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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 …

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…

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…

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 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…

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…

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

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 $(n+1)!+1=dk$..............…

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…

If $ab=0$ what is the possible integer value of $a$ and $b$. Answer   The only zero divisor in the Ring of integers is zero. Hence the simplest answer to this question i…