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ISI B.Stat Paper 2017 Subjective| Problems & Solutions

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.
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Discussion
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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 $$ \frac {12 + 9 \sqrt {15} }{5} $$
I.S.I. 2017 geometry problem
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Discussion
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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}$

  1. Find $f'(1)$

2. Evaluate \( \displaystyle{\lim_{n \to \infty } \left [ 100 u - u \sum_{k=1}^{100} f \left (1 + \frac {k}{u} \right ) \right ] }\)
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Discussion
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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 $R$.
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Discussion
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Problem 5 :

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

  1. 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 $( i, j \in A )$

  1. 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.

  1. 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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