## 06 May Mathletics 2013 (Category IV) - Assam Academy of Mathematics

**(Class XI and XII)**

Marks: 100

Time: 3 Hours

[Answer in English. Two students of the group will discuss the solution of the problems and then write the answer in a khata for the group. No third person can help the group to solve the problems.]

*Answer the following ten questions*

Marks: 10X10=100

1. Prove that every number of the sequence

49, 4489, 444889, 44448889, ........... is a perfect square.

2. If and Find the value of K.

3. Show that is divisible by for all

4. Find the number of integral solution of the equation

5. If in a triangle ABC, the line joining the circumcentre O and incentre I is parallel to BC. Prove that CosB+CosC=1.

6. If are n unequal positive real numbers then show that

7. For what real value of a, is one of the roots of the equation

greater and the other smaller than unity?

8. Let be a function such that

and

Find

9. If a, b, c are any three integers, then show that

is divisible by 7.

(Hint: Any cube divided by 7 leaves remainder 0, 1 or 6.)

10. Find the largest number in the infinite sequence:

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