Math topic中文

Linear law

transformation of given relationships, including $y = ax^{n}$ and $y = kb^{x}$ , to linear form to determine the unknown constants from a straight line graph

Practice questions

1The mass, m mg, of a radioactive substance decreases with time, t hours. Measured values of m and t are given in the tab…
2A drug is administered to a patient through an injection. The concentration of drug, $x$ mg/ $l$ in a patient's bloodstre…
3The table shows experimental values of two variables, $x$ and $y$ . | $x$ | 2 | 4 | 6 | 8 | |---|---|---|---|---| | $y$ |…
4Measured values of $x$ and $y$ are given in the following table. | $x$ | 1.2 | 1.6 | 2.0 | 2.6 | 3.5 | 4.5 | |---|---|--…
5The table shows experimental values of $x$ and $y$ . | $x$ | 2 | 3 | 4 | 5 | 6 | |---|---|---|---|---|---| | $y$ | 11.8 |…
6The diagram shows part of the curve $y = 4 - e^{-2x}$ which crosses the axes at $A$ and at $B$ . Find the coordinates of…
7A straight line graph, obtained by plotting $\frac{x}{y}$ against $x^2$ , passes through the points $(-2, 5)$ and $(4, 8)$ …
8A pot of melted chocolate is cooled from its initial temperature to a temperature of $T^\circ\text{C}$ in $x$ minutes an…
9The table below shows experimental values of two variables $x$ and $y$ . It is known that $x$ and $y$ are related by an e…
10The variables x and y are known to be related by the formula $y = px + \frac{q}{x}$ where p and q are constants. The tab…
11The population $P$ , in millions, of a country was recorded on January of the various years and the results are shown in…
12A researcher conducted a Particle Filtration Efficiency (PFE) test on surgical masks and reusable masks which have been…
13The diagram shows part of the curve $y = 27 + \frac{8}{(x-2)^3}$ which crosses the axes at points $A$ and $B$ . Explain w…
14The number, n, of a certain type of bacteria increases with time t minutes. Recorded values of n and t are shown in the…
15State one aspect of the graph that may be misleading and explain how this may lead to a misinterpretation of the graph.…
16The diagram shows the curve $y = \cos x + \sin x$ for $0 \le x \le \pi$ . The curve meets the $y$ -axis at the point $A$ a…
17The approximate mean distance $x$ (in millions of kilometres) from the centre of the Sun and the period of the orbit $T$ …
18The population of a certain species of bacteria, $N$ , is inversely proportional to the square root of the temperature, $t^\circ\text{C}$ …
19Find $\dfrac{\mathrm{d}}{\mathrm{d}x}\left(3x^{2}\ln x\right). Hence find$ \int x\ln x,\mathrm{d}x $. The diagram shows the line$ …
20The variables $x$ and $y$ are related by $y = \frac{2x+8}{2x+1}$ . When values of $x(1-y)$ are plotted against $y$ , a str…
21The time taken, t in minutes, to download a particular gaming software is related to internet speed, x Mbps. The variabl…
22The diagram shows part of a straight line obtained by plotting $x^2 y$ against $x$ . Find an expression for $y$ in terms…
23A solid was heated and left to cool in a container in a room. The difference between its temperature and the surrounding…
24Complete the table for $y = 9x^3 + 5x^2 - 3x + 1$ . Give your answer correct to 1 decimal place. | $x$ | $-1$ | $-0.75$ |…
25The table shows experimental values of two variables, x and y, which are connected by an equation of the form $y = \frac{k}{x^{n}}$ …
26An object moving in a certain medium, with speed $v$ m/s, experiences a resistance to motion of $R$ newtons. $R$ and $v$ …
27It is known that x and y are related by an equation, $y = ab^x + 4$ where a and b are constants. Draw a straight line gr…
28Solve the equation $(\ln x)^2 + \frac{2}{\log_x e} = 3$ . Mrs Tan decides to track the relationship between the age, $x$ …
29The table shows experimental values of two variables, $x$ and $y$ . | $x$ | 0.5 | 1.0 | 1.5 | 2.0 | |---|---|---|---|---|…
30The mass, m mg, of a radioactive substance decreases with time, t hours. It is known that m and t are related by the equ…
31The temperature, $\theta^\circ C$ , of a cup of hot chocolate milk, as it cools to room temperature, can be modelled by t…
32Mr Lu invested a sum of money in a bank in 1980. The value of the investment, $V$ in thousands dollars can be modelled b…
33The table shows experimental values of two variables, $x$ and $y$ , which are connected by the equation $y = ae^{bx - 1}$ …
34Sketch, on the axes below, the graphs of $y = -3\sin x$ and $y = 1 - \frac{2x}{\pi}$ for $0 \le x \le 2\pi$ . Find the nu…
35Designer bag X was first released in 1950. The price, $V$ , of the bag is related to $t$ , the number of years since 1950,…
36The diagram shows part of the curve $y = 5 + \frac{4}{e^{2x}}$ intersecting the $y$ -axis at point $P$ . The tangent to th…
37The table below shows experimental values of two variables $x$ and $y$ , which are connected by an equation of the form $\frac{a}{y} = \frac{b}{x} + 1$ …
38On the Venn diagram, shade the region which represents $(A \cup B') \cap (A' \cup B)$ . $\mathcal{E} = \{(x, y): x + y \le 10\}$ …
39An auction house claimed that it is worthwhile to invest in their art pieces as the value of one of their art pieces has…
40The diagram shows part of the curve $y = -\sqrt{10 - 6x}$ meeting the x-axis at the point A. The normal to the curve at…
41The diagram shows part of the curve $y = 1 - \sin x$ passing through the point $P$ . The curve touches the $x$ -axis at th…
42The table shows experimental values of two variables x and y. It is known that x and y are related by the equation $y = A k^x$ …
43The table shows the population, $P$ , in thousands, of a small town decreases with time, $t$ years. Show your working cle…
44Sketch the graphs of $y=-2\cos 3x+1$ and $y= -\dfrac{2}{\pi}x+1$ on the same axes, for $0 \le x \le \pi$ . Explain how th…
45In an experiment to investigate the relationship between light intensity I lux on a surface and distance d metres from a…
46The table shows the experimental values of two variables x and y. Express the equation in a form suitable to draw a stra…
47The diagram below shows the curve $y = x^3 - 12x^2 + 20x$ , which crosses the $x$ -axis at the origin $O$ and the points $A$ …
48The diagrams $A$ , $B$ , $C$ , $D$ , $E$ and $F$ show six graphs of different functions. Select the diagram that best shows…
49The function $f$ is defined, for $0 \le x \le 2\pi$ , by $f(x) = 1 - 2 \cos 2x$ . State the period and amplitude of $f$ . S…
50The diagram shows part of a straight line graph obtained by plotting $\frac{1}{y}$ against $\frac{1}{x}$ . Express $y$ in…
51For the equation $py = xy + 1$ where $p$ is an unknown constant, may be represented by a straight line, expressed in the…