}}=0$. None of the above.
If $a_k$ is an increasing function of $k$, i.e. $a_1<a_2<\ldots<a_k \ldots$. Then which of the following is $\text{TRUE.}$$\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \fr...
admin
46.4k
points
107
views
admin
asked
Dec 5, 2022
Calculus
tifr2011
calculus
limits
+
–
1
votes
0
answers
60
TIFR ECE 2011 | Question: 10
Let $f(x)=|x|$, for $x \in(-\infty, \infty)$. Then $f(x)$ is not continuous but differentiable. $f(x)$ is continuous and differentiable. $f(x)$ is continuous but not differentiable. $f(x)$ is neither continuous nor differentiable. None of the above.
Let $f(x)=|x|$, for $x \in(-\infty, \infty)$. Then$f(x)$ is not continuous but differentiable.$f(x)$ is continuous and differentiable.$f(x)$ is continuous but not differe...
admin
46.4k
points
101
views
admin
asked
Dec 5, 2022
Calculus
tifr2011
calculus
continuity-and-differentiability
+
–
1
votes
0
answers
61
TIFR ECE 2011 | Question: 9
Consider two independent random variables $X$ and $Y$ having probability density functions uniform in the interval $[-1,1]$. The probability that $X^{2}+Y^{2}>1$ is $\pi / 4$ $1-\pi / 4$ $\pi / 2-1$ Probability that $X^{2}+Y^{2}<0.5$ None of the above
Consider two independent random variables $X$ and $Y$ having probability density functions uniform in the interval $[-1,1]$. The probability that $X^{2}+Y^{2}>1$ is$\pi /...
admin
46.4k
points
96
views
admin
asked
Dec 5, 2022
Probability and Statistics
tifr2011
probability-and-statistics
probability
probability-density-function
+
–
1
votes
0
answers
62
TIFR ECE 2011 | Question: 8
Let $f(x, y)$ be a function in two variables $x, y$. Then which of the following is true $\max _{x} \min _{y} f(x, y) \leq \min _{y} \max _{x} f(x, y)$. $\max _{x} \min _{y} f(x, y) \geq \min _{y} \max _{x} f(x, y)$ ... $\max _{x} \min _{y} f(x, y)=\min _{y} \max _{x} f(x, y)+\min _{y} \min _{x} f(x, y)$. None of the above.
Let $f(x, y)$ be a function in two variables $x, y$. Then which of the following is true$\max _{x} \min _{y} f(x, y) \leq \min _{y} \max _{x} f(x, y)$.$\max _{x} \min _{y...
admin
46.4k
points
92
views
admin
asked
Dec 5, 2022
Calculus
tifr2011
calculus
maxima-minima
+
–
1
votes
0
answers
63
TIFR ECE 2011 | Question: 11
What is the value of $\lambda$ such that $\operatorname{Prob}\{X>\operatorname{mean}\{X\}\}=1 / e$, where $\text{PDF}$ of $X$ is $p_{X}(x)=\lambda e^{-\lambda x}, x \geq 0, \lambda>0?$ $1$ $1 / e$ $1 / \sqrt{e}$ $1 / e^{2}$ All of the above
What is the value of $\lambda$ such that $\operatorname{Prob}\{X>\operatorname{mean}\{X\}\}=1 / e$, where $\text{PDF}$ of $X$ is $p_{X}(x)=\lambda e^{-\lambda x}, x \geq ...
admin
46.4k
points
90
views
admin
asked
Dec 5, 2022
Probability and Statistics
tifr2011
probability-and-statistics
probability
poisson-distribution
+
–
1
votes
0
answers
64
TIFR ECE 2011 | Question: 4
Let $\lim _{n \rightarrow \infty} x_{n}=x$. Then which of the following is $\text{TRUE.}$ There exists an $n_{0}$, such that for all $n>n_{0},\left|x_{n}-x\right|=0$. There exists an $n_{0}$ ... $n>n_{0},\left|\frac{x_{n}}{x}\right| \leq \epsilon$ for any $\epsilon>0$. None of the above.
