by Ashani Dasgupta

State True or False: The set of nilpotent matrices of \( M_3 (\mathbb{R} ) \) spans \( M_3 (\mathbb{R} ) \) considered as an \( \mathbb {R} \) – vector space ( a matrix A is said to be nilpotent if there exists \( n \in \mathbb{N} \) such that \( A^n = 0 \) )....
by Ashani Dasgupta

Let, \( a \geq b \geq c > 0 \) be real numbers such that for all natural number n, there exist triangles of side lengths \( a^n,b^n,c^n \) Prove that the triangles are isosceles. Hint 1 - Triangular Inequality If a, b, c are sides of a triangle, triangular inequality...
by Ashani Dasgupta

Let \(a, b, c\) are natural numbers such that \(a^{2}+b^{2}=c^{2}\) and \(c-b=1\). Prove that (i) a is odd. (ii) b is divisible by 4 (iii) \( a^{b}+b^{a} \) is divisible by c Hint 1 - Isolate aHint 2 - Eliminate cHint 3 - A bit of Modular Arithmetic Notice that \( a^2...
by Ashani Dasgupta

Let \( \{a_n\}_{n\ge 1} \) be a sequence of real numbers such that $$ a_n = \frac{1 + 2 + … + (2n-1)}{n!} , n \ge 1 $$ . Then \( \sum_{n \ge 1 } a_n \) converges to ____________ Hint 1 - Sum of oddsHint 2 - Break in partialsHint 3 - Something goes to e! Notice...
by Ashani Dasgupta

ProblemHint 1Hint 2Hint 3Final Answer Suppose \(a,b\) are positive real numbers such that \(a\sqrt{a}+b\sqrt{b}=183\). \(a\sqrt{b}+b\sqrt{a}=182\). Find \(\frac{9}{5}(a+b)\). This problem will use the following elementary algebraic identity: $$ (x+y)^3 = x^3 + y^3 +...
by Ashani Dasgupta

There is an intuitive definition of perpendicularity. It does not involve angle. Instead, it involves the notion of distance. Consider a point P and a line L not passing through it. If you wish to walk from P to L, the path of shortest distance is the perpendicular...