Math topic中文

Solving equation

Solving an equation exactly or approximately using a graphing calculator or a graphing software.

Practice questions

1[The volume of a cone of base radius $r$ and height $h$ is given by $V = \frac{1}{3}\pi r^2 h$ .] A manufacturer makes a…
2The sum of the first $n$ terms of a series is given by $S_n = 3n(n + 2)$ . Show that this series follows an arithmetic pr…
3A drilling company plans to install a straight pipeline $AB$ through a mountain. Points $(x, y, z)$ are defined relative…
4[Cambridge/9709/2010] The diagram shows parts of the curves $y = 9 - x^3$ and $y = \frac{8}{x^3}$ and their points o…
5The sum of the first $n$ terms of a series is given by $S_n = 3n(n + 2)$ . Show that this series follows an arithmetic pr…
6A curve $C_1$ has the equation $p^2x^2 - y^2 = p^2$ where $p > 1$ . Sketch $C_1$ , stating the coordinates of any points o…
7Sketch, on the same diagram, the graphs of $y = |2x + 3|$ and $y = -2x^2 - 8x - 3$ . (You are not required to label any a…
8The sum of the first 100 terms of an arithmetic progression is 10,000. The first, second and fifth terms of this progres…
9Solve the equation $i z^5 = -32$ , giving your roots in the form $re^{i\theta}$ , where $r>0$ and $-\pi<\theta\le\pi$ . Ske…
10Given that $z = 2 - \text{i}$ is a root of the equation $4z^4 - 12z^3 + 17z^2 + pz + q = 0$ , where $p$ and $q$ are real,…
11Find $\displaystyle \int \frac{3e^x}{5 - 0.3e^x} \, dx$ $\displaystyle \int \cos(\ln x) \, dx$ , where $x > 0$ the exact…
12NJC JC1 Promo 9758/2019/Q6 A vessel is formed by removing a smaller cone of radius 5 m from a bigger cone whose semi-ver…
13Find the exact solution of the inequality $\frac{2x+1}{x^2-1} \le 1.$ A curve $G$ has equation $ay^2 + bx^3 + cx = 2$ , w…
14A sequence of numbers $u_{1}, u_{2}, u_{3}, \ldots$ has sum $S_{n}$ where $S_{n} = \sum_{r=1}^{n} u_{r}$ . It is given th…
15The following diagram shows the dimensions of a trapezoidal prism with fixed volume $4k\sqrt{3}$ units $^{3}$ , with varia…
16The curve $C$ has equation $y = \frac{x - 2}{kx^2 + x - 2}$ , where $k > 1$ . Find the equation of the tangent at the poin…
17[The volume of a right circular cone of radius $r$ and height $h$ is $\frac{1}{3}\pi r^2 h$ .] A student makes a printer…
18Solve $z^{4} = -4 - 4\sqrt{3}\,i$ , expressing your answers in the form $re^{i\theta}$ , where $r > 0$ and $-\pi < \theta \le \pi$ …
19Sketch, on the same axes, the graphs of $y=5-\dfrac{1}{x-1}$ and $y=\dfrac{1}{(x-5)^2}$ . Solve the inequality $5-\dfrac{1}{x-1}<\dfrac{1}{(x-5)^2}$ …
20Figure 1 shows an open cylindrical tank. Figure 2 shows the cross-sectional view of the tank. The external radius of the…
21A Hwa Chong student taking part in Project Day needs to design a rectangular poster to showcase his project. According t…
22[Cambridge/H2 Maths 2007] The complex number $w$ is such that $ww^* + 2w = 3 + 4i$ , where $w^*$ is the complex conju…
23Find the exact roots of the equation $\left|\dfrac{4x-5}{x-2}\right| = 3x. On the same axes, sketch the curves with equations$ …
24For events $A$ and $B$ it is given that $\mathrm{P}(A) = 0.55$ , $\mathrm{P}(B) = 0.65$ and $\mathrm{P}(A \cup B) = 0.93$ …
25A bag contains one green disc, $r$ red discs and $2r$ blue discs, where $r>1$ . Tim and Sim play a game by taking turns t…
26A random variable $X$ has the probability distribution given in the following table. | $x$ | 1 | 4 | 6 | 8 | |---|---|--…
27The function $g$ is defined by $g : x \mapsto \ln(x - \sqrt{3})$ , for $x \in \mathbb{R}$ , $x > \sqrt{3}$ . Explain how yo…
28The diagram shows an ellipse, centred at the origin with semi-minor axis $a$ and semi-major axis $2a$ , where $a > 0$ . Th…
29A curve $C$ is given parametrically by the equations $x = 2\cos^3 \theta$ , $y = 2\sin^3 \theta$ where $-\frac{\pi}{2} < \theta < \frac{\pi}{2}$ …
30Fig. 1 shows a piece of card, $ABCD$ , in the form of a rhombus of sides $a$ cm. The card is made up of 2 identical equil…
