Cambridge Maths Academy

수학 모음 (Maths collection) 전체보기 1. Area of a regular octagon Question. Find the area of a regular octagon with side length $a$. (4-9 Higher GCSE by Michael White, Ch13 Geometry 4, p.426 Q19.) We consider the area of the triangle with an angle $\theta$ subtended at the centre. $$ \begin{align} \theta=\frac{360}{8}=45^\circ \end{align} $$ The height $h$ of the triangle is given by $$ \begin{align} \..

수학 모음 (Maths collection) 전체보기 Question. Misha invests ￡4000 in a bank with a compound interest rate of 3.75% per annum. At the end of each year Misha has to pay 20% tax on the interest made during that year. After 4 years Misha finds a ruby stone valued at 4500. Has he go enough money in the bank to afford to buy the ruby stone? If not, how much more would he need to save? Solution. For GCSE/mid..

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..

수학 모음 (Maths collection) 전체보기 This question comes from a test paper for A-level maths for Michaelmas Term at Eton College. Q16. (a) Solve the equation $$ \begin{align} \log_{10}\left(3^x-5^{2-x}\right)=2+\log_{10}(2)-x\log_{10}(5), \end{align} $$ giving your answer as exact. [3 marks] (b) Solve the equation $$ \begin{align} \left(5\sqrt{5}\right)^{3x}-31=30\left(\frac1{\sqrt[4]{5}}\right)^{9x}, ..

수학 모음 (Maths collection) 전체보기 This question comes from a test paper for A-level maths for Michaelmas Term at Eton College. Q15. (a) Express $\cos(2\theta)+\sin(2\theta)$ in the form $R\sin(2\theta+\alpha)$ where $R>0$ and $0 < \alpha < \frac{\pi}{2}$. [3 marks] (b) Hence show that $$ \begin{align} \frac{1-\sqrt{2}}{2}\le \cos\theta(\cos\theta+\sin\theta) \le\frac{1+\sqrt{2}}{2} \end{align} $$ fo..