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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.