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ISI 2015 BStat - BMath Objective Problems

This post contains solutions of Indian Statistical Institute, ISI 2015 BStat - BMath Objective Problems. Try to solve them.

This is a work in progress. Candidates, please submit objective problems in the comments section (even if you partially remember them). If you do not remember the options, that is fine too. We can work with the questions.

(these problems and solutions are suggested by students; they will cross-checked soon)

  1. For positive x, find the maximum value of x^2 + 4x + \frac{1}{x} ^2 + \frac{4}{x} .
    1. ANS: 10
  2. Find the integer solutions of x^2 + y^2 = 2015
    1. ANS:
  3. x^2 + y^2 \le 4 , \tan^4 x + \cot^4 x + 1 = 3 \sin^2 y ; Find the number of solutions to this equation.
    1. ANS: 4
  4. ABCD...I is a 9 digit number (all digits distinct). A+B+C = 9, D, E, F, G are consecutive odd numbers in decreasing order. Last three-digit are three even numbers in decreasing order. Find A.
    1. ANS: 8
  5. Consider functions which map from set A to set B. Set A has 10 elements. Set B has two elements, one even and one odd. Suppose a in A, bin B , if a is mapped to b then we write f(a) = b. Suppose the function f has the property \displaystyle { \sum_{a \in A} f(a) } = even. How many such functions are there?
  6. g(x) = \int_{-x^3}^{x^3} f(x) dx , f(t) > 1 for all t, then
    1. g(x) = c has a unique solution for all x,
    2. g(x) = c has no solution for some c
    3. g(x) has a minima or maxima at x = 0
  7. A=(0, 8) and B= (12, 0) be two points. From any point P sum of the distances from the given points AP + BP = d_p . How many lattice points P= (x, y) , x>0, y> 0 (both x, y are integer) are there such that d_p is minimized.
  8. 2n + 1 integers are there. Sum of any n numbers is always less than the rest n+1 numbers.
    1. all integers must be equal;
    2. all are positive;
    3. all integers are negative;
    4. such a set does not exist;
  9. Find the angle between the tangents to the hyperbolae x^2 - y^2 = 1 and xy=1 at their points of intersection.
    1. Ans: \frac{\pi}{2}
  10. There are two circles which are represented by the equations x^2 + y^2 - 9 = 0 and x^2 +y^2 -20x + 96 = 0 . Points P and Q lie on circle 1 and circle 2 respectively. Point P lies above x-axis and point Q lies below x-axis. PQ is joined such that it is a common tangent to both the circles. Find the length of PQ.
    1. ANS: 5 \sqrt 3
  11. The lengths of two sides of a triangle are 2 cm and 3 cm. Find the maximum possible area of the triangle
    1. ANS: 3
  12. A set is called multiplicatively closed if the product of any two elements of the set is also an element of the set. Consider the following two sets:S1 = {a + ib : a, b are integers}
    S2 = {a + wb : a, b are integers; w is a non real cube root of unity}Then:(1) Both S1 and S2 are multiplicatively closed.
    (2) Neither is closed.
    (3) Only S1 is closed.
    (4) Only S2 is closed.
  13. The rate of increase of a population is directly proportional to the current population. If it takes 50 years for the population to double, find the time it takes to triple.(1) 50 log (3, base 2)
    (2) 50 log (3/2)
  14. A function y=f(x) is defined in (0, 2) U (4, 6) such that dy/dx = 1
    (1) f(x) = x
    (2) f(x) is always increasing
    (3) f(5.5) – f(4.5) = f(1.5) – f(0.5)
  15.  How many points P of integral coordinates are there on Y axis which are there such that triangles AOP and BOP are both right angled and such that A is (10,0) and B is (7,0) and O is origin ?A) infinite
    B) 2
    C) 4
  16. How many surjective functions are there from A to B such that A has 6 distinct elements and B has 3 elements.A)52
  17. What is the least area of the triangle bounded by a tangent to the ellipse \frac {x^2}{a^2} + \frac {y^2}{b^2} , X axis and Y axis ?A) ab
    B) (a^2+b^2-ab)
  18. How many complex numbers z are there such that (z conjugate= z^2 )A) 2
    B) 4
  19. How many factors are there which are either perfect squares, perfect cubes or both of 2^{14} \times 3^9 \times 5^5 A)252
  20. How many triples of prime numbers a < b < c are there such that a+b+c=30A) 2

I.S.I. Entrance 2015 Subjective problems

Solving a few Diophantine Equations - Video

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