## 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.]

Marks: 10X10=100

1. Prove that every number of the sequence

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

2. If $(2-3lambda+lambda)^K=A_0+A_1lambda+A_2lambda^2+dots +A_{2k}lambda^{2k}$ and $A_0+A_2+A_4+dots +A_{2k}=648.$ Find the value of K.

3. Show that $2^{3^n}+1$ is divisible by $3^{n+1}$ for all $nin N.$

4. Find the number of integral solution of the equation

$2x+y+z=20,$ $x,y,zge 0.$

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 $a_1,a_2,dots ,a_n$ are n unequal positive real numbers then show that

$frac{(1+a_1+a_1^2)(1+a_2+a_2^2)dots (1+a_n+a_n^2)}{a_1a_2dots a_n}ge 3^n.$

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

$(2a+1)x^2-ax+a-2=0$ greater and the other smaller than unity?

8. Let $f$ be a function such that

$f(1)+2f(2)+3f(3)+dots +nf(n)=n(n+1)f(n),nge 2$ and $f(1)=1.$

Find $f(1000).$

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

$abc(a^3-b^3)(b^3-c^3)(c^3-a^3)$ 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:

$1,^2sqrt{2},^3sqrt{3},^4sqrt{4},dots ,^nsqrt{n},dots$

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#### Gonit Sora

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