Consider a cube $ C $ centered at origin in $R^3$ . The number of invertible linear transformations of $R^3$ which map $ C $ onto itself is
(a) $72$
(b) $48$
(c) $24$
(d) $12$
I'm totally stuck on this question, don't know how to think...
Please help.
TIFR GS-2018- PhD Screening Test.
Suppose $p$ is a degree $3$ polynomial such that $p(0)=1$, $p(1)=2$,
and $p(2)=5$. Which one of the following cannot equall $p(3)?$
(A)$0$
(B)$2$
(C)$6$
(D)$10$
Suppose $p(x)=ax^3+bx^2+cx+d$
$$p(0)=1 \implies d=1$$
$$p(1)=2 \implies ...
Following is a list of problems from an exam for admission into Ph.D program. I have just compiled all previous questions on uniform convergence of sequence of functions and i tried to work out . I would be thankful if some one can check the solutions and please suggest if there are any better wa...
If $A$ is the closure in $\mathcal{C}[0,1]$ of the set $B$ where
$$B=\{ f\in C'[0,1] : |f(x)|\leq 1, |f'(t)|\leq 1 \forall t\in[0,1]\}$$
Then Which of the following is true?
$A$ is closed.
$A$ is compact.
$A$ is connected.
$A$ is dense.
for sure $A$ is closed as it is closure of some set...
Let, $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ be an entire function and let $r$ be a positive real number. Then, which is(/are) correct?
(a) $\sum_{n=0}^{\infty}|a_{n}|^{2}r^{2n}\le \sup_{|z|=r} |f(z)|^{2}$.
(b) $\sup_{|z|=r} |f(z)|^{2}\le \sum_{n=0}^{\infty}|a_{n}|^{2}r^{2n}$.
(c) $\sum_{n=0}^{\i...
Let $a,b,c$ be a positive real number such that $b^2+c^2<a<1$. Let
$A=\begin{bmatrix} 1&b&c\\ b&a & 0\\ c & 0 & 1\end{bmatrix}$. Then
(1) all eigen values of $A$ are positive
(2) all eigenvalues of $A$ are negative
(3) all eigenvalues of $A$ are either positive or negative
(4) all eigenvalu...
Let $\{a_n\} , \{b_n\}$ be bounded sequences of positive numbers. Suppose if $\{a_n\}$ is increasing to $a$ ,then which one is true ?
$$\sup_{n\ge1}a_n b_n=a\left(\sup_{n\ge1}b_n\right), \quad \text{ or } \quad \sup_{n\ge1}a_n b_n<a\left(\sup_{n\ge1}b_n\right)$$
I think that second ...
The solution of the initial value problem
$ (x-y) u_{x} + (y-x-u) u_{y} = u $
with the initial condition $u(x,0) = 1$ satisfies
$ u^2(x-y+u) + (y-x-u) = 0$
$ u^2(x+y+u) + (y-x-u) = 0$
$ u^2(x-y+u) - (x+y+u) = 0$
$ u^2(y-x+u) + (x+y-u) = 0$
This is what I am able to do
The characteristic equ...
Let $X= \{ (x,y)\in \mathbb{R}^2: x^2+y^2<5\}$ and K=$\{(x,y)\in \mathbb{R}^2: 1\le x^2+y^2\le2 \quad\text{or}\quad 3\le x^2+y^2\le 4\}$
Then,
1.$X\setminus K$ has three connected components?
2.$X\setminus K$ has no relatively compact connected component in $X$.
3.$X\setminus...
Let $\{a_n\}$ be an unbounded, strictly increasing sequence of positive real numbers and let $x_k=\frac{a_{k+1}-a_{k}}{a_{k+1}}$. Which of the following statements is/are correct? (CSIR NET December 2014)
For all $n\geq m, \sum\limits^{n}_{k=m}x_k>1-\bf{\frac{a_m}{a_n}}$
There exist ...
