Cambridge Maths Academy

We have: $$ \begin{align} 1^1&=1 &&& 0^1&=0 \\ 1^0&=1 &&& 0^0&=1 \end{align} $$ Are you surprised by $0^0=1$? Proof. To prove this, we consider $$ \begin{align} y=x^x \end{align} $$ and take the limit $x\rightarrow0$. It is not straightforward to do this directly with $x^x$ so we take the (natural or any) logarithm on both sides: $$ \begin{align} \ln y=\ln x^x=x\ln x \end{align} $$ Let $x=e^{-n}..

수학 모음 (Maths collection) 전체보기 Question. The line $L$ has equation $y = c − mx$, with $m > 0$ and $c > 0$. It passes through the point $R (a, b)$ and cuts the axes at the points $P(p, 0)$ and $Q(0, q)$, where $a, b, p$ and $q$ are all positive. Find $p$ and $q$ in terms of $a, b$ and $m$. As $L$ varies with $R$ remaining fixed, show that the minimum value of the sum of the distances of $P$ and $Q..

In mathematics, we encounter terms, equations, expressions, inequalities and identities. $7xy$ is a term. $5x-2=8$ is an equation. $7x^2+3x$ is an expression. $5x-2

수학 모음 (Maths collection) 전체보기 Question. Rebecca has 9 cards, each with a number on them. The numbers are: $$ \begin{align} 2,\quad 2,\quad 3,\quad 4,\quad 5,\quad 6,\quad 6,\quad 7,\quad 9 \end{align} $$ She picks three cards at random without replacement. Rebecca multiplies three numbers to get a score. Calculate the probability that the score is an even number. Solution. Let's consider: the ca..

원뿔의 표면적을 구하기 위해서 전개도를 고려한다. 전개도는 원뿔을 잘라 펼친 그림의 의미한다. 전개도에서 부채꼴(sector)의 넓이와 원의 넓이를 구한 다음 둘의 넓이를 더하면 된다. 부채꼴의 넓이: 부채꼴 안에 있는 각도 $\theta$를 구해야 하는데, 이는 호의 길이를 고려하여 구할 수 있다. $$ \begin{align} L=2\pi\ell\times\frac{\theta}{360}&=2\pi r \\ \Rightarrow\quad \ell\times\frac{\theta}{360}&=r \end{align} $$ 부채꼴(sector)의 넓이: $$ \begin{align} A_{\rm sector}&=\pi\ell^2\times\frac{\theta}{360} \\ &=\pi\ell\times..

수학 모음 (Maths collection) 전체보기 To find the surface area of a circular cone, we need to consider the area of the circular sector and the area of the circle. The area of the circular sector: We find the angle $\theta$ by considering the arc length, i.e. $$ \begin{align} L=2\pi\ell\times\frac{\theta}{360}&=2\pi r \\ \Rightarrow\quad \ell\times\frac{\theta}{360}&=r \end{align} $$ The area is then $$ ..

Pure mathematics Year 1 Table of contents Introduction Factorial $n!$ and arrangements Permutations (${}_nP_r$) Combination: n choose r An alternative derivation of the formula for 'n choose r' 0! = 1 revisited Pascal's triangle using combinations Examples Edexcel P1 Ch8 Exercise 8B 1. Introduction While Pascal's triangle is a convenient tool, for large values of the index, $n$, it takes a long ..

Pure mathematics Year 1 Table of contents Binomial, trinomial and polynomial Binomial expansion Pascal's triangle Examples Edexcel P1 Ch8 Exercise 8A 1. Binomial, trinomial and polynomial The binomial expansion tells us how to expand $(a+b)^n$. The word 'bi-nomial' means 'two terms' - here, they are $a$ and $b$. Similarly, monomial means 'one term', trinomial means 'three terms' and polynomial m..