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1
Variation on Birthday Problem
So, I have read the birthday paradox problem, and now I came across below question: Assuming the following: there are no leap years, all years have $n = 365$ days and that people's birthdays are uniformly distributed across the $n$ days of the year. (i) How many ... $n=23$, this works out to be 0.53 and Yes it seems to me I am done. Please correct me If I am wrong.
asked
Nov 12, 2019
in
Probability
by
Ayush Upadhyaya
Boss
(
29.1k
points)

174
views
probability
+2
votes
0
answers
2
The Interesting combination sum problems
Find the number of possible solutions for $x,y,z$ for each the following cases. $Case\ 1.$ Case of unlimited repetition. $x + y +z = 10$ and $x \geq 0\ , y \geq 0,\ z \geq 0 $ $Case\ 2 $ Case of unlimited repetition with variable lower bounds $x + y +z = 10$ and ... variable. $x + y +z = 10$ and $8 \geq x \geq 1\ , \ 20 \geq y \geq 2 \ , 12 \geq z \geq 3\ $
asked
Nov 1, 2019
in
Combinatory
by
Satbir
Boss
(
24k
points)

214
views
permutationandcombination
0
votes
0
answers
3
Gravner probability
Each day, you independently decide, with probability p, to flip a fair coin. Otherwise, you do nothing. (a) What is the probability of getting exactly 10 Heads in the first 20 days? (b) What is the probability of getting 10 Heads before 5 Tails?
asked
Oct 23, 2019
in
Probability
by
ajaysoni1924
Boss
(
10.8k
points)

92
views
gravner
probability
engineeringmathematics
+1
vote
0
answers
4
ISI2014DCG21
Suppose that the function $h(x)$ is defined as $h(x)=g(f(x))$ where $g(x)$ is monotone increasing, $f(x)$ is concave, and $g’’(x)$ and $f’’(x)$ exist for all $x$. Then $h(x)$ is always concave always convex not necessarily concave None of these
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

27
views
isi2014dcg
calculus
functions
maximaminima
convexconcave
+2
votes
0
answers
5
ISI2014DCG31
For real $\alpha$, the value of $\int_{\alpha}^{\alpha+1} [x]dx$, where $[x]$ denotes the largest integer less than or equal to $x$, is $\alpha$ $[\alpha]$ $1$ $\dfrac{[\alpha] + [\alpha +1]}{2}$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

29
views
isi2014dcg
calculus
integration
definiteintegrals
+2
votes
0
answers
6
ISI2014DCG32
Consider $30$ multiplechoice questions, each with four options of which exactly one is correct. Then the number of ways one can get only the alternate questions correctly answered is $3^{15}$ $2^{31}$ $2 \times \begin{pmatrix} 30 \\ 15 \end{pmatrix}$ $2 \times 3^{15}$
asked
Sep 23, 2019
in
Combinatory
by
Arjun
Veteran
(
431k
points)

78
views
isi2014dcg
permutationandcombination
0
votes
0
answers
7
ISI2014DCG33
Let $f(x)$ be a continuous function from $[0,1]$ to $[0,1]$ satisfying the following properties. $f(0)=0$, $f(1)=1$, and $f(x_1)<f(x_2)$ for $x_1 < x_2$ with $0 < x_1, \: x_2<1$. Then the number of such functions is $0$ $1$ $2$ $\infty$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

27
views
isi2014dcg
calculus
functions
limits
0
votes
0
answers
8
ISI2014DCG42
Let $f(x)=\sin x^2, \: x \in \mathbb{R}$. Then $f$ has no local minima $f$ has no local maxima $f$ has local minima at $x=0$ and $x=\pm\sqrt{(k+\frac{1}{2} ) \pi}$ for odd integers $k$ and local maxima at $x=\pm\sqrt{(k+\frac{1}{2} ) \pi}$ for even integers $k$ None of the above
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

22
views
isi2014dcg
calculus
maximaminima
0
votes
0
answers
9
ISI2014DCG43
Let $f(x) = \begin{cases}\mid \:x \mid +1, & \text{ if } x<0 \\ 0, & \text{ if } x=0 \\ \mid \:x \mid 1, & \text{ if } x>0. \end{cases}$ Then $\underset{x \to a}{\lim} f(x)$ exists if $a=0$ for all $a \in R$ for all $a \neq 0$ only if $a=1$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

15
views
isi2014dcg
calculus
functions
limits
0
votes
0
answers
10
ISI2014DCG45
Which of the following is true? $\log(1+x) < x \frac{x^2}{2} + \frac{x^3}{3} \text{ for all } x>0$ $\log(1+x) > x \frac{x^2}{2} + \frac{x^3}{3} \text{ for all } x>0$ $\log(1+x) > x \frac{x^2}{2} + \frac{x^3}{3} \text{ for some } x>0$ $\log(1+x) < x \frac{x^2}{2} + \frac{x^3}{3} \text{ for some } x>0$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