Let $\lim _{n \rightarrow \infty} x_{n}=x$. Then which of the following is $\text{TRUE.}$There exists an $n_{0}$, such that for all $n>n_{0},\left|x_{n}-x\right|=0$.There...
admin
46.4k
points
73
views
admin
asked
Dec 5, 2022
Calculus
tifr2011
calculus
limits
+
–
1
votes
0
answers
65
TIFR ECE 2022 | Question: 5
Let $Q$ be a unit square in the plane with corners at $(0,0),(0,1),(1,0)$ and $(1,1)$. Let $B$ be a ball of radius $1$ in the plane centered at the origin $(0,0)$. Let $Q+B$ denote the set of all vectors in the plane of the form $v+w,$ where $v \in Q$ and $w \in B$. The area of $Q+B$ is: $5+\pi$ $4+\pi$ $3+\pi$ $2+\pi$ $1+\pi$
Let $Q$ be a unit square in the plane with corners at $(0,0),(0,1),(1,0)$ and $(1,1)$. Let $B$ be a ball of radius $1$ in the plane centered at the origin $(0,0)$. Let $Q...
admin
46.4k
points
148
views
admin
asked
Nov 30, 2022
Vector Analysis
tifrece2022
vector-analysis
vector-in-planes
+
–
1
votes
0
answers
66
TIFR ECE 2022 | Question: 11
A drunken man walks on a straight lane. At every integer time (in seconds) he moves a distance of $1$ unit randomly, either forwards or backwards. What is the expectation of the square of the distance after $100$ seconds from the initial position? Hint: ... sum of independent and identically distributed random variables. $100$ $\frac{\sqrt{300}}{4}$ $40$ $200$ $20 \pi$
A drunken man walks on a straight lane. At every integer time (in seconds) he moves a distance of $1$ unit randomly, either forwards or backwards. What is the expectation...
admin
46.4k
points
141
views
admin
asked
Nov 30, 2022
Probability and Statistics
tifrece2022
probability-and-statistics
probability
random-variable
expectation
+
–
1
votes
0
answers
67
TIFR ECE 2022 | Question: 9
Suppose you throw a dart and it lands uniformly at random on a target which is a disk of unit radius. What is the probability density function $f(x)$ ... None of the above.
Suppose you throw a dart and it lands uniformly at random on a target which is a disk of unit radius. What is the probability density function $f(x)$ of the distance of t...
admin
46.4k
points
130
views
admin
asked
Nov 30, 2022
Probability and Statistics
tifrece2022
probability-and-statistics
probability
probability-density-function
+
–
1
votes
0
answers
68
TIFR ECE 2010 | Question: 16
Consider two independent random variables $\text{X}$ and $\text{Y}$ having probability density functions uniform in the interval $[0,1]$. The probability that $\text{X + Y}>1.5$ is $1 / 4$ $1 / 8$ $1 / 3$ $\operatorname{Pr}\{\text{X + Y} <0.25\}$ None of the above
Consider two independent random variables $\text{X}$ and $\text{Y}$ having probability density functions uniform in the interval $[0,1]$. The probability that $\text{X + ...
admin
46.4k
points
112
views
admin
asked
Nov 30, 2022
Probability and Statistics
tifr2010
probability-and-statistics
probability
probability-density-function
+
–
1
votes
0
answers
69
TIFR ECE 2022 | Question: 8
Let $a, b, c$ be real numbers such that the following system of equations has a solution \[\begin{aligned} x+2 y+3 z &=a & & (1)\\ 8 x+10 y+12 z &=b & & (2)\\ 7 x+8 y+9 z &=c-1 & & (3) \end{aligned}\] Let $A$ be a ... 1 & 0 \\ -1 & 0 & 1 \end{array}\right]\] What is the value of $\operatorname{det}(A)$? $1$ $2$ $3$ $4$ $5$
Let $a, b, c$ be real numbers such that the following system of equations has a solution\[\begin{aligned}x+2 y+3 z &=a & & (1)\\8 x+10 y+12 z &=b & & (2)\\7 x+8 y+9 z &=c...
