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.

8 Replies to “I.S.I. B.Stat and B.Math Entrance 2017”

  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?

    1. 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.

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