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February 20, 2021

How to Pursue Mathematics after High School?

For Students who are passionate for Mathematics and want to pursue it for higher studies in India and abroad.

Here, you will find all the questions of ISI Entrance Paper 2009 from Indian Statistical Institute's B. Math Entrance. You will also get the solutions soon of all the previous year problems.

Problem 1:

Let $x, y, z$ be non-zero real numbers. Suppose $\alpha, \beta, \gamma$ are complex numbers such that $|\alpha|=|\beta|=|\gamma|=1 .$ If $x+y+z=0=\alpha x+\beta y+\gamma z,$ then prove that $\alpha=\beta=\gamma$

Problem 2:

Let $c$ be a fixed real number. Show that a root of the equation
$$
x(x+1)(x+2) \cdots(x+2009)=c
$$
can have multiplicity at most 2. Determine the number of values of $c$ for which the equation has a root of multiplicity 2 .

Problem 3:

Let $1,2,3,4,5,6,7,8,9,11,12, \ldots$ be the sequence of all the positive integers integers which do not contain the digit zero. Write $\{a_{n}\}$ for this sequence. By comparing with a geometric series, show that $\sum_{n} \frac{1}{a_{n}}<90 .$

Problem 4:

$$
\begin{aligned}
&\text { Find the values of } x, y \text { for which } x^{2}+y^{2} \text { takes the minimum value where }(x+\
&5)^{2}+(y-12)^{2}=14
\end{aligned}
$$

Problem 5:

Let $p$ be a prime number bigger than $5 .$ Suppose, the decimal expansion of $1 / p$ looks like $0 . \overline{a_{1} a_{2} \cdots a_{r}}$ where the line denotes a recurring decimal. Prove that $10^{r}$ leaves a remainder of 1 on dividing by $p$.

Problem 6:

Let $a, b, c, d$ be integers such that $a d-b c$ is non-zero. Suppose $b_{1}, b_{2}$ are integers both of which are multiples of $a d-b c .$ Prove that there exist integers simultaneously satisfying both the equalities $a x+b y=b_{1}, c x+d y=b_{2}$.

Problem 7:

Compute the maximum area of a rectangle which can be inscribed in a triangle of area $M$

Problem 8:

Suppose you are given six colours and, are asked to colour each face of a cube by a different colour. Determine the different number of colourings possible.

Problem 9:

Let $f(x)=a x^{2}+b x+c$ where $a, b, c$ are real numbers. Suppose $f(-1), f(0), f(1) \in$ $[-1,1] .$ Prove that $|f(x)| \leq 3 / 2$ for all $x \in[-1,1]$

Problem 10:

Given odd integers $a, b, c,$ prove that the equation $a x^{2}+b x+c=0$ cannot have a solution $x$ which is a rational number.

Some useful link :

What to do to shape your Career in Mathematics after 12th? 

From the video below, let's learn from Dr. Ashani Dasgupta (a Ph.D. in Mathematics from the University of Milwaukee-Wisconsin and Founder-Faculty of Cheenta) how you can shape your career in Mathematics and pursue it after 12th in India and Abroad. These are some of the key questions that we are discussing here:

  • What are some of the best colleges for Mathematics that you can aim to apply for after high school?
  • How can you strategically opt for less known colleges and prepare yourself for the best universities in India or Abroad for your Masters or Ph.D. Programs?
  • What are the best universities for MS, MMath, and Ph.D. Programs in India?
  • What topics in Mathematics are really needed to crack some great Masters or Ph.D. level entrances?
  • How can you pursue a Ph.D. in Mathematics outside India?
  • What are the 5 ways Cheenta can help you to pursue Higher Mathematics in India and abroad?

Want to Explore Advanced Mathematics at Cheenta?

Cheenta has taken an initiative of helping College and High School Passout Students with its "Open Seminars" and "Open for all Math Camps". These events are extremely useful for students who are really passionate for Mathematic and want to pursue their career in it.

To Explore and Experience Advanced Mathematics at Cheenta
Register here

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