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MATHMAA

# 17thKVS-JMO2014

## 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 .

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 .