Cambridge Maths Academy

수학 모음 (Maths collection) 전체보기 Question 1. Given $x^2+y^2=r^2$, show that $$ \begin{align} \frac{{\rm d}y}{{\rm d}x}&=-\frac{x}{y} \\ \\ \frac{{\rm d}^2y}{{\rm d}x^2}&=-\frac{r^2}{y^3} \end{align} $$ Solution. 더보기 We use implicit differentiation which is an application of chain rule and product rule as we will see here. $$ \begin{align} && x^2+y^2&=r^2 \\ &\Rightarrow& \frac{{\rm d}}{{\rm d}x} \l..

수학 모음 (Maths collection) 전체보기 Question. In the diagram, $ABC$ is the line with equation $$ \begin{align} y=-\frac12x+5 \end{align} $$ It is also given that $AB=BC$ and that $D$ is the point with coordinates $(-13,0)$. Find an equation of the line through $A$ and $D$. Solution. $B$ is the $y$-intercept of $y=-\frac12x+5$: $$ \begin{align} B=(0,5) \end{align} $$ $C$ is the $x$-intercept of $y=-\fr..

수학 모음 (Maths collection) 전체보기 Question. Triangle $HJK$ is isosceles with $HJ=HK$ and $JK=\sqrt{80}$. (i) $H$ is the point with coordinates $(-4,1)$. (ii) $J$ is the point with coordinates $(j,15)$ where $j < 0$. (iii) $K$ is the point with coordinates $(6,k)$. (iv) $M$ is the midpoint of $JK$. (v) The gradient of $HM$ is 2. Find the value of $j$ and the value of $k$. Solution. From (i), we can l..

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..

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 $$ ..