Math topic中文

Modulus

Concept of $|x|$ , and use of the relations $|x - a| < b \Leftrightarrow a - b < x < a + b$ and $|x - a| > b \Leftrightarrow x < a - b \text{ or } x > a + b$ .

Practice questions

1已知函数 $y=f(x)$ 是定义在 $R$ 上的偶函数，且在区间 $[0,+\infty)$ 上单调递增，若实数 $m$ 满足 $f({{\log}_{3}}m)\geqslant f(1)$ ，则 $m$ 的取值范围为___.
2Sketch, on the same diagram, the graphs of $y = |2x + 3|$ and $y = -2x^2 - 8x - 3$ . (You are not required to label any a…
3Find $\displaystyle \int \frac{3e^x}{5 - 0.3e^x} \, dx$ $\displaystyle \int \cos(\ln x) \, dx$ , where $x > 0$ the exact…
4已知函数 $f(x)=x^{2}-ax(a\neq 0)$ ， $g(x)=\ln x$ ， $f(x)$ 图象与 $x$ 轴异于原点的交点 $M$ 处的切线为 $l_{1}$ ， $g(x-1)$ 与 $x$ 轴的交点 $N$ 处的切线为…
5On a single diagram, sketch the graphs of $y = |2x - p|$ and $y = qx$ where the following conditions are satisfied, indi…
6Prove by induction that, for all positive integers $n$ and for $x\in\mathbb{R}$ , $x\ne 0,\ x\ne 1$ , $\frac{1}{x-1}-\frac{1}{x}-\frac{1}{x^2}-\dots-\frac{1}{x^n}=\frac{1}{x^n(x-1)}\,.$ …
7Find the exact roots of the equation $\left|\dfrac{4x-5}{x-2}\right| = 3x. On the same axes, sketch the curves with equations$ …
8A home interior designer is looking to design a pull-down wardrobe lift. To test his concept, the designer built a small…
9The points $A$ and $B$ have coordinates $(0, 2, 0)$ and $(1, 0, 2)$ respectively, and the plane $\pi_1$ has equation $\mathbf{r} \cdot \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} = 4$ …
10Solve the inequality $|y-5|<1$ . Hence solve the inequality $|3^{x}-5|<1$ , giving 3 significant figures in your answer.
11已知集合 $A=\{x\in z||x|\leqslant 2\}$ ， $B=\left\{ x\left|\dfrac{x-2}{x+2}\leqslant 0\right.\right\}$ ，则 $(\quad)$ A. $A∩B=A$ …
12在数轴上，点 $M$ ， $N$ 表示的数分别是 $x_{1}$ ， $x_{2}$ ，我们把 $x_{1}$ ， $x_{2}$ 之差的绝对值叫做点 $M$ ， $N$ 之间的距离，即 $MN=|x_{1}-x_{2}|.(1)$ 已知数…
13Write down constants $A$ and $B$ such that for all values of $x$ , $x - 1 = A(2x - 4) + B.$ By expressing $x^2 - 4x + 5$ …
14已知 $|\log _{a} \dfrac {3}{4}| < 1$ ，求 $a$ 的取值集合．
15A 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…
16已知函数 $f(x)$ 是定义在 $R$ 上的偶函数，且当 $x\geqslant 0$ 时， $f(x)=x^{2}+x.$ 若 $f(a)$ 十 $f(-a) < 4$ ，测实数 $a$ 的取值范围为___．
17已知 $f(x)=\left|2x+1\right|+\left|x-1\right|.(Ⅰ)$ 求 $f(x)$ 在 $[-1,1]$ 上的最大值 $m$ 及最小值 $n$ ； $(Ⅱ)$ 在 $(Ⅰ)$ 的条件下，设 $a,b\in R$ …
18Tetrahedral dice have four faces. Two fair tetrahedral dice, one red and one blue, have faces numbered 0, 1, 2, and 3 re…
19The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto \sqrt{5+2x}$ , for $x \in \mathbb{R}$ , $x > -\frac{5}{2}$ …
