Attempt the following questions. Once you are done with them, send a me pdf containing your answers. Send it to me by the end of the week. The marks for each questions are mentioned in the bracket following the question. There is no time limit, you can take as long as you need. You may consult books and other resources, but I would highly advice against searching the questions online and looking up solutions.
Total Marks = 85
Consider the set of all fractions, $F = \{ \frac mn | m,n \in \Z\}$ where we consider the fractions $\frac 12$ and $\frac 24$ to be different. In other words, $F \neq \mathbb{Q}$. Define an equivalence relation on $F$ such that there is a bijective mapping between the set of all equivalence classes and $\mathbb{Q}$. Provide explicitly, this bijective mapping. [5]
Determine whether the set of all binary sequences is countable or not. [3]
Prove the law of trichotomy — for every real number, only of the following statements is true — $x=0 \quad x<0 \quad x>0$. Explicitly mention all the axioms/results that you are using to prove this. [7]
Consider the following argument : “Let $C$ be a relation that is both transitive and symmetric. Then, $aCb$ implies that $bCa$ and the two together imply that, by the transitive property, $aCa$. Thus $C$ is reflexive.” Is this argument sound? If there’s a flaw then describe where. [5]
Use induction to prove the multinomial theorem. [5]
Find the value of $\sum_{i=0}^\infty \frac{i^3}{3^i}$. Try to generalize your result to $\sum_{i=0}^\infty \frac{i^n}{n^i}$ where $n$ is some positive integer. Can you prove your claim using induction? [3+5]
Suppose $a,b,c \in \{ -3,-2,-1,0,1,2,3\}$ are three distinct elements and the equation of a straight line is given by $ax+by+c=0$. Suppose the slope of the line is a positive real number. How many such lines are possible? [8]
How many ways are there to choose four distinct positive integers $x_1, x_2, x_3, x_4$ from the set $S =\{ 1,2,\dots,500\}$such that the $\{x_i\}$ forms an increasing geometric sequence with a positive integer as the common ratio? [8]
The product of two of the roots of the quartic equation $x^4 -18x^3+kx^2+200x-1984=0$ is $-32$. Determine the value of $k$. [5]
Construct a polynomial in $\mathbb{Q}[x]$ such that one of its roots is $\sin(10^\circ)$. [8]
Find all values of $a,b$ such that the following system of polynomial equations has a unique solution. [8]
$$ \begin{align*}
xyz+z &= 2 \\ xyz^2 + z &= 2 \\ x ^2 + y^2 +z^2 &= 4
\end{align*} $$
Consider a regular pentagon — label its vertices as $ABCDE$ in counterclockwise order (note that it doesn’t matter what vertex you start the labeling from). Suppose, point $E$ and $D$ are rotated about point $B$ by $15^\circ$ each, the former in clockwise direction, being mapped to point $E'$ ; and the latter is rotated in the counterclockwise direction, being mapped to point $D'$. Find the distance between points $E'$ and $D'$. [10]
Find the closed form for $\sum_{k=0}^m \binom{m}{k}/\binom nk$. [5]