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17thKVS-JMO2014 - Question Paper

This page contains questions of 17thKVS-JMO2014 . This exam held on 21 - 09 - 2014 . Exam duration 3 hrs and maximum marks 100 . Given 10 questions to solve . All questions are compulsory. Each question carries 10 marks. Use of electronic gadgets is not allowed .

Solutions of these are in next page .

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NOTE :  All questions are compulsory. Each question carries 10 marks . Use of electronic gadgets is not allowed .

Q1) Prove that for no integer n, \(n^{6}\) + 3 \(n^{5}\) -5 \(n^{4}\) - 15 \(n^{3}\) + 4 \(n^{2}\) + 12n + 3 is a perfect square .

2Q) Two dice are thrown once simultaneously. Let E be the event "Sum of numbers appearing on the dice ." what are the members of E ? Can you load these dice(not necessarily in the same way) such that all members of E are equally likely ? Give justification.

3Q) Let sin(x) + sin(y) = a and cos(x) + cos(y) = b, Show that tan(x/2) and tan(y/2) are two roots of the equation :

  \((a^{2}+b^{2} + 2b)t^{2}\) - 4at + \((a^{2} + b^{2} -2b)\) = 0 .

4Q) In a triangle ABC, angle A is twice the angle B. Show that \(a^{2}\) = b(b+c) , where a, b and c are the sides opposite to angle A, B and C respectively.

5Q) A, B, C, D are four points on a circle with radius R such that AC is Perpendicular to BD and meets BD at E. Prove that \(EA^{2}\) + \(EB^{2}\) + \(EC^{2}\) + \(ED^{2}\) = 4\(R^{2}\).

6Q) Suppose \(A_{1}A_{2}A_{3}........A_{n}\) is an n-sided regular polygon such that \(\frac{1}{A_{1}A_{2}}\) = \(\frac{1}{A_{1}A_{3}}\) + \(\frac{1}{A_{1}A_{4}}\) . Determine the number of sides of the polygon. 

 7Q) Find all positive integers a, b for which the number \(\frac{\sqrt{2} + \sqrt{a}}{\sqrt{3} + \sqrt{b}}\) is a rational number .

 8Q) If a, b, c are positive real numbers, prove that :

 \(\frac{\sqrt{a+b+c} + \sqrt{a}}{b+c}\) + \(\frac{\sqrt{a+b+c} +\sqrt{b}}{c+a}\) + \(\frac{\sqrt{a+b+c}+\sqrt{c}}{a+b}\) \(\geq\) \(\frac{9+3\sqrt{3}}{2\sqrt{a+b+c}}\) .

9Q) Find integers a and b such that \(x^{2}\) - x - 1 divides    a\(x^{17}\) + b\(x^{16}\) +1 = 0.

 10Q) Consider the equation \(x^{4}\) - 18\(x^{3}\) + k\(x^{2}\) + 174x - 2015 = 0. If the product of two of  the four roots of the equation is -31, find the value of k .

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