Math topic中文

Interpreting

Interpreting a differential equation and its solution in terms of a problem situation.

Practice questions

1In a large town, the number of people infected by a particular virus $t$ days after the virus was first discovered is $x$ …
2A freshly brewed cup of tea, initially at a temperature of $85^\circ\text{C}$ , is left out to cool in a room with a room…
3In a large town, the number of people infected by a particular virus $t$ days after the virus was first discovered is $x$ …
4In harvesting of renewable natural resources, it is desirable that policies are formulated to allow maximal harvest of t…
5Given that $z = e^{2x}\frac{dy}{dx}$ , express $\frac{dz}{dx}$ in terms of $x$ , $\frac{dy}{dx}$ and $\frac{d^{2}y}{dx^{2}}$ …
6In military exercises, parachutists jump from stationary helicopters and their motion is tracked by sensors tagged to th…
7The Stoke's Law states that when a spherical object of radius $R$ is falling at velocity $v$ in a fluid, the resistance…
8A heated metal block at $160^\circ\text{C}$ is left to cool in a laboratory with an ambient temperature of $30^\circ\text{C}$ …
9By using the substitution $u = x - y$ , solve the differential equation $\dfrac{dy}{dx} + 4(x - y)^2\cos^2 x = \sin^2 x.$ …
10A chemical substance Uru starts to form when $N$ units of Vibranium react with $M$ units of Kryptonite, where $M < N$ . L…
11Show, by using the substitution $w=\dfrac{y}{t^{2}}$ , that the general solution of the differential equation $t\dfrac{dw}{dt}=w\!\,^{2}t^{2}+2wt-2w$ …
12In a chemical reaction, a compound X is formed from two compounds Y and Z. The masses in grams of X, Y and Z present at…
13In a city of 5 million people, the number of customers of a new company increases at a rate proportional to the number o…
14A family of curves is given by the differential equation $x\dfrac{dy}{dx}-y=x(x-y)$ . By substituting $y=vx$ , show that t…
15In a particular chemical reaction, every 2 grams of $U$ and 1 gram of $V$ are combined and converted to form 3 grams of…
16[Cambridge] In a chemical reaction a compound $X$ is formed from a compound $Y$ . The masses in grams of $X$ and $Y$ …
17Glucose is a type of sugar in our blood and our body uses it for energy. Having low sugar in the blood for long periods…
18To prepare for the General Election, a political party studies the effectiveness of spreading information on Instantglam…
19Show that the differential equation $\frac{\text{d}y}{\text{d}x} - \frac{2y}{x} = x^3$ may be reduced by the substitutio…
20The S-I model is used to study the spread of infectious diseases across different scenarios. In this question, it is ass…
21Section A: Pure Mathematics [40 marks] (i) Prove that the substitution $u = x^2 + y^2$ reduces the differential equation…
22The variables $w$ , $x$ and $y$ are connected by the following differential equations: $e^{2x}\frac{dw}{dx}=w^{2}$ and…
23Given that $z = e^{2x}\frac{dy}{dx}$ , express $\frac{dz}{dx}$ in terms of $x$ , $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ .…
24An epidemiologist is studying the spread of a disease, dengue fever, which is spread by mosquitoes, in town $A$ . $P$ is…
25Botanists are studying how excessive logging is affecting the population of forest trees in a country in South-East Asia…
26In a city of 5 million people, the number of customers of a new company increases at a rate proportional to the number o…
27A heated metal block at $160^\circ$ C is left to cool in a laboratory with an ambient temperature of $30^\circ$ C. The te…
28A wafer fabrication company uses the floating-zone method to purify polysilicon ingots, each having a uniform cross-sect…
29In the theory of learning, the amount of material memorised by a person, $y$ units, in time $t$ minutes satisfies the di…
30In a chemical reaction, compounds X and Y react to form a product. Let $x$ and $y$ denote the concentrations, in mol per…
31[Cambridge/9709/2004] A rectangular reservoir has a horizontal base of area 1000 $\text{m}^2$ . At time $t = 0$ , it i…
32Jim places an 8 m rod $XY$ against a vertical wall, as shown in the diagram below. The points $O, X$ and $Y$ are coplana…
33Using integration by parts, or otherwise, find $\displaystyle \int \frac{1}{y}\ln y\,dy$ . Given that $y>1$ and $\displaystyle \frac{dy}{dx}=\frac{y}{\ln y}$ …
34Baby Peter is pulling a toy wagon on a flat ground which is modelled by the $xy$ -coordinate plane. Peter is initially at…
35A metal ball is released from the surface of the liquid in a tall cylinder. The ball falls vertically through the liquid…
36The molecular geometry is the three-dimensional arrangement of atoms in a molecule. Fig. 1 shows the molecular geometry…
37[Cambridge/H2 Maths 2007] The current $I$ in an electric circuit at time $t$ satisfies the differential equation $4\frac{\text{d}I}{\text{d}t} = 2 - 3I.$ …
38In a chemical reaction, a compound $X$ is formed from two compounds $Y$ and $Z$ . The masses in grams of $X$ , $Y$ and $Z$ …
39Jesse leans a 6 metre rod $XY$ in the position as shown below. At time $t$ seconds, the two ends $X$ and $Y$ are $x$ and…
40In a chemical reaction, compounds X and Y react to form a product. Let x and y denote the concentrations, in mol per kil…
41By using the substitution $x = a\tan\theta$ , show that $\displaystyle \int \frac{1}{x\sqrt{x^2 + a^2}}\,dx = \frac{1}{a}\ln\frac{x}{\sqrt{x^2 + a^2} + a} + c ,$ …
42A Red Lion skydiver jumps off from a helicopter. He leaves the helicopter with zero speed, and his speed $v\text{ ms}^{-1}$ …
43A team of naturalists is studying the change in population of wild boars on an island. It is suggested that the populati…
44In a city of 5 million people, the number of customers of a new company increases at a rate proportional to the number o…
45Find the general solution of the differential equation cosec x $\frac{d^{2}y}{dx^{2}} = x$ . Find the general solution o…
46The diagram below shows the Gateway Arch, which is a monument in St. Louis, Missouri, United States. The arch stands at…
47Oil spill from a boat has a significant impact on the population of fish in a lake. The population (in thousands) $t$ da…