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 $$(2-3lambda+lambda)^K=A_0+A_1lambda+A_2lambda^2+dots +A_{2k}lambda^{2k}$$ 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:

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