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25 votes
5 answers
11k views

Let $A$ and $B$ be $n \times n$ real matrices such that $AB=BA=0$ and $A+B$ is invertible

12 votes
3 answers
393 views

If $a_1,a_2,\dotsc,a_n>0 $, then $\lim\limits_{x \to \infty} \left[\frac {a_1^{1/x}+a_2^{1/x}+\dotsb+a_n^{1/x}}{n}\right]^{nx}=a_1 a_2 \dotsb a_n$

12 votes
4 answers
1k views

How to solve : $\,8^x=6x$

11 votes
1 answer
1k views

The residue at $\infty$

10 votes
2 answers
15k views

Initial value problem $dy/dx=y^{1/3}$, $y(0)=0$ has one of the following solution

9 votes
4 answers
7k views

$f:\mathbb R\rightarrow \mathbb R$ be a continuous function such that $\int_{0}^{\infty}f(x)dx$ exists.

9 votes
2 answers
1k views

The number of limit points of the set $\left\{\frac1p+\frac1q:p,q \in \Bbb N\right\}$ is which of the following:

9 votes
4 answers
4k views

Why are analytic functions functions of $z$ and not of $\bar{z}$?

8 votes
9 answers
3k views

If $x \neq 0,y \neq 0,$ then $x^2+xy+y^2$ is .....

8 votes
2 answers
563 views

The minimum value of $|z+1|+|z-1|+|z-i|$ for $z \in \mathbb C?$

7 votes
4 answers
5k views

$A$ and $B$ are $3\times 3$ real matrices such that $\operatorname{rank}(AB)=1$, then $\operatorname{rank}(BA$) can not be which of the following?

7 votes
3 answers
1k views

The set of limit points of the sequence $1,\frac12,\frac14,\frac34,\frac18,\frac38,\frac58,\frac78,\frac1{16},\frac3{16},\ldots$

7 votes
1 answer
1k views

Let $f$ be a non-constant entire function such that $\left \lvert f(z) \right\lvert=1$ for every $z$ with $\left \lvert z \right\lvert=1$.

6 votes
1 answer
258 views

A be a $3\times3$ real valued matrix such that $A^{3}=I$ but $A \neq I$ .Then trace(A)=?

6 votes
3 answers
218 views

Find the limit of $ x_n = \prod_{j=2}^{n} \left(1 - \frac{2}{j(j+1)}\right)^2$

6 votes
4 answers
1k views

The minimum value of $a^2+b^2+c^2+\frac1{a^2}+\frac1{b^2}+\frac1{c^2}?$

6 votes
2 answers
170 views

$w_1,w_2$ are distinct complex numbers such that $|w_1|=|w_2|=1$ and $w_1+w_2=1$

6 votes
2 answers
12k views

How to find the shortest path between opposite vertices of a cube, traveling on its surface?

5 votes
3 answers
322 views

Let $\,f \colon \Bbb R \to \Bbb R$ be a continuous function such that $|f(x)-f(y)|\ge \frac12 |x-y|$

5 votes
2 answers
529 views

$q$ be a real polynomial of real variable $x$ of the form $q(x)=x^n+a_{n-1}x^{n-1}+....+a_1x-1 .\,\,$ [duplicate]

5 votes
2 answers
206 views

Finding the number of symmetric, positive definite $10 \times 10$ matrices having...

5 votes
3 answers
133 views

How to prove $\lim_{n \to \infty} \cos \frac {\pi}{2^2}\cos \frac {\pi}{2^3}\cos \frac {\pi}{2^4}......\cos \frac {\pi}{2^n}=\frac {2}{\pi}$

5 votes
1 answer
2k views

$f$ be an analytic function defined on $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$

5 votes
2 answers
152 views

How many complex numbers $z=x+iy$ are there such that $x+y=1$ and $e^{i(x^2+y^2)}=1.$

5 votes
3 answers
119 views

$\lim_{n \to \infty} n \int_0^1 x^np(x) \, dx=$? , where $p(x)$ is a polynomial

4 votes
4 answers
185 views

For $n >1$,let $\displaystyle f(n)$ be the number of $n \times n$ real matrices $A$ such that $A^2+I=0.$

4 votes
3 answers
253 views

The value of $(2^n+3^n+4^n)^{1/n}$ as $n \rightarrow \infty?$

4 votes
3 answers
189 views

Let $f:\mathbb R \rightarrow \mathbb R$ be a differentiable even function

4 votes
1 answer
5k views

Finding the order of the quotient group $G/Z(G)$

4 votes
3 answers
526 views

comparison between two sequences

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