# Recent questions tagged tifrmaths2018

1
The set of real numbers in the open interval $(0,1)$ which have more than one decimal expansion is empty non-empty but finite countably infinite uncountable
2
How many zeroes does the function $f\left ( x \right )=e^{x}-3x^{2}$ have in $\mathbb{R}$? $0$ $1$ $2$ $3$
3
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ ... ? $f$ is continuous everywhere except at $0$ $f$ is continuous only at the irrationals $f$ is continuous only at the non-zero rationals $f$ is not continuous anywhere
4
Suppose $p$ is a degree $3$ polynomial such that $p\left ( 0 \right )=1,p\left ( 1 \right )=2,$ and $p\left ( 2 \right )=5$. Which of the following numbers cannot equal $p\left ( 3 \right )$ ? $0$ $2$ $6$ $10$.
5
Let $A$ be the set of all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ that satisfy the following two properties: $f$ has derivatives of all orders, and for all $x,y \in \mathbb{R}$, $f\left ( x+y \right )-f\left ( y-x \right )=2x{f}'\left ( y \right ).$ ... less than or equal to $2$. There exists $f \in A$ which is not a polynomial. There exists $f \in A$ which is a polynomial of degree $4$.
6
Denote by the set all $n \times n$ complex matrices $A$ ($n\geq 2$ a natural number) having the property that $4$ is the only eigenvalue of $A$. Consider the following four statements. $\left ( A-4I \right )^{n}=0,$ $A^{n}=4^{n}I,$ $\left ( A^{2}-5A+4I \right )^{n}=0,$ $A^{n}=4nI.$ How many of the above statements are true for all $A \in$ ? $0$ $1$ $2$ $3$
7
Let $A$ be the set of all continuous functions $f:\left [ 0,1 \right ]\rightarrow \left [ 0,\infty \right )$ satisfying the following condition: $\int_{0}^{x}f\left ( t \right )dt\geq f\left ( x \right ), \:for\:all \:x\in\left [ 0,1 \right ].$ Then which of the following statements is true? $A$ has cardinality $1$. $A$ has cardinality $2$. $A$ is infinite. $A$ is empty.
8
Consider the following four sets of maps $f:\mathbb{Z}\rightarrow \mathbb{Q}$: $\{f:\mathbb{Z}\rightarrow \mathbb{Q} \mid f$ is bijective and increasing$\}$, $\{f:\mathbb{Z}\rightarrow \mathbb{Q} \mid f$ is onto and increasing$\}$ ... $\}$. How many of these sets are empty? $0$ $1$ $2$ $3$
9
What are the last $3$ digits of $2^{2017}$? $072$ $472$ $512$ $912.$
10
The minimal polynomial of $\begin{pmatrix} 2 & 1 & 0 & 0\\ 0& 2 & 0 & 0\\ 0& 0 & 2 & 0\\ 0& 0 & 1 & 5 \end{pmatrix}$ is $\left ( x-2 \right )\left ( x-5 \right ).$ $\left ( x-2 \right )^{2}\left ( x-5 \right ).$ $\left ( x-2 \right )^{3}\left ( x-5 \right ).$ none of the above.
11
Consider a cube $C$ centered at the origin in $\mathbb{R^{3}}$. The number of invertible linear transformation of $\mathbb{R^{3}}$ which map $C$ onto itself is $72$ $48$ $24$ $12$
12
The number of rings of order $4$, up to isomorphism, is: $1$ $2$ $3$ $4.$
13
For a sequence $\left \{ a_{n} \right \}$ of real numbers, which of the following is a negation of the statement $\underset{n\rightarrow \infty }{lim}\:a_{n}=0$'? There exists $\varepsilon > 0$ ... $a \in \mathbb{R}$, and every $\varepsilon > 0$, there exist infintely many $n$ such that $\left | a_{n}-a \right |> \varepsilon$.
14
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be continuous. Then which of the following statements implies that $f\left ( 0 \right )=0$? $\underset{n\rightarrow \infty }{lim}\int_{0}^{1}f\left ( x \right )^{n}dx=0.$ $\underset{n\rightarrow \infty }{lim}\int_{0}^{1}f\left ( x/n\right )dx=0.$ $\underset{n\rightarrow \infty }{lim}\int_{0}^{1}f\left ( nx\right )dx=0.$ None of the above.