20已知不等式 $a+ \sqrt{3}b\leqslant \left|{x}^{2}-1\right|$ 对于满足条件 ${a}^{2}+{b}^{2}=1$ 的任意实数 $a$ ， $b$ 恒成立，实数 $x$ 的取值范围是．
21不等式|x|＞2的解集是（） A. x＞2 B. {x|x＞2} C. {x|x＞2或x＜-2} D. x＞2或x＜-2
22The two planes $p_{1}$ and $p_{2}$ , given by the equations $2x+y=1$ and $8x+ay+z=4$ respectively, meet at a line $L$ whi…
23大家知道|5|=|5-0|，它在数轴上表示5的点与原点（即表示0的点）之间的距离．又如式子|6-3|，它在数轴上的意义是表示6的点与表示3的点之间的距离．即点A、B在数轴上分别表示数a、b，则A、B两点的距离可表示为：|AB|=|a-b|．…
24已知函数 $f\left(x\right)$ 是定义在 $R$ 上的偶函数，且 $f\left(0\right)=0$ ，当 $x>0$ 时， $f\left(x\right)={{\log}_{2}}\left(x+1\right).(1)$ …
25In this question you should state clearly the values of the parameters of any normal distributions that you use. The mas…
26Find, in terms of $a$ , the roots of the equation $\dfrac{1}{(x-a)^2}=|x-a|$ . On the same axes, sketch the curves with eq…
27Do not use a calculator in answering this question. Solve the inequality $\frac{6x^2 + 2x - 3}{2x - 1} \ge 2(x + 1).$ …
28Dendrologists are specialised scientists who study trees and woody plants. Their work is diverse and can encompass vario…
29The plane $\Pi$ has equation $2x-3y+6z=7$ and the line $l$ has equation $\mathbf{r}=8\mathbf{i}-2\mathbf{j}+8\mathbf{k}+t(3\mathbf{i}-\mathbf{j}+2\mathbf{k}).$ …
30By using the substitution $y = |x|$ , solve the inequality $\frac{|x|+3}{x^2+1} > 1.$
31Sketch the curve $C$ with equation $y = \frac{x-2}{x-1}$ , stating the exact coordinates of any points of intersection wi…
32The diagram shows the graphs of $y = |a - bx|$ and $y = |mx - 6|$ where $a$ , $b$ and $m$ are positive constants. The gra…
33To build a Virtual Reality (VR) experience, a virtual world is created to contain geometric models. With reference to th…
34Section A: Pure Mathematics [40 marks] Sketch, on the same axes, the graphs of $y = 3x$ and $y = |x^2 - a^2|$ where $a > 0$ …
35State a sequence of transformations that will transform the curve with equation $y^{2}-x^{2}=1$ on to the curve with equ…
36 $f(x)$ 是定义在 $[-2,2]$ 上的偶函数，且 $f(x)$ 在 $[0,2]$ 上单调递减，若 $f(1-m) < f(m)$ 成立，求实数 $m$ 的取值范围 ___ ．
37The plane $\pi_1$ has equation $\mathbf{r}\cdot\begin{pmatrix}2 \\ 5 \\ 1\end{pmatrix}=11.$ The plane $\pi_2$ passes t…
38已知点P的坐标为（2-a，3a+6），且点P到两坐标轴的距离相等，则a= ．
39Given that $a \in \mathbb{R}$ and $a > 1$ , use an algebraic method to solve the inequality $\frac{ax - a + 2}{(x - a)(1 - x)} \le 1 .$ …
40Given that both $a$ and $b$ are positive constants, on the same diagram, sketch the graphs of $y = \frac{2}{|x - a|} \quad \text{and} \quad y = b|x - a|.$ …
