Math topic中文

Simplify solve

simplifying expressions and solving simple equations involving exponential and logarithmic functions

Practice questions

1Answer all questions. Given that $a = \sqrt{\frac{b^2+4ac}{2a}}$ , find the values of $b$ when $a = 4$ and $c = -1$ . Answ…
2Mr Phua bought a new car. After $t$ months, its value $V$ is given by $V = 250000e^{-pt}$ , where $p$ is a constant. The…
3The diagram shows part of the curve $y = 3e^{-\frac{1}{2}x} - 6$ which crosses the $x$ -axis at $P$ and the $y$ -axis at $Q$ …
4The decay of a certain radioactive isotope can be modelled by the exponential equation $N = N_0 e^{-at}$ after $t$ wee…
5Given that $\sqrt{125^x} = \dfrac{5^{1-x}}{25}$ , find the value of $\sqrt{125^x}$ . Show that $\log_2 x - \log_{16} x = \dfrac{3\lg x}{4\lg 2}$ …
6The solution of the equation $3^{2x-1}=4(3^x)$ can be expressed as $\dfrac{\ln p}{\ln q}$ . Find the value of $p$ and o…
7Jane bought an electric car in January 2025. After purchase, the value of the car, $V, diminishes over time and can be…
8A drug is administered to a patient through an injection. The concentration of drug, $x$ mg/ $l$ in a patient's bloodstre…
9A motorist, travelling at a constant velocity of V m/s, passed a fixed point X and saw few vehicles ahead. He stepped on…
10Given that $u = 2^x$ , express $2^{x+1} = 5(2^x) + 3$ as an equation in $u$ . Hence find the value of $x$ for which $2^{x+1} = 5(2^x) + 3$ …
11It is given that $\frac{2^x \times 32(2^x)}{8^{x+1}} = \frac{9(5^{2x}) - 5^{2x+1}}{5^x - 5^{x-1}}$ . Evaluate $10^x$ with…
12Radiocarbon dating, or carbon-14 dating, is a scientific method that can accurately determine the age of organic materia…
13Use the substitution $u = 7^x$ , solve the equation $7^{2x+2}-4(7^{x+1})-5=0$ and show that the solution may be written i…
14Given that $y = \frac{4}{5}\left(\frac{x}{12} - 1\right)^6$ and that both $x$ and $y$ vary with time, find the value of…
15Solve the equation $2^{x+1}+2^{-x}=3$ . The equation of a curve is $y=2x^2-4ax+b$ where $a$ and $b$ are non-zero constant…
16Given that $\frac{\sqrt[3]{2^2} \times 3^{\frac{1}{5}}}{6^3} \times \left(\frac{2}{3}\right)^{-2} = 2^a \times 3^b$ , fin…
17Given the curve $y = \frac{1}{2}e^{2x} - \frac{1}{3}e^{-3x} + \frac{1}{3}$ and the normal to the curve at point $P\left(\ln 2, \frac{55}{24}\right)$ …
18已知等比数列 $\{a_{n}\}(a_{1}\neq a_{2})$ 的公比为 $q$ ，且 $a_{7}$ ， $a_{1}$ ， $a_{4}$ 成等差数列，则 $q=()$ A. $1$ 或 $- 32$ B. $- 32$ …
19Given that $u = \left(\dfrac{3}{2}\right)^x$ , express $9^x + 6^x = 6(4^x)$ as a quadratic equation in $u$ . Hence, or oth…
20Solve the equation $x + 1 + \sqrt{5 - x} = 0$ . Solve the equation $\log_x(2x-1) = \log_{x^2}(4-5x)$ .
21The point A lies on the curve $y = x \ln x^2$ , $x > 0$ . The tangent to the curve at A is parallel to the line $y - 3x = 1$ …
22Solve the equation $\log_3 x - 3\log_x 9 = 1$ .
23Solve the equation $2^{4p+1} + 20(4^{p-1}) = 3$ . Explain why $2^{4p+1} + 20(4^{p-1}) = k$ has no solution if $k < -3\frac{1}{8}$ …
24By using an appropriate substitution, solve the equation $e^{\frac{1}{2}x} = 2 + 24 e^{-\frac{1}{2}x}$ .
25Show that the equation $3e^{2x}+7e^{x}=6$ has only one solution and find its value correct to 2 significant figures. Fac…
26Find the value of $10^x$ , given that $4^{x+1} \times 5^{2x-3} = 10^{3x}$ . Hence, solve $4^{x+1} \times 5^{2x-3} = 10^{3x}$ …
27Given that $y = 3e^{2x} + 2e^{-x}$ , and that $\frac{\text{d}^2 y}{\text{d}x^2} + \frac{\text{d}y}{\text{d}x} + y = Ae^{2x} + Be^{-x}$ …
28Given that $\frac{49^{2x-3}}{7^{3x}} = \frac{7^{2x+1}}{343^{x-1}}$ , find the value of $\sqrt[3]{343^x}$ .
29Solve the following equations. $5^x - 5^{\frac{x}{2}+1} = 6$ , $2\lg(x - 3) - \lg(x + 7) = \frac{1}{\log_{100} 10}$ .
