CMI BSc Math entrance 2014 model Problem Set

This post contains problem from Chennai Mathematics Institute, CMI BSc Math Entrance 2014 Model Problem set.

In each problem you have to fill in 4 blanks as directed. Points will be given based only on the filled answer, so you need not explain your answer. Each correct answer gets 1 point and having all 4 answers correct will get 1 extra point for a total of 5 points per problem.

But each wrong/illegible/unclear answer will get minus 1 point. Negative points from any problem will be counted in your total score, so it is better not to guess! If you are unsure about a part, you may leave it blank without any penalty. If you write something and then want it not to count, cross it out and clearly write no attempt" next to the relevant part.

1. Let f(x) be a function such that f(x+y) = f(x) + f(y). For each statement below write TRUE or FALSE
   1. If domain and codomain of f(x) be positive integers then the only  function that satisfies the above relation is f(x) = cx
   2. If domain and codomain of f(x) be integers then the only  function that satisfies the above relation is f(x) = cx
   3. If domain and codomain of f(x) be rational numbers then the only  function that satisfies the above relation is f(x) = cx
   4. If domain and codomain of f(x) be real numbers then the only  function that satisfies the above relation is f(x) = cx
2. Let f(x) be a function that is defined for rationals and irrationals separately in the interval [0, 1]. For each statement below write TRUE or FALSE
   1. f(x) = x if x is rational, f(x) = (1-x) otherwise; then f(x) is discontinuous at all points
   2. f(x) = 1/q if x = p/q (a rational) otherwise f(x) = 0 and f(0) = 0; then f(x) is continuous at x=0
   3. f(x) = x if x is irrational and f(x) = (1-x) otherwise; then f(x) is continuous at all rational points
   4. f(x) = 1/q if x = p/q (a rational) otherwise f(x) = 0 and f(0) = 0; then f(x) is continuous at all rational points
3. Suppose ABC is any triangle, P is any point inside it and RS be any line segment inside it (where R, S may or may not lie on the sides). For each statement below write TRUE or FALSE
   1. PX + PY +PZ (X, Y, Z are feet of perpendiculars dropped on sides of the triangle from P) is smaller than AT (where AT is altitude dropped from A)
   2. RS is always smaller than at least one side of the triangle.
   3. RS is always smaller than PX + PY +PZ
   4. If P is now taken to be any point on the plane of the triangle ABC then area of triangle XYZ ( (X, Y, Z are feet of perpendiculars dropped on sides of the triangle from P) is always larger than a constant c.
4. Suppose A is a set of n elements. B, C are subsets of this set. For each of the cases specified below, write an expression for the number of such elements. Do NOT try to simplify your answers.
   1. Ordered pair of subsets (A, B)
   2. Ordered pair of disjoint subsets (A, B)
   3. Ordered pair of disjoint non empty subsets (A, B)
   4. Unordered pair of disjoint non empty subsets {A, B}
5. Calculate the following limits (or write if they do not exist). Note that [x] denotes the greatest integer smaller than x
   1. $latex \mathbf{ \lim_{n to \infty} \sqrt { n^2 + 2n} - [\sqrt {n^2 + 2n} ] }$
   2. $latex \mathbf{ \lim_{x to 0 } \sin(x) \sin( \frac {1}{x}) }$
   3. $latex \mathbf{ \lim_{n to \infty} prod_{i=1}^n \frac{n^3 -1}{n^3 +1} }$
   4. $latex \mathbf{ \lim_{n to \infty} (1 - \frac{1}{n^2} )^n }$
6. Let $latex \mathbf {A_1,A_2,...., A_n}$ be a regular n gon inscribed in a unit circle and P be a point inside it. Find the values of the given expressions:
   1.  $latex \mathbf{ \overline{A_1A_2}\cdot\overline{A_1A_3}\cdots\overline{A_1A_n} }$
   2. $latex \mathbf{ \sin\frac{\pi}{n}\sin\frac{2\pi}{n}\cdots \sin \frac{(n-1)\pi}{n}}$
   3. $latex \mathbf{ \sin\frac{\pi}{2n}\sin\frac{3\pi}{2n}\cdots \sin \frac{(2n-1)\pi}{2n}}$
   4. $latex \mathbf{ \overline{PB_1}+\overline{PB_2}+\cdots+\overline{PB_n}}$ where $latex \mathbf{B_1,B_2,...., B_n}$ are feet of perpendiculars of from P on the n sides
7. Let $latex \mathbf{ x^2 + bx + c , b, c \in \mathbb{Z}}$ be a quadratic equation. Then find the values of the given expressions
   1. Probability of getting a rational root between (0,1) given b, c are from the set {-100, -99, ... , 99, 100}
   2. Probability of getting a rational root between [0,1] given b, c are from the set {-100, -99, ... , 99, 100}
   3. Probability of getting a rational root between (0, 1) given b, c are any integer
   4. Probability of getting a rational root between (1, 10) given b, c are any integer

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