31In a particular soccer league of 20 teams, each team plays a total of 38 games over the course of a season. Teams are aw…
32Sketch the graphs of $y = 3e^x$ and $y = x + 3$ on the same diagram. Indicate clearly the coordinates of the points of i…
33Solve the inequality $|y-5|<1$ . Hence solve the inequality $|3^{x}-5|<1$ , giving 3 significant figures in your answer.
34The curve C is given by the equation $\dfrac{y^2}{x^2}=\dfrac{3a-x}{a+x},\quad x \ne 0,$ where $a$ is a positive const…
35The function f, with domain the set of all real values, is given by $\text{f}(x) = \begin{cases} 4 - x^2 & \text{for } 0 < x \le 2, \\ 2x - 4 & \text{for } 2 < x \le 4, \end{cases}$ …
36Section A: Pure Mathematics [60 marks] The curve $C$ has equation $y = \frac{x^2 - 2x + 4}{x - b}$ , $x \ne b$ . Given tha…
37The complex number $z$ satisfies the equation $\mathrm{f}(z) = 0$ , where $\mathrm{f}(z) = (1 - a\mathrm{i})z^2 - 2\mathrm{i}z - 10 - 20\mathrm{i}$ …
38Mr Dough's factory produces packets of flour for sale in local supermarkets packed in 2 sizes -- Regular-sized packets a…
39Do not use a calculator in answering this question. Given that $x = 1 + 3\mathrm{i}$ is a root of the equation $x^3 + ax^2 + 18x + b = 0,$ …
40Kelly has a box of spherical beads labelled with the 26 letters from A to Z, all of which have the same size. For each l…
41In a large town, the number of people infected by a particular virus $t$ days after the virus was first discovered is $x$ …
42A hyperbola has the equation $a(x-2)^2 - b(y+1)^2 = c$ where $a$ , $b$ , and $c$ are positive integers. It passes through…
43A Dunman High alumnus plans to donate money to build two koi ponds in the middle of Zheng Xin Yuan. The ponds are in the…
44A ball is rolling in a straight line such that its distance away from the starting point, $s$ cm, can be modelled using…
45The equations of planes $p_1$ and $p_2$ are given by $p_1:\;\mathbf{r}\cdot \begin{pmatrix}\alpha\\7\\2\end{pmatrix} = \alpha + 20,\; \alpha \in \mathbb{R},$ …
46Happie, the owner of a store selling novelty items, is organizing a publicity stunt for his store. Using a stock of ultr…
47An arithmetic sequence has first term $a$ and common difference $0.5$ , where $a$ is an integer. Find the smallest value…
48Sketch the graph of $y = \frac{7-2x}{3x^2 + x - 14}$ . Give the equations of the asymptotes and the coordinates of the tu…
49Find, in a simplified form, the cartesian equation of the locus of the point P representing the complex number z where $|z+3|=2\,\Re(z)$ …
50The diagram shows a shot put being projected with a velocity $v\text{ ms}^{-1}$ from the point $O$ at an angle $\theta$ …
51The following diagram shows the graphs of two differentiable functions $y=f(x)$ and $y=g(x)$ . At $x = c$ , the distance b…
52The curves $C_1$ and $C_2$ have equations $y = \frac{2x^2 + 9}{x^2 - 4}$ and $\frac{y^2}{25} - \frac{x^2}{9} = 1$ respec…
53A curve $C$ has parametric equations $x = 3t^2, \quad y = 6t^3, \quad \text{where } t \text{ is a real parameter.}$ Sk…
54You are given that $C = \int_{0}^{\pi} \mathrm{e}^{-2x} \cos x \; \mathrm{d}x$ and $S = \int_{0}^{\pi} \mathrm{e}^{-2x} \sin x \; \mathrm{d}x$ …
55It is given that $x^2 y - \tan^{-1} y = \frac{3}{4}\pi$ . Find $\frac{\text{d}y}{\text{d}x}$ in terms of $x$ and $y$ . Hen…
56Without using a calculator, solve the inequality $\frac{3x^2 + 9x - 8}{x - 1} < 2.$ Hence solve the following inequali…
57Anand writes a computer programme to simulate a population of organisms in a controlled environment. It is assumed that…
58 $X$ is a normal random variable with mean 1 and variance 1, and $Y$ is also a normal random variable with mean $\mu$ and…
59A function f is defined as $f(x)=\ln(5-x),\; x<5.$ Find the volume of revolution when the region bounded by the curve of…
60On the first day of last month, the temperature of a machine in a manufacturing plant was found to be $90^\circ\text{C}$ …