15
Consider the following maps from $\mathbb{R}^{2}$ to $\mathbb{R}^{2}$: the map $\left ( x,y \right )\mapsto \left ( 2x+5y+1,x+3y \right )$, the map $\left ( x,y \right )\mapsto \left ( x+y^{2},y+x^{2} \right )$, and the map given in polar coordinates as ... $r\neq 0$, with the origin mapping to the origin. The number of maps in the above list that preserve areas is: $0$ $1$ $2$ $3.$
16
True/False Question : Let $A$ be a countable subset of $\mathbb{R}$ which is well-ordered with respect to the usual ordering on $\mathbb{R}$ (where ‘well-ordered’ means that every nonempty subset has a minimum element in it). Then $A$ an order perserving bijection with a subset of $\mathbb{N}$.
17
True/False Question : $\underset{x\rightarrow 0}{lim}\:\frac{sin\:x}{log\left ( 1+tan\:x \right )}=1$.
18
True/False Question : For any closed subset $A\subset \mathbb{R}$, there exists a continuous function $f$ on $\mathbb{R}$ which vanishes exactly on $A$.
19
True/False Question : Let $f$ be a nonnegative continuous function on $\mathbb{R}$ such that $\int_{0}^{\infty }f\left ( t \right )dt$ is finite. Then $\underset{x\rightarrow \infty }{lim}\:f\left ( x \right )=0.$
20
True/False Question : The function $f\left ( x \right )=cos\left ( e^{x} \right )$ is not uniformly continuous on $\mathbb{R}$.
21
True/False Question : Let $A$ be a $3 \times 3$ real symmetric matrix such that $A^{6}=I$. Then, $A^{2}=I$.
22
True/False Question : In the vector space $\left \{ f \mid f : \left [ 0,1 \right ] \rightarrow \mathbb{R}\right \}$ of real-valued functions on the closed interval $\left [ 0,1 \right ]$, the set $S=\left \{ sin\left ( x \right ) , cos\left ( x \right ),tan\left ( x \right )\right \}$ is linearly independent.
23
True/False Question ; Let $f$ be a twice differentiable function on $\mathbb{R}$ such that both $f$ and ${f}''$ are strictly positive on $\mathbb{R}$. Then $\underset{x\rightarrow \infty }{lim}\:f\left ( x \right )=\infty$.
24
True/False Question : Let $G,H$ be finite groups. Then any subgroup of $G \times H$ is equal to $A \times B$ for some subgroups $A<G$ and $B<H$.
25
True/False Question : Let $g$ be a continuous function on $\left [ 0,1 \right ]$ such that $g\left ( 1 \right )=0$. Then the sequence of functions $f_{n}\left ( x \right )=x^{n}g\left ( x \right )$ converges uniformly on $\left [ 0,1 \right ]$.
26
True/False Question : Let $A,B,C \in M_{3} \left(\mathbb{R}\right)$ be such that $A$ commutes with $B$, $B$ commutes with $C$ and $B$ is not a scalar matrix. Then $A$ commutes with $C$.
27
True/False Question : If $A \in M_{n}\left ( \mathbb{R} \right )$ ( with $n\geq 2$) has rank $1$, then the minimal polynomial of $A$ has degree $2$.
True/False Question : Let $V$ be the vector space over $\mathbb{R}$ consisting of polynomials of degree less than or equal to $3$. Let $T:V \rightarrow V$ be the operator sending $f\left(t\right)$ to $f\left(t+1 \right)$, and $D:V \rightarrow V$ the operator sending $f\left(t \right)$ to $df\left(t \right)/dt$. Then $T$ is a polynomial in $D$.
True/False Question : Let $V$ be the subspace of the real vector space of real valued functions on $\mathbb{R}$, spanned by cost and sin $t$. Let $D:V \rightarrow V$ be the linear map sending $f\left ( t \right ) \in V$ to $df\left ( t \right )/dt$. Then $D$ has a real eigenvalue.
True/False Question : The set of nilpotent matrices in $M_{3}\left ( \mathbb{R} \right )$ spans $M_{3}\left ( \mathbb{R} \right )$ considered as an $\mathbb{R}$-vector space (a matrix $A$ is said to be nilpotent if there exists $n \in \mathbb{N}$ such that $A^{n}=0$).