Cambridge Maths Academy

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..

Core pure mathematics 2 Table of contents Introduction Examples Exercise 7D 1. Introduction We can use given boundary conditions to find a particular solution to a second-order differential equation. Since there are two arbitrary constants, we will need two boundary conditions to determine the complete particular solution. 2. Examples Example 1. Find $y$ in terms of $x$, given that $$ \begin{ali..

Core pure mathematics 2 Table of contents Complementary function + Particular integral An example Another example Yet another example An investigation: more general cases Exercise 7C 1. Complementary function + Particular integral Second-order differentail equations of the form $$ \begin{align} a \frac{ \textrm{d}^2y }{ \textrm{d}x^2 } + b \frac{ \textrm{d}y }{ \textrm{d}x } + c y = \textrm f(x)..

Core pure mathematics 2 Table of contents First example Arbitrary constants = constants of integration Linear and non-linear differential equations Finding the general solution with the auxiliary equation More examples Exercise 7B 1. First example A second-order differential equation contains second derivatives. Example 1. Find the general solution to the differential equation $$ \begin{align} \..

Core pure mathematics 2 Table of contents First example - the method of separable variables Removing the modulus sign First-order DEs with the method of the integrating factor (I.F.) Second example Two important questions for the integrating factor More examples Exercise 7A First-order differential equations with constant coefficients 1. First example - the method of separable variables If a fir..

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..