Recent questions tagged tifrmaths2011

A gardener throws $18$ seeds onto an equilateral triangle shaped plot of land with sides of length one metre. Then at least two seeds are within a distance of $25$ centim...

Let $f$ be a continuous integrable function of $\mathbb{R}$ such that either $f(x) 0$ or $f(x) + f(x + 1) 0$ for all $x \in \mathbb{R}$. Then $\int_{-\infty}^{\infty} f...

Any non-singular $k \times k$-matrix with real entries can be made singular by changing exactly one entry.

Suppose a box contains three cards, one with both sides white, one with both sides black, and one with one side white and the other side black. If you pick a card at rand...

Let $P$ be a degree $3$ polynomial with complex coefficients such that the constant term is $2010$. Then $P$ has a root $\alpha$ with $|\alpha| 10$.

If $A$ and $B$ are $3 \times 3$ matrices and $A$ is invertible, then there exists an integer $n$ such that $A + nB$ is invertible.

In the ring $\mathbb{Z}/8\mathbb{Z}$, the equation $x^{2}=1$ has exactly $2$ solutions.

State True or FalseLet $A$ be a $2 \times 2$-matrix with complex entries. The number of $2 \times 2$-matrices $A$ with complex entries satisfying the equation $A^{3}$ is ...

The function$f(x)= \begin{cases}0 & \text{if x is rational} \\ x& \text{if x is irrational} \end{cases}$is not continuous anywhere on the real line.

There exists a group with a proper subgroup isomorphic to itself.

Any continuous function from the open unit interval $(0, 1)$ to itself has a fixed point.

The derivative of the function$\int_{0}^{\sqrt{x}} e^{-t^{2}}dt$at $x = 1$ is $e^{-1}$ .

The polynomial $x^{4}+7x^{3}-13x^{2}+11x$ has exactly one real root.

If $A, B$ are closed subsets of $[0,\infty)$, then$A + B = \left\{x + y | x \in A, y \in B\right\}$is closed in $[0,\infty)$

$\log x$ is uniformly continuous on $( \frac{1}{2}, \infty)$.

$A$ is $3 \times 4$-matrix of rank $3$. Then the system of equations,$Ax = b$has exactly one solution.

$e^{\sqrt{2}} 3$

For any real number $c$, the polynomial $x^{3}+x+c$ has exactly one real root.