23
views
isi2014dcg
calculus
functions
logarithms
0
votes
0
answers
11
ISI2014DCG47
The value of the definite integral $\int_0^{\pi} \mid \frac{1}{2} + \cos x \mid dx$ is $\frac{\pi}{6} + \sqrt{3}$ $\frac{\pi}{6}  \sqrt{3}$ $0$ $\frac{1}{2}$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

19
views
isi2014dcg
calculus
integration
definiteintegrals
0
votes
0
answers
12
ISI2014DCG48
If $x$ is real, the set of real values of $a$ for which the function $y=x^2ax+12a^2$ is always greater than zero is $ \frac{2}{3} < a \leq \frac{2}{3}$ $ \frac{2}{3} \leq a < \frac{2}{3}$ $ \frac{2}{3} < a < \frac{2}{3}$ None of these
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

13
views
isi2014dcg
calculus
functions
quadraticequations
0
votes
0
answers
13
ISI2014DCG50
$\underset{x \to 0}{\lim} \dfrac{x \tan x}{1 \cos tx}$ is equal to $0$ $1$ $\infty$ $2$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

24
views
isi2014dcg
calculus
limits
0
votes
0
answers
14
ISI2014DCG53
The value of the integral $\displaystyle{}\int_{1}^1 \dfrac{x^2}{1+x^2} \sin x \sin 3x \sin 5x dx$ is $0$ $\frac{1}{2}$ $ – \frac{1}{2}$ $1$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

32
views
isi2014dcg
calculus
integration
definiteintegrals
0
votes
0
answers
15
ISI2014DCG70
For the matrices $A = \begin{pmatrix} a & a \\ 0 & a \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, $(B^{1}AB)^3$ is equal to $\begin{pmatrix} a^3 & a^3 \\ 0 & a^3 \end{pmatrix}$ ... $\begin{pmatrix} a^3 & 0 \\ 3a^3 & a^3 \end{pmatrix}$ $\begin{pmatrix} a^3 & 0 \\ 3a^3 & a^3 \end{pmatrix}$
asked
Sep 23, 2019
in
Linear Algebra
by
Arjun
Veteran
(
431k
points)

38
views
isi2014dcg
linearalgebra
matrices
inverse
0
votes
0
answers
16
ISI2015MMA23
Let $X$ be a nonempty set and let $\mathcal{P}(X)$ denote the collection of all subsets of $X$. Define $f: X \times \mathcal{P}(X) \to \mathbb{R}$ by $f(x,A)=\begin{cases} 1 & \text{ if } x \in A \\ 0 & \text{ if } x \notin A \end{cases}$ Then $f(x, A \cup B)$ ... $f(x,A)+f(x,B)\:  f(x,A) \cdot f(x,B)$ $f(x,A)\:+ \mid f(x,A)\:  f(x,B) \mid $
asked
Sep 23, 2019
in
Set Theory & Algebra
by
Arjun
Veteran
(
431k
points)

11
views
isi2015mma
sets
functions
nongate
0
votes
0
answers
17
ISI2015MMA26
$\displaystyle{}\underset{n \to \infty}{\lim} \frac{1}{n} \bigg( \frac{n}{n+1} + \frac{n}{n+2} + \cdots + \frac{n}{2n} \bigg)$ is equal to $\infty$ $0$ $\log_e 2$ $1$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

18
views
isi2015mma
calculus
limits
nongate
0
votes
0
answers
18
ISI2015MMA30
Suppose that a function $f$ defined on $\mathbb{R} ^2$ satisfies the following conditions: $\begin{array} &f(x+t,y) & = & f(x,y)+ty, \\ f(x,t+y) & = & f(x,y)+ tx \text{ and } \\ f(0,0) & = & K, \text{ a constant.} \end{array}$ Then for all $x,y \in \mathbb{R}, \:f(x,y)$ is equal to $K(x+y)$ $Kxy$ $K+xy$ none of the above
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

16
views
isi2015mma
calculus
functions
nongate
0
votes
0
answers
19
ISI2015MMA31
Consider the sets defined by the real solutions of the inequalities $A = \{(x,y):x^2+y^4 \leq 1 \} \:\:\:\:\:\:\:\: B = \{ (x,y):x^4+y^6 \leq 1\}$ Then $B \subseteq A$ $A \subseteq B$ Each of the sets $A – B, \: B – A$ and $A \cap B$ is nonempty none of the above
asked
Sep 23, 2019
in
Set Theory & Algebra
by
Arjun
Veteran
(
431k
points)

18
views
isi2015mma
sets
nongate
0
votes
0
answers
20
ISI2015MMA38
A real $2 \times 2$ matrix $M$ such that $M^2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \varepsilon \end{pmatrix}$ exists for all $\varepsilon > 0$ does not exist for any $\varepsilon > 0$ exists for some $\varepsilon > 0$ none of the above is true
asked
Sep 23, 2019
in
Linear Algebra
by
Arjun
Veteran
(
431k
points)

