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.
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.
