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ISI Entrance Paper 2017 – from Indian Statistical Institute’s B.Stat Entrance

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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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 $$\frac {12 + 9 \sqrt {15} }{5}$$

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3. Suppose $f : \mathbb{R} \to \mathbb{R}$ is a function given by $$f(x) = \left\{\def\arraystretch{1.2}% \begin{array} 1 & \text{if x=1}\\ e^{(x^{10} -1)} + (x-1)^2 \sin \left (\frac {1}{x-1} \right ) & \text{if} x \neq 1\ \end{array} \right \}$$
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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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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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 $n^2 -12n + 36 = g(n)$
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 $p_1 , p_2, p_3$.
7. Let $A = \{ 1, 2, … , n \}$. For a permutation P = { P(1) , P(2) , … , 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 that 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
8. Let k, n and r be positive integers.
1. Let $Q(x) = x^k + a_1 x^{k+1} + …+ a_n x^ {k+n}$ 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 $P(x) = b_0 + b_1 x + … + b_r x^r$  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-acis at the origin (i.e. P changes sign at x = 0) and only if m is an odd integer.