Name of the Organisation : Indian Statistical Institute
Name of the Paper : Mathematical Olympiad Question Paper 2015
Year : 2015
Name of the Subject : Mathematics
Website : http://www.isical.ac.in/index.php
Download Sample Question Paper : https://www.pdfquestion.in/uploads/8241maths.pdf
Mathematical Olympiad Question Paper :
Instructions :
1. This is a Two-Hour test.
2. There are 16 questions in this question paper. Each question carries 5 marks.
3. Answer all questions. Please use a BALL-POINT PEN WITH BLUE INK to write the answer
Related : Indian Statistical Institute MSQE ISI Admission Test Sample Question 2017 : www.pdfquestion.in/11333.html
4. (a) Each question has exactly one correct answer. No mark will be awarded if more than one answer is given to a question.
(b) For each answered question, you’ll get full credit if and only if the answer is complete and fully correct. Otherwise, you’ll get no credit.
(c) There is no negative marking and no credit for unanswered question. There is no partial marking.
5. Please report just the answer to a question in the space appearing immediately after it.
6. If you wish to change an answer just strike it off with your pen and then write your new answer legibly. Needless to say that the new answer should be written in pen.
7. (a) All rough work MUST be done in the blank pages and nowhere else. No extra paper will be provided.
(b) Rough work will NOT be considered for any credit.
8. Use of any calculators, protractors, log tables, trigonometric tables is not allowed.
9. Mobile phones or any other electronic devices are strictly prohibited in the exam hall.
Answer All the Question :
1. The sum of two positive integers is 52 and their LCM is 168. Find the GCD of the numbers.
2. Find the sum of the last two digits of the number 7100 – 3100.
3. Let A;B;C be distinct digits of a 3-digit number such that
4. Let M be the maximum value of 4x – 3y – 2z subject to 2×2 + 3y2 + 4z2 = 1. Then M2 equals
5. Consider the set A of positive integers defined by A: = { p-q:3<q<p & p,q are primes } Then the GCD of the integers in A equals
6. Consider all the 5-digit numbers containing each of the digits 1, 2, 3, 4, 5 exactly once, and not divisible by 6. The sum of all these numbers equals
7. In how many ways can you distribute 100 identical chocolates among 10 children so that the number of chocolates everyone gets is a multiple of 3, allowing some chocolates to be undistributed? [You should consider 0 to be a multiple of 3.
8. The number 23104 * 791 is divisible by 63. The missing digit (*) equals
9. The sum of all 3-digit numbers, whose digits are all odd, is
10. The number of positive integers which divide 1015 but do not divide 1510 equals
11. If the number of ways of choosing 2 boys and 2 girls in a class for a game of mixed doubles is 1620, the number of ways of choosing 2 students from the class equals
12. The number of diagonals of a convex octagon equals
13. Let 4ABC with AB = BC and \BAC = 30. Let A0 be the reflection of A across the line BC; B0 be the reflection of B across the line CA; C0 be the reflection of C across the line AB. Then \A0B0C0 equals
14 . Find the sum of the series
15. Let A be an m n real matrix. (a) Show that N(A) \ Im(AT ) = f0g, where AT is the transpose of A, Im(A) is the image of A and N(A) = fv 2 Rn : Av = 0g. (b) If for two suitable matrices B and C, AATB = AATC then show that ATB = ATC.
16. Let V be a nite dimensional vector space over R. Suppose that a subset A V has the following property: For any nite set of scalars a1; a2; : : : ; an
17. Let Xn = number of heads obtained from n independent coin tosses with probability of head p. Let pn be the probability that Xn is an even number.
18. Let C be a closed subset of Rn and r be a positive real number. Show that the set
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