41设集合S={x||x|＜5}，T={x|（x+7）（x-3）＜0}．则S∩T=（） A. {x|-7＜x＜5 } B. {x|3＜x＜5 } C. {x|-5＜x＜3 } D. {x|-7＜x＜-5 }
42关于x的方程||x-2|-1|=a恰有三个整数解，则a的值为．
43已知实数 $a > 0$ ，设函数 $f(x)=\left|x+ \dfrac{1}{a}\right|+\left|x-a\right|$ ． $($ Ⅰ $)$ 证明： $f(x)\geqslant 2$ ； $($ Ⅱ $)$ …
44The line $l$ has equation $1 - x = \frac{z - \alpha}{2}, y = 1$ , where $\alpha$ is a constant. Given that the point of i…
45若|x-3|=2，则x的值为．
46The plane $\Pi_1$ has equation $\mathbf{r} = \mathbf{i} + 2\mathbf{j} + \mathbf{k} + \theta(2\mathbf{j} - \mathbf{k}) + \phi(3\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}).$ …
47设集合A＝{x||x－a|<1，x∈R}，B＝{x|1<x<5，x∈R}，若A∩B＝∅，则实数a的取值范围是( ) A. {a|0≤a≤6} B. {a|a≤2，或a≥4} C. {a|a≤0，或a≥6} D. {a|2≤a≤4}
48Relative to the origin $O$ , two points $A$ and $B$ have position vectors given by $\mathbf{a} = 2\mathbf{i} - \mathbf{j}$ …
49Without using a calculator, solve the inequality $\dfrac{4x}{x-3} \ge 1. Hence find the range of values of$ x $that satisfies$ …
50设全集为 $R$ ， $A=\{x|x^{2}-5x-6>0\}$ ， $B=\{x||x-5|< a\}(a$ 为常数 $)$ ， $11\in B$ ，则 $(\quad)$ A. $(∁_{R}A)∪B=R$ B. $A∪(∁_{R}B)=R$ …
51In this question, $k$ is a positive constant. Using an algebraic method, solve the inequality $\frac{2x^2 - 2k^2}{x} < x + k$ …
52Without using a calculator, solve $1 + \frac{6}{x} \ge x$ , $1 + \frac{6}{|x|} \ge |x|$ .
53设不等式 $\left|{{4}^{x}}-a\right|>\left|1-{{2}^{x\text{+3}}}\right|$ 对所有 $x\in[1,2]$ 均成立，则实数 $a$ 的取值范围是 $(\quad)$ A. $a< -15$ …
54Without the use of a calculator, solve the inequality $\frac{(x^2 - 2x + 4)(x - 3)}{(x + 2)} \ge 0.$ Hence solve the i…
55Without the use of a calculator, solve the inequality $\frac{3x^2 + 1}{2x^2 + 3x - 2} > 1$ . Hence solve the inequality $\frac{3x^2 + 1}{2x^2 + 3|x| - 2} > 1$ …
56Sketch, on the same diagram, the curves with equations $y = \ln(x-2)$ and $y = \frac{1}{\text{e}^x} + 3$ , stating the eq…
57In a small town, there are two clinics, Lee's Clinic and Hope Clinic. At Lee's Clinic, the waiting time of a patient fol…
58By using the substitution $y = |x|$ , solve the inequality $\frac{|x| + 3}{x^2 + 1} > 1$ .
59Show that $\frac{3}{2r-1} - \frac{4}{2r+1} + \frac{1}{2r+3} = \frac{8r+20}{(2r-1)(2r+1)(2r+3)}$ . Hence show that $\sum_{r=1}^{n} \frac{2r+5}{(2r-1)(2r+1)(2r+3)} = \frac{2}{3} - \frac{n+2}{(2n+1)(2n+3)}$ …
60The planes $\Pi_1$ and $\Pi_2$ have equations $\mathbf{r} \cdot (\mathbf{i} - 2\mathbf{j} + \mathbf{k}) = 4$ and $\mathbf{r} = 3\mathbf{i} - \mathbf{j} + \mathbf{k} + \lambda(-\mathbf{i} + \mathbf{j} + \mathbf{k}) + \mu(4\mathbf{i} + \mathbf{j})$ …