25
views
isi2015mma
linearalgebra
matrices
0
votes
0
answers
21
ISI2015MMA40
Let $x_1, x_2, x_3, x_4, y_1, y_2, y_3$ and $y_4$ be fixed real numbers, not all of them equal to zero. Define a $4 \times 4$ matrix $\textbf{A}$ ... $(\textbf{A})$ equals $1$ or $2$ $0$ $4$ $2$ or $3$
asked
Sep 23, 2019
in
Linear Algebra
by
Arjun
Veteran
(
431k
points)

36
views
isi2015mma
linearalgebra
matrices
rankofmatrix
0
votes
0
answers
22
ISI2015MMA42
Let $\lambda_1, \lambda_2, \lambda_3$ denote the eigenvalues of the matrix $A \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos t & \sin t \\ 0 &  \sin t & \cos t \end{pmatrix}.$ If $\lambda_1+\lambda_2+\lambda_3 = \sqrt{2}+1$ ... $\{  \frac{\pi}{4}, \frac{\pi}{4} \}$ $\{  \frac{\pi}{3}, \frac{\pi}{3} \}$
asked
Sep 23, 2019
in
Linear Algebra
by
Arjun
Veteran
(
431k
points)

40
views
isi2015mma
linearalgebra
matrices
eigenvalue
0
votes
0
answers
23
ISI2015MMA55
Let $\{a_n\}$ be a sequence of real numbers. Then $\underset{n \to \infty}{\lim} a_n$ exists if and only if $\underset{n \to \infty}{\lim} a_{2n}$ and $\underset{n \to \infty}{\lim} a_{2n+2}$ exists $\underset{n \to \infty}{\lim} a_{2n}$ ... $\underset{n \to \infty}{\lim} a_{3n}$ exist none of the above
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

14
views
isi2015mma
calculus
limits
0
votes
0
answers
24
ISI2015MMA57
Suppose $a>0$. Consider the sequence $a_n = n \{ \sqrt[n]{ea} – \sqrt[n]{a}, \:\:\:\:\: n \geq 1$. Then $\underset{n \to \infty}{\lim} a_n$ does not exist $\underset{n \to \infty}{\lim} a_n=e$ $\underset{n \to \infty}{\lim} a_n=0$ none of the above
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

14
views
isi2015mma
calculus
limits
0
votes
0
answers
25
ISI2015MMA58
Let $\{a_n\}, n \geq 1$, be a sequence of real numbers satisfying $\mid a_n \mid \leq 1$ for all $n$. Define $A_n = \frac{1}{n}(a_1+a_2+\cdots+a_n)$, for $n \geq 1$. Then $\underset{n \to \infty}{\lim} \sqrt{n}(A_{n+1}A_n)$ is equal to $0$ $1$ $1$ none of these
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

13
views
isi2015mma
calculus
limits
nongate
0
votes
0
answers
26
ISI2015MMA59
In the Taylor expansion of the function $f(x)=e^{x/2}$ about $x=3$, the coefficient of $(x3)^5$ is $e^{3/2} \frac{1}{5!}$ $e^{3/2} \frac{1}{2^5 5!}$ $e^{3/2} \frac{1}{2^5 5!}$ none of the above
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

11
views
isi2015mma
calculus
taylorseries
nongate
0
votes
0
answers
27
ISI2015MMA60
Let $\sigma$ be the permutation: $\begin{array} {}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 3 & 5 & 6 & 2 & 4 & 9 & 8 & 7 & 1, \end{array}$ $I$ be the identity permutation and $m$ be the order of $\sigma$ i.e. $m=\text{min}\{\text{positive integers }n: \sigma ^n=I \}$. Then $m$ is $8$ $12$ $360$ $2520$
asked
Sep 23, 2019
in
Combinatory
by
Arjun
Veteran
(
431k
points)

19
views
isi2015mma
permutationandcombination
0
votes
0
answers
28
ISI2015MMA69
Consider the function $f(x) = \begin{cases} \int_0^x \{5+ \mid 1y \mid \} dy & \text{ if } x>2 \\ 5x+2 & \text{ if } x \leq 2 \end{cases}$ Then $f$ is not continuous at $x=2$ $f$ is continuous and differentiable everywhere $f$ is continuous everywhere but not differentiable at $x=1$ $f$ is continuous everywhere but not differentiable at $x=2$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

20
views
isi2015mma
calculus
continuity
differentiation
definiteintegrals
nongate
0
votes
0
answers
29
ISI2015MMA72
The map $f(x) = a_0 \cos \mid x \mid +a_1 \sin \mid x \mid +a_2 \mid x \mid ^3$ is differentiable at $x=0$ if and only if $a_1=0$ and $a_2=0$ $a_0=0$ and $a_1=0$ $a_1=0$ $a_0, a_1, a_2$ can take any real value
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

12
views
isi2015mma
calculus
differentiation
0
votes
0
answers
30
ISI2015MMA73
$f(x)$ is a differentiable function on the real line such that $\underset{x \to \infty=}{\lim} f(x) =1$ and $\underset{x \to \infty=}{\lim} f’(x) =\alpha$. Then $\alpha$ must be $0$ $\alpha$ need not be $0$, but $\mid \alpha \mid <1$ $\alpha >1$ $\alpha < 1$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

14
views
isi2015mma
calculus
limits
differentiation
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