Another math exam

posted on 29 Apr 2009 02:47 by monoguy in Math

Let $A,B\subset \mathbb{R}$ be bounded above. Is $A+B:=\{a+b\mid a\in A,b\in B\}$ bounded above too? Is the converse true? (give your answer along with proofs.)
Let $X=\left\{\frac{m}{2^n} \mid m,n\in \mathbb{N}, 0 < m \leq 2^n \right\},$ $f:\left[0,1\right]\rightarrow \mathbb{R}$ such that $f(x)=0$ for all $x\in X.$ Suppose that $f$ is continuous. Show that $f(x)=0$ $\forall x \in \left[0,1\right].$
Classify all groups of order 6 with proof.
Let $\mathbb{Z}$ be a ring with usual operations. Find all maximal ideals of $\mathbb{Z}.$ Is the ring $2\mathbb{Z}$ isomorphic to $3\mathbb{Z}$?
Let $f$ be a function with the tangent line from $\left(a,f(a)\right)$ crossing $X$-axis at $a+3.$ If $f(0)=3,$ then find such $f.$
Evaluate $\int_{0}^{1} \int_{y}^{1} \frac{8xy}{\sqrt{1+x^4}}dxdy.$
Find the maximum of $f(x,y) = 7x^2 + 2xy +3y^2$ subject to the constraint $x^2+y^2=1.$
Given the definition of countibility. Let $A$ be a countable set and $B$ a finite set. Show that $A-B$ is countable.
Linear Algebra
1. Given explicitly $(A+2I)x = 0.$ Compute the eigenvector from this equation, which will turn out to be $\left(\begin{array}{c}0\\0\\0\\\end{array}\right)$. What happened? Is $2$ an eigenvalue? What can we say about the solutions of $Ax=b$ where $b$ is also a given vector?
2. Given explicitly a matrix $M_{2 \times 3}$. Describe geometrically about the null space of the matrix. (Is it a point, a line, a plane or $R^3$?)
3. Let $P^2$ be the space of polynomials of degree less than $3.$ Let $T : P^2 \rightarrow P$ maps a polynomial $p \in P^2$ to $a \cdot p'' + b \cdot p' + c \cdot p,$ where $p'',$ $p'$ denote their derivatives. Find the transformation matrix of $T$ with respect to the usual basis.