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Evaluate $x$ in the equation $\frac{x^2-10}{x-3}+\frac{1}{x-3}=\frac{1}{5}$ Solution First we need to find the Lowest Common Multiple (LCM) of the left hand …

Echoes of Victory from KMUTT - An epistle of accolades on Dr. Jamilu Abubakar Academia is a field of patience and resilience, only those who are willing to …

Let $f:A\rightarrow{A}$ with $f(x)=\frac{x+2}{x-3}$ for $A=\mathbb{R}/\{1,3\}$ Show that $f$ is a one-to-one function. Proof: we are given $f:A\rightarrow{A}…

Let $a,b\in\mathbb{R}$. Prove th inequality $\frac{a}{b}+\frac{b}{a}\geq{2}$ Proof We are given $\frac{a}{b}+\frac{b}{a}\geq{2}$, . Let $(a-b)^2\geq{0}$ so t…

a) We are to find area of $\bigtriangleup{ABC}$ This is quite simple, all we need to do is to use the idea of similar triangles, to do this, simply extract the $\bigtria…

Proof that for any integer $n>1$ the number $n^5+n^4+1$ is not a prime. Simply rewrite $n^5+n^4+1$ as $n^5+n^4+n^3-n^3-n^2-n+n^2+n+1$ Factor out common terms  $n^3(n^…

The product of the third and the sixth terms of an arithmetic sequence is $406$. The ninth term of the sequence divided by the fourth term gives a quotient of $2$ and a …

Find the sum of all natural numbers between $1$ and $1000$ that are not divisible by $13$. Let $S$ be the unknown sum and $M$ multiples of $13$ less than $1000$, then $S…

Lecture series by Dr Kamran Khan on Complex Integration Kamran Khan is an assistant professor at the department of mathematics, Aligarh Muslim Universi…

Solve the equation $2\cosh{2x}+10\sinh{2x}=5$ ................. 🆔 This is a hyperbolic trigonometric function equation, in order to solve the given equation, we must re…

Find the integer solution $x^2+3x+9=9y^2$.............(1) This is a beautiful quadratic Diophantine's equation that requires an integer solution. So how …

Find the value of $x$ $\frac{x^{\sqrt{x}}} {\sqrt{x}}=\frac{{\sqrt{x}}^x}{\sqrt{x}}$ Since both sides of the equation have a common denominator, multiply through by $\sq…

Find the integer solution for $3^x+27^y+81^z=10935$ This is a rather thickish question so we have to be very careful in our simplifications. The left hand side of the eq…

olve the logarithmic equations $\log_3(x+1)=\log_4(x+8)$ The bases of this equation is different so Let $\log_3(x+1)=y$...............................(i) And Let $\log_4…

Solve the indicial equation $4^\frac{1}{x}-6^\frac{1}{x}=9^\frac{1}{x}$ To solve the equation, simply divide through by any of the term since the indices are all given i…

Solving quadratic equations using the Poh-Shen Log method. The Poh-Shen Loh method is another method for solving quadratic equations, the method was pioneered by Prof. P…

An object is thrown vertically upwards with an initial velocity, $u$ of $20m/s$. The motion of the object follows the differential equation $\frac{ds}{dt}=u-gt$, where $…

Below is a beautiful proposition on matrix eigenvalues. It says find the minimal eigenvalue of the matrix $A^{2021}$. We find the eigenvalue using the determinant for $|…

The question below was obtained from one of the past questions for the Skoltech institute entrance examination for admission into computational science masters program. …

Prove of the Cassini-Samson Identity The Cassini-Samson Identity is an identity that involves the Fibonnaci numbers, it states that $F_{n-1}F_{n+1}-F_{n}^2=(-1)^n$ for $…

Determine the value of the following series $\sum^{\infty}_{n=1}(\frac{4}{n^2+4n+3}-9^{-n+2}4^{n+1})$ Solution To determine their value, we must check if the series is a…

The question says evaluate the polynomial $\frac{x^2}{ax+b}$ using long division. Therefore, $f(x)=x^2=(ax+b)(\frac{x}{a}-\frac{b}{a^2})+\frac{b^2}{a^2}$.

Lucas number: another beautiful patterns of sequence Formula for obtaining the Lucas numbers The Lucas numbers are numbers that form Fibonacci integer sequence, each Luc…

The harmonic functions and construction of analytic functions.  In this video series Dr. Kamran Khan takes us through harmonic functions and construction of analytic fun…

How to curry and uncurry a function $\mathbb{R}\rightarrow\mathbb{R}^N$.  The functions $A\rightarrow{B}^C$ corresponds to functions of the form $A\times{C}\rightarrow{B…

How do you proof $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$? It is not always true. To see the clear picture, we solve some clear examples. Remember, $\sqrt{\frac{a}…

Find $x$ if $\sec^2{x}+\tan^2{x}=3$ From our trigonometric identities $\sec^2{x}=\frac{1}{\cos^2{x}}$ $\tan^2{x}=\frac{\sin^2{x}}{\cos^2{x}}$ Now, $\frac{1}{\cos^2{x}}+\…

Verify that $(n!+1,(n+1)!+1)=1$ Let $(n!+1,(n+1)!+1)=d$ Then there exists $(h,k)=1$ such that $n!+1=dh$.....................(i) $n!=dh-1$ and $(n+1)!+1=dk$..............…

Polar form of Cauchy-Riemann equations In this lecture series, Dr. Kamran Khan takes us through polar forms of Cauchy-Riemann equation, how they are derived and example…

Example of a function that satisfies the Cauchy-Riemann equation. In this lecture Dr. Kamran Khan takes his time to solve a problem that satisfies the Cauchy-Riemann equ…

Analytic functions and the Cauchy-Riemann equation - Lecture II In this lesson Professor Kamran Khan takes us through the Cauchy Riemann equation which is an equation th…

Introduction to complex analysis by Dr. Kamran Khan. Kamran Khan is an assistant professor at the department of mathematics, Aligarh Muslim University, Aligarh India. Ka…

Homomorphism of Groups: properties and examples. In abstract mathematics (algebra), homomorphism can be defined as a map or relation between two algebraic structures of…

The finite and infinite groups, the Lagrange theorem, its proof and a corollary. Today i will be discussing about the finite and infinite groups, and just like the set t…

What is an ordinal number?  Ordinal numbers was introduced in 1883 by George cantor, it is the process of arranging natural numbers in orderly(increasing or decreasing o…

The cyclic group $Z_8$(integer modulo 8) has two nontrivial subgroups. Let $G$ be a cyclic group $Z_8$ under group addition, $G=\{0,2,4,6,1,3,5,7\}$. Then the subgrou…

Subgroup: the subsets in group theory. In our last lesson, we gave a brief analogy of groups , in this lesson we will be discussing subgroups, how they are formed and t…

Group theory and its examples Group theory studies abstract structures called groups. Algebraic structures such as rings, fields, and vector spaces, can all be seen as g…