30The population of bacteria, $P$ , after $t$ hours can be modelled by $P = P_0(2^t)$ , where $P_0$ is a constant. Find the…
31The diagram shows part of the curve $y = 4 - e^{-2x}$ which crosses the axes at $A$ and at $B$ . Find the coordinates of…
32Sketch the graph of $y = 3^x$ . In order to solve the equation $\log_9(5-x) = \frac{x}{2}$ , a suitable straight line has…
33The diagram shows part of the curve $y = e^{x-1} - 2$ . The point $P(2,\ e-2)$ lies on the curve and the normal to the cu…
34A metal cube is heated to a temperature of $212^\circ\text{C}$ before being dropped into a liquid. As the cube cools, it…
35Given that $4(5^{x+3}) = 20^{3-x}$ , evaluate $10^x$ without using a calculator.
36Show that the solution of the equation $2^{3x+4}\times 5^{2x-1}=16^{x}\times 5^{3x}$ is $\lg\dfrac{16}{5}$ . Express $2\log_{2}x-\log_{2}(x-4)=3$ …
37A pot of melted chocolate is cooled from its initial temperature to a temperature of $T^\circ\text{C}$ in $x$ minutes an…
38In an experiment, 5 million cells were stored in a container. It was observed that 40% of the cells are dying every minu…
39By considering the general term in the binomial expansion of $\left(x^{4}-\frac{1}{k x^{2}}\right)^{6}$ , where $k$ is a…
40Find the area of the shaded region. The figure above is not drawn to scale.
41The table below shows experimental values of two variables $x$ and $y$ . It is known that $x$ and $y$ are related by an e…
42Solve the equations. $\log_2(x+4) = 2\log_2 x - 1$ $10\log_y 5 + 3 = \log_5 y$
43若曲线 $y=e^{-x}$ 上点 $P$ 处的切线平行于直线 $2x+y+1=0$ ，则点 $P$ 的坐标为 ___ ．
44The mass, $m$ grams, of a radioactive substance detected in a piece of stone is given by the formula $m = ae^{-kt}$ , whe…
45Find the values of $x$ and $y$ which satisfy the equations $2^x \div 4^y = \frac{1}{8},$ $9^y (\sqrt{3})^{2x} = 3\sqrt{3}.$ …
46The population $P$ , in millions, of a country was recorded on January of the various years and the results are shown in…
47Sketch the graph of of $y = \log_2(3x + 1)$ and label the point where $x = \frac{1}{3}$ . Explain why $x > -\frac{1}{3}$ .…
48Solutions to this question by accurate drawing will not be accepted. $ABCD$ is a trapezium where $AD$ is parallel to…
49The number, n, of a certain type of bacteria increases with time t minutes. Recorded values of n and t are shown in the…
50Given that $\log_a 8 + 3\log_a N = 3$ and $N > \frac{1}{4}$ , find the range of values of $a$ . Solve the equation $\log_3 x - \log_x 3x - 2 = 0$ …
51已知 $\tan\alpha$ ， $\tan\beta$ 是方程 $\lg(3{{x}^{2}}-x-2)=0$ 的两个实数根，则 $\tan(\alpha+\beta)=(\quad)$ A. $2$ B. $\dfrac{1}{5}$ …
52The equation of a polynomial is given by $\mathrm{p}(x) = 4x^3 + x - 5$ . Find the remainder when $\mathrm{p}(x)$ is divi…
53To prepare for the General Election, a political party studies the effectiveness of spreading information on Instantglam…
54Given that $\frac{(\log_y x)^2}{\log_x y} + 64 = 0$ , express $y$ in terms of $x$ . Solve the equation $\log_2 x + 3 = 2\log_2(x - 1)$ …
55Solve $9^{x+1} - 3^{x+2} = 2\left(3^{x+1} - 2\right)$ . Given that $\log_3(2x-1) = \log_9(2x-1) + 1$ , find the value of $x$ …
56Find $\dfrac{\mathrm{d}}{\mathrm{d}x}\left(3x^{2}\ln x\right). Hence find$ \int x\ln x,\mathrm{d}x $. The diagram shows the line$ …
57Find $\dfrac{d}{dx}\left(\dfrac{\ln x}{4x}\right). Hence, determine the range of values of$ x $for which the curve$ y=\d…
58Given that $4(5^{x+3}) = 20^{3-x}$ , evaluate $10^x$ without using a calculator.
59Solve the equation $2^{x+1} + 3(2^{-x}) = 7$ .
60Show that the equation $2(e^x - 3) = e^{\frac{1}{2}x}$ has only one solution and find this value correct to 3 significan…