admin
46.4k
points
124
views
admin
asked
Nov 30, 2022
Linear Algebra
tifrece2022
linear-algebra
system-of-equations
+
–
1
votes
0
answers
70
TIFR ECE 2010 | Question: 20
The function $f(t)$ is a convolution of $t^{2}$ with $\exp \left(-t^{2} / 2\right) / \sqrt{2 \pi}$. Its derivative is $2 t$ $t^{2}$ $2 t+t e^{-t^{2} / 2}$ Does not have a simple closed form expression None of the above
The function $f(t)$ is a convolution of $t^{2}$ with $\exp \left(-t^{2} / 2\right) / \sqrt{2 \pi}$. Its derivative is$2 t$$t^{2}$$2 t+t e^{-t^{2} / 2}$Does not have a sim...
admin
46.4k
points
108
views
admin
asked
Nov 30, 2022
Calculus
tifr2010
calculus
derivatives
+
–
1
votes
0
answers
71
TIFR ECE 2021 | Question: 4
The first-order differential equation $\frac{d y(t)}{d t}+2 y(t)=x(t)$ describes a particular continuous-time system initially at rest at origin i.e., $x(0)=0$. Consider the following statements? System is memoryless. System is causal. System is stable. Which of the ... correct. All $(1), (2)$ and $(3)$ are correct. Only $(2)$ and $(3)$ are correct. None of the above
The first-order differential equation $\frac{d y(t)}{d t}+2 y(t)=x(t)$ describes a particular continuous-time system initially at rest at origin i.e., $x(0)=0$. Consider ...
admin
46.4k
points
119
views
admin
asked
Nov 30, 2022
Differential Equations
tifrece2021
differential-equations
first-order-differential-equation
+
–
1
votes
0
answers
72
TIFR ECE 2022 | Question: 14
Let a bag contain ten balls numbered $1,2, \ldots, 10$. Let three balls be drawn at random in sequence without replacement, and the number on the ball drawn on the $i^{\text {th }}$ choice be $n_{i} \in\{1,2, \ldots, 10\}.$ What is the probability that $n_{1} < n_{2} < n_{3} ?$ $\frac{1}{3}$ $\frac{1}{12}$ $\frac{1}{4}$ $\frac{1}{6}$ None of the above
Let a bag contain ten balls numbered $1,2, \ldots, 10$. Let three balls be drawn at random in sequence without replacement, and the number on the ball drawn on the $i^{\t...
admin
46.4k
points
118
views
admin
asked
Nov 30, 2022
Probability and Statistics
tifrece2022
probability-and-statistics
probability
conditional-probability
+
–
1
votes
0
answers
73
TIFR ECE 2022 | Question: 7
Two players $\mathrm{A}$ and $\mathrm{B}$ of equal skill are playing a match. The first one to win $4$ rounds wins the match. Both players are equally likely to win each round independent of the outcomes of the other rounds. After $3$ rounds, $\mathrm{A}$ has won $2$ ... probability that $\mathrm{A}$ wins the match? $5 / 8$ $2 / 3$ $11 / 16$ $5 / 7$ None of the above
Two players $\mathrm{A}$ and $\mathrm{B}$ of equal skill are playing a match. The first one to win $4$ rounds wins the match. Both players are equally likely to win each ...
admin
46.4k
points
111
views
admin
asked
Nov 30, 2022
Probability and Statistics
tifrece2022
probability-and-statistics
probability
conditional-probability
+
–
1
votes
0
answers
74
TIFR ECE 2010 | Question: 2
For $x \in[0, \pi / 2], \alpha$ for which $\sin (x) \geq x-\alpha x^{3}$ is $\alpha>1 /(2 \pi)$ $\alpha \geq 1 / 6$ $\alpha \leq 1 /(2 \pi)$ $\alpha=1 / 4$ None of the above
For $x \in[0, \pi / 2], \alpha$ for which $\sin (x) \geq x-\alpha x^{3}$ is$\alpha>1 /(2 \pi)$$\alpha \geq 1 / 6$$\alpha \leq 1 /(2 \pi)$$\alpha=1 / 4$None of the above
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Calculus
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calculus
maxima-minima
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75
TIFR ECE 2010 | Question: 3
Consider two independent random variables $\text{X}$ and $\text{Y}$ having probability density functions uniform in the interval $[0,1]$. When $\alpha \geq 1$, the probability that $\max (\text{X, Y})>\alpha \min (\text{X, Y})$ is $1 /(2 \alpha)$ $\exp (1-\alpha)$ $1 / \alpha$ $1 / \alpha^{2}$ $1 / \alpha^{3}$
Consider two independent random variables $\text{X}$ and $\text{Y}$ having probability density functions uniform in the interval $[0,1]$. When $\alpha \geq 1$, the probab...
