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. 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..

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..

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 Question. Evaluate the following integrals: $$ \begin{align} {\rm (a)}&& &\int\frac{1}{\sqrt{1-3x^2}}\,\textrm{d}x \\ {\rm (b)}&& &\int\frac{x}{4x^2+8x+13}\,\textrm{d}x \\ {\rm (c)}&& &\int_0^1\arcsin x\,\textrm{d}x \end{align} $$ Solution. (a) By substitution, $$ \begin{align} x&=\frac1{\sqrt{3}}\sin u \\ \Rightarrow\quad \textrm{d}x&=\frac1{\sqrt{3}}\cos u\,\textr..

수학 모음 (Maths collection) 전체보기 Question. Find $\frac{dy}{dx}$ for: $$ \begin{align} \textrm{(a)}&\qquad y = \arcsin(1-2x) \\ \textrm{(b)}&\qquad y = \arctan\left(x^2+1\right) \end{align} $$ Solution. (a) For $y=\arcsin(1-2x)$, we have $$ \begin{gather} -1\le 1-2x \le 1 \\ -\frac{\pi}{2}\le y \le\frac{\pi}{2} \end{gather} $$ We re-express the equation $$ \begin{align} y&=\arcsin(1-2x) \\ \Rightarr..

수학 모음 (Maths collection) 전체보기 There are goldfish in two bowls $P$ and $Q$. During one minute the probability of one goldfish jumping from bowl $P$ to $Q$ is $\frac25$. During one minute the porbability of one goldfish jumping from bown $Q$ to $P$ is $\frac13$. Calculate the probability that: (a) the number of goldfish in each bowl at the end of 1 minute is equal to the number of goldfish in each..

수학 모음 (Maths collection) 전체보기 Question. Lucy and Anton are partying with 8 friends. A photo of 6 people in a line is taken. How many different photos are possible if: (a) Lucy is always in the photo; (b) Lucy and Anton are always in the photo; (c) Only one of Lucy or Anton are in the photo. (Higher GCSE Maths 4-9 by Michael White, E8.1 Listing possible outcomes Q14.) Solution. (a) There are a to..