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Explore the Back-StoryHere, you will find all the questions of ISI Entrance Paper 2017 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1 :**

Let the sequence \( \{ a_n\} _{n \ge 1 } \) be defined by $$ a_n = \tan n \theta $$ where \( \tan \theta = 2 \). Show that for all n \( a_n \) is a rational number which can be written with an odd denominator.

...........................**Discussion**

...........................

**Problem 2 : **

Consider a circle of radius $6$ as given in the diagram below. Let $B, C, D$ and $E$ be points on the circle such that $BD$ and $CE$, when extended, intersect at $A$. If $AD$ and $AE$ have length $5$ and $4$ respectively, and $DBC$ is a right angle, then show that the length of $BC$ is $$ \frac {12 + 9 \sqrt {15} }{5} $$

...........................**Discussion**

...........................

**Problem 3 : **

Suppose $f : \mathbb{R} \to \mathbb{R}$ is a function given by $ f(x) = \begin{cases} 1 & \text{ if } x = 1\\ e^{(x^{10} - 1)} + (x-1)^2 \sin (\frac{1}{x-1}) & \text{ if } x \neq 1 \end{cases}$

- Find $f'(1)$

2. Evaluate \( \displaystyle{\lim_{n \to \infty } \left [ 100 u - u \sum_{k=1}^{100} f \left (1 + \frac {k}{u} \right ) \right ] }\)

...........................**Discussion**

...........................

**Problem 4 : **

Let $S$ be the square formed by the four vertices $(1, 1), (1, -1), (-1, 1)$, and $(-1, -1)$. Let the region $R$ be the set of points inside $S$ which are closer to the center than to any of the four sides. Find the area to the region $R$.

...........................**Discussion**

...........................

**Problem 5 : **

Let $g: \mathbb{N} \to \mathbb{N}$ with $g(n)$ being the product of the digits of $n$.

- Prove that \( g(n) \le n \) for all \( n \in \mathbb{N} \)

2. Find all \( n \in \mathbb {N}\) for which \( n^2 -12n + 36 = g(n) \)

**Problem 6 : **

Let $p_1 , p_2, p_3$ be primes with $p_2 \neq p_3 $ such that $4 + p_1 p_2 $ and $4 + p_1 p_3$ are perfect squares. Find all possible values of $p_1 , p_2, p_3$.

**Problem 7 : **

Let $A = 1, 2 \cdots n $. For a permutation $P = P(1) , P(2) \cdots P(n)$ of the elements of $A$, let $P(1)$ denote the first element of $P$. Find the number of all such permutations $P$ so that for all $( i, j \in A )$

- if $i < j < P(1)$ then $j$ appears before $i$ in $P$

2. if $P(1) < i< j$ then $i$ appears before $j$ in $P$

**Problem 8 : **

Let $k, n$ and $r$ be positive integers.

- Let $Q(x) = x^k + a_1 x^{k+1} + \cdot+ a_n x^ {k+n}$ be a polynomial with real coefficients. Show that the function $ \frac {Q(x)}{x^k}$ is strictly positive for all real $x$ satisfying $ 0 < |x| < \frac {1} { 1 + \sum _{i=1}^n |a_i| } $.

2. Let $P(x) = b_0 + b_1 x + \cdot + b_r x^r $ be a non-zero polynomial with real coefficients. Let $m$ be the smallest number such that $b_m \neq 0 $. Prove that the graph of $y = P(x) $ cuts the $x$-axis at the origin (i.e. $P$ changes sign at $x = 0$) and only if $m$ is an odd integer.

Here, you will find all the questions of ISI Entrance Paper 2017 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1 :**

Let the sequence \( \{ a_n\} _{n \ge 1 } \) be defined by $$ a_n = \tan n \theta $$ where \( \tan \theta = 2 \). Show that for all n \( a_n \) is a rational number which can be written with an odd denominator.

...........................**Discussion**

...........................

**Problem 2 : **

Consider a circle of radius $6$ as given in the diagram below. Let $B, C, D$ and $E$ be points on the circle such that $BD$ and $CE$, when extended, intersect at $A$. If $AD$ and $AE$ have length $5$ and $4$ respectively, and $DBC$ is a right angle, then show that the length of $BC$ is $$ \frac {12 + 9 \sqrt {15} }{5} $$

...........................**Discussion**

...........................

**Problem 3 : **

Suppose $f : \mathbb{R} \to \mathbb{R}$ is a function given by $ f(x) = \begin{cases} 1 & \text{ if } x = 1\\ e^{(x^{10} - 1)} + (x-1)^2 \sin (\frac{1}{x-1}) & \text{ if } x \neq 1 \end{cases}$

- Find $f'(1)$

2. Evaluate \( \displaystyle{\lim_{n \to \infty } \left [ 100 u - u \sum_{k=1}^{100} f \left (1 + \frac {k}{u} \right ) \right ] }\)

...........................**Discussion**

...........................

**Problem 4 : **

Let $S$ be the square formed by the four vertices $(1, 1), (1, -1), (-1, 1)$, and $(-1, -1)$. Let the region $R$ be the set of points inside $S$ which are closer to the center than to any of the four sides. Find the area to the region $R$.

...........................**Discussion**

...........................

**Problem 5 : **

Let $g: \mathbb{N} \to \mathbb{N}$ with $g(n)$ being the product of the digits of $n$.

- Prove that \( g(n) \le n \) for all \( n \in \mathbb{N} \)

2. Find all \( n \in \mathbb {N}\) for which \( n^2 -12n + 36 = g(n) \)

**Problem 6 : **

Let $p_1 , p_2, p_3$ be primes with $p_2 \neq p_3 $ such that $4 + p_1 p_2 $ and $4 + p_1 p_3$ are perfect squares. Find all possible values of $p_1 , p_2, p_3$.

**Problem 7 : **

Let $A = 1, 2 \cdots n $. For a permutation $P = P(1) , P(2) \cdots P(n)$ of the elements of $A$, let $P(1)$ denote the first element of $P$. Find the number of all such permutations $P$ so that for all $( i, j \in A )$

- if $i < j < P(1)$ then $j$ appears before $i$ in $P$

2. if $P(1) < i< j$ then $i$ appears before $j$ in $P$

**Problem 8 : **

Let $k, n$ and $r$ be positive integers.

- Let $Q(x) = x^k + a_1 x^{k+1} + \cdot+ a_n x^ {k+n}$ be a polynomial with real coefficients. Show that the function $ \frac {Q(x)}{x^k}$ is strictly positive for all real $x$ satisfying $ 0 < |x| < \frac {1} { 1 + \sum _{i=1}^n |a_i| } $.

2. Let $P(x) = b_0 + b_1 x + \cdot + b_r x^r $ be a non-zero polynomial with real coefficients. Let $m$ be the smallest number such that $b_m \neq 0 $. Prove that the graph of $y = P(x) $ cuts the $x$-axis at the origin (i.e. $P$ changes sign at $x = 0$) and only if $m$ is an odd integer.

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If i have made a slight calculation error by writing f'1 =-500e and the follow up question as well when the answer doesnt have e, how much will be deducted?

That depends on the process that you have used to solve the problem.

If your process is correct then they will more or less deduct 1-2 marks at max.

Thanks a lot!

7. (n-1)! (n+1)C3

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[…] Back to the question paper […]

complete solution of isi b math 2017 objective and subjective paper. And result date