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Probability and Statistics
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probability-and-statistics
probability
probability-density-function
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76
TIFR ECE 2022 | Question: 13
Calculate the minimum value attained by the function \[\sin (\pi x)-\sqrt{2} \pi x^{2}\] for values of $x$ which lie in the interval $[0,1]$. $\frac{1}{\sqrt{2}}\left(1-\frac{\pi}{8}\right)$ $0$ $1-\frac{\pi}{2 \sqrt{2}}$ $-\frac{1}{\sqrt{2}}\left(1+\frac{9 \pi}{2}\right)$ $-\sqrt{2} \pi$
Calculate the minimum value attained by the function\[\sin (\pi x)-\sqrt{2} \pi x^{2}\]for values of $x$ which lie in the interval $[0,1]$.$\frac{1}{\sqrt{2}}\left(1-\fra...
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Calculus
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calculus
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77
TIFR ECE 2010 | Question: 17
Let $a_{1} \geq a_{2} \geq \cdots \geq a_{k} \geq 0$. Then the limit \[ \lim _{n \rightarrow \infty}\left(\sum_{i=1}^{k} a_{i}^{n}\right)^{1 / n} \] is $0$ $\infty$ $a_{k}$ $a_{1}$ $\left(\sum_{i=1}^{k} a_{k}\right) / k$
Let $a_{1} \geq a_{2} \geq \cdots \geq a_{k} \geq 0$. Then the limit\[\lim _{n \rightarrow \infty}\left(\sum_{i=1}^{k} a_{i}^{n}\right)^{1 / n}\]is$0$$\infty$$a_{k}$$a_{1...
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calculus
limits
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78
TIFR ECE 2020 | Question: 7
Given $n$ independent Bernoulli random variables, taking value $1$ with probability $p$ and $0$ with probability $1-p$. Then, which of the following is the value of $E\left[\left(z_{1}+\right.\right.$ $\left.\left.\ldots z_{n}\right)^{2}\right] ?$ $0$ $n p+n(n-1) p^{2}$ $n^{3} p^{2}$ $n^{2} p^{2}+n p$ None of the above
Given $n$ independent Bernoulli random variables, taking value $1$ with probability $p$ and $0$ with probability $1-p$. Then, which of the following is the value of $E\le...
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Probability and Statistics
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probability-and-statistics
probability
random-variable
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0
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79
TIFR ECE 2022 | Question: 15
Consider the difference below for $m \geq 5$: \[\sum_{n=1}^{m-1} \frac{1}{(1+n)^{2}}-\int_{x=1}^{m} \frac{1}{(1+x)^{2}} d x .\] Which statement about the difference is $\text{TRUE}?$ It is positive for infinitely many $m \geq 5$ ... is positive for infinitely many $m$ It is positive for all $m \geq 5,$ and is decreasing as $m$ increases It is negative for all $m \geq 5$
Consider the difference below for $m \geq 5$:\[\sum_{n=1}^{m-1} \frac{1}{(1+n)^{2}}-\int_{x=1}^{m} \frac{1}{(1+x)^{2}} d x .\]Which statement about the difference is $\te...
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80
TIFR ECE 2021 | Question: 3
Consider the following statements: $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$. $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}=1$. $\lim _{x \rightarrow 0} \frac{1-\cos x}{x}=1$. Which of the following is $\text{TRUE?}$ Only Statement $1$ ... $1$ and $3$ are correct. All of Statements $1, 2,$ and $3$ are correct. None of the three Statements $1,2,$ and $3$ are correct.
Consider the following statements:$\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$.$\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}=1$.$\lim _{x \rightarrow 0} \frac{1-\cos x}...
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Calculus
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calculus
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