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I.S.I. B.Stat and B.Math Entrance 2017

  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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  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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  3. Suppose \( f : \mathbb{R} \to \mathbb{R} \) is a function given by $$ f(x) = \left\{\def\arraystretch{1.2}% \begin{array} 1 & \text{if x=1}\\ e^{(x^{10} -1)} + (x-1)^2 \sin \left (\frac {1}{x-1} \right ) & \text{if} x \neq 1\ \end{array} $$
    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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  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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  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.
May 14, 2017

8 comments

  1. 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?

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

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