[caption id="attachment_7349" align="alignleft" width="320"] Image Source : Shutterstock[/caption] The 2012 exam question papers of Mathematical Sciences, Assamese, English, Economics, Education, History, Philosophy, Political Science, Geography, Manipuri, Bodo, Bengali, Hindi, Anthropology, Sanskrit, Sociology, Commerce, Chemical Science, Life Science and Physical Science for SET conducted by SLET Commission, Assam...

01 September 2013 (Class XI and XII)   Marks: 10 X 10 = 100 Time: 1.30 pm to 4.30 pm Answer the following ten questions 1. Prove that $$4(x_1^4+ x_2^4+dots + x_{14}^4)=7(x_1^3+ x_2^3+dots + x_{14}^3)$$ has no solution in positive integers. (Hint: Suppose on the contrary $$sum_{k=1}^{14}(x_k^4-frac{7}{4}x_k^3)=0.$$ Also use $$sum (x_k-1)^4.$$)   2. Find...

01 September 2013 (Class IX and X)   Marks: 10 X 10 = 100 Time: 1.30 pm to 4.30 pm Answer the following ten questions 1. Show that there does not exist a function $$f:Nrightarrow N$$ which satisfy (a) $$f(2)=3,$$ (b) $$f(mn)=f(m)f(n)$$ for all m,n in N; (c) $$f(m)<f(n)$$ whenever $$m<n.$$ (Hint: Suppose the contrary....

01 September 2013 (Class VII and VIII)   Marks: 5 X 20 = 100 Time: 1.30 pm to 4.30 pm Answer the following questions 1. If the radius of a circle is increased by 100%, determine the increase percent in the area of the circle. এটা বৃত্তৰ ব্যাসাৰ্দ্ধ 100% বৃদ্ধি হ’লে বৃত্তটোৰ...

01 September 2013 (Class V and VI)   Marks: 5 X 20 = 100 Time: 1.30 pm to 4.30 pm Answer the following questions 1. Show that 52563744 is divided by 24 without direct division. হৰণ প্ৰক্ৰিয়া প্ৰয়োগ নকৰাকৈ প্ৰমাণ কৰা যে 52563744 সংখ্যাটো 24 ৰে বিভাজ্য।   2. Find the remainder when $$7^{84}$$...

1. Introduction Fixed point theory is very simple, but is based on fundamentals in Mathematics. For any continuous function $$f:Xrightarrow X$$ a fixed point of $$f$$ is a point $$xin X$$ satisfying the identity $$f(x)=x.$$ Two fundamental theorems concerning fixed points are Banach Theorem and Brouwer Theorem. Banach theorem states that if $$X$$ is a complete metric space and $$f$$ is a contraction then $$f$$ has a unique fixed point. In Brouwer theorem, $$X$$ must be the closed unit ball in a Euclidean space. Then any $$f$$ has a fixed point. But in this case, the set of fixed points is not necessarily a one-point set. In Banach theorem, a metric on $$X$$ is used in the crucial assumption that $$f$$ is a contraction. The unit ball in a Euclidean space is also a metric space and the metric topology determines the continuity of continuous functions, however the essence of Brouwer theorem is a topological property of the unit ball, namely the unit ball is compact and contractible. Banach theorem and Brouwer theorem tell us a difference between two major branches of fixed point theory, metric space fixed point theory and topological fixed point theory. It is impossible to distinguish two fixed point theories in an exact way, and it is not easy to determine a certain topics belong to which branch. In general, the fixed point theory is regarded as a branch of topology. But due to deep influence on topics related to nonlinear analysis or dynamic systems, many parts of the fixed point theory can be considered as a branch of analysis.

[caption id="attachment_3363" align="alignleft" width="418"] Google Doodle depicting Fermat's Last Theorem[/caption] We can find infinitely many solutions that can solve the equations $$x+y=z$$ and $$x^2+y^2=z^2$$ in integers, but what about the equation $$x^3+y^3=z^3$$ or more generally $$x^n+y^n=z^n$$, where n is an integer greater than 2 and x,y,...