CMI 2014 B.Sc. Entrance Paper

This post contains problem from Chennai Mathematics Institute, CMI 2014 B.Sc. Entrance Paper. Try to solve them out.

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4 Point Problems

1. Find the minimum value for x for which $latex \mathbf{ 50!/ (24)^n }$ is not an integer.
2. Find the slope of the line L which satisfies the following conditions: (i) L is tangent to the graph of $latex \mathbf{y = x^3}$ (ii) L passes through the point (0, 2000)
3. If $latex \mathbf{f(x) = (x-a)(x-b)^3(x-c)^5(x-d)^7}$ is a polynomial with 16 real roots such that 4 are distinct. How many real roots will its derivative have? And how many are distinct?
4. Find the area of the 12 sided regular polygon inscribed in the unit circle. Find the greatest integer lesser than or equal to the area of the polygon with 2014 sides.
5. Given that the sum of the lengths of the 12 sides of a cuboid is 60, find the range of possible volumes. If the total surface area is 56 square units, find the length of the longest diagonal and the volume, if possible.
6. There is a regular 100-gon. Choose any three vertices of it. What is the probability that this triangle is a right triangle
7. Consider $latex \mathbf{ e^{i \theta } , e^{2i \theta } , ... , e^{13 i \theta } }$. Let $latex \mathbf{ \sum_{t=1}^{13} e^{t i \theta} }$ . Then find the maximum and minimum of |A|.
8. How many triangles are possible in each of these cases?
   (a) $latex \mathbf{ \angle A = 95^o , \angle B = 55^{o} , \angle C = 30^{0} }$
   (b) a=95, b=55, c=30
   (c) $latex \mathbf{ \angle A = 95^o , \angle B = 55^{0} , c= 30 }$
   (d) $latex \mathbf{ b = 95 , b = 55 , \angle C = 30^{0} }$

3 Point Problems

1. If $latex \mathbf{f(x) = x^2e^x}$ when $latex \mathbf{x ge 0}$, and $latex \mathbf{f(x) = x e^{(-x)}}$ when x < -1, answer the following with true or false:
   f(x) is continuous at all points
   f(x) is differentiable at all points
   f(x) is one-one
   f(x) takes all possible real values

10 point for problem 1 and rest are 15 point Problems

1. $latex \mathbf{ A= {(x, y), x^2 + y^2 \le 144 , sin(2x+3y) \le 0 } } $ . Find the area of A.
   Solution
2. Let $latex \mathbf{ x \in \mathbb{R} , x^{2014} - x^{2004} , x^{2009} - x^{2004} \in \mathbb{Z} }$ . Then show that x is an integer. (Hint: First show that x is a rational number)
   Solution
3. Let A = {1, ... , k} and B = {1, ... , n}. Find the number of maps from A to B .
   Define $latex \mathbf{ P_k } $ be the set of subsets of A. Let f be a map from $latex \mathbf{P_k to B }$ such that if $latex \mathbf{ U , V \in P_k }$ then $latex \mathbf{ f(U \cup V) }$= $latex \mathbf{\text{max} { f(U) , f(V) } }$ . Find the number of such functions. (For example if k = 3 and n =4 then answer is 100).
   Solution
4. (1) Let S, T be two circles intersecting at X, Y. Let AB and CD be two chords of circle S such that AX and DY meet on the circumference of circle T at M and BY, CX meet on the circumference of T at N.
   (2) There is a triangle EFI in which EF|| GH.  GF , EH  are joined to meet at L.  Then a cevian is drawn from I to EF passing through L which cut GH at J(say) & EF at K(say).
   Then prove that j is midpoint of GH & K is midpoint of EF.
   Hint: use ceva’s theorem our assume GHFE as cyclic quadrilateral.
   (3) Using part (1) and (2) and an unmarked straight edge find the center of a given circle
5. Suppose f is a function continuous over [-1,1] and differentiable at 0. Also, define $latex \mathbf{ g(x)=\frac{f(x)-f(0)}{x} }$ for $latex \mathbf{ x \in [-1,0) \cup (0,1] }$.
   i) If g is to be continuous over [-1,1], what should the value of g(0) be?
   ii) Prove that the $latex \mathbf{ \lim_{r to 0+} \int_{-1}^{-r} \frac{f(x)}{x} dx + \int_r^1 \frac{f(x)}{x} dx }$ exists.
   iii) Give an example to show that (ii) need not hold when f is not differentiable at 0.
6. i) For a polynomial F(x), define the discrete derivative of F at x as F(x)-F(x-1). Determine the leading term of F’(x) in terms of leading term of F(x).
   ii) Define $latex \mathbf{p_0(x)=1, p_1(x)=x, p_2(x)=\frac{x(x-1)}{2!} }$ and so on. Show that any n degree polynomial can be written as $latex \mathbf{ \sum_{r=0}^n b_r \cdot p_r(x)}$ for real values of $latex \mathbf{b_r}$.
   iii) Let G(x) be a polynomial which achieves integer values for all integral x. Using (i) and (ii), show that if G is written in the summation form as mentioned in (ii), $latex \mathbf{b_i}$ is an integer for all values of i.