题库网 (tiku.one)

 找回密码
 立即注册

 

      

上传图片附件

未使用图片

小贴士: 允许的图片文件格式为: gif, jpg, jpeg, png, webp,上传完成后会在上方生成预览,用鼠标连续双击缩略图,或拖动缩略图,该图片就被绑定至本题,显示在题目下方

本次作答已使用

小贴士: 此栏目显示的是当前作答使用的所有图片,绑定到某一题目的图片同时会显示在该题目下方; 删除使用的图片会将其转移到<未使用图片>类别


习题练习:IB MAI HL Number and Algebra Topic 1.4 Complex Numbers

 作者: admin   总分: 14分  得分: _____________

答题人: 游客未登录  开始时间: 24年01月23日 23:24  切换到: 整卷模式

标记此题
1#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  In this question give all angl;e0n/bfiw +xc-rg9 dzes in radians.

Let z=1+2i and w=4+i.

1.Find z+w=x+yi;x=  ,y=  .

2.Find:
2.1.∣z+w∣;

2.2.arg(z+w).

3.Find θ, the angle shown on the diagram below.


参考答案:     查看本题详细解析

标记此题
2#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Let z=2+i and w=1−2i rhv+(7bs) o pac vwr6s1ty3p9.

1.Find zw=x-yi ; x=  ,y=  .

2.Illustrate z , w and zw on the same Argand diagram.

3.let θ be the angle between zw and w. Find θ, giving your answer in radians.
Hence we get θ(≈)  .

参考答案:     查看本题详细解析

标记此题
3#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The complex numbers z and w correspond to the points A and B as shown on the huyk6a//lkb 1(bqdmr 3;q g; qdiagra/g1b6h kqmq bk/dyq;;lu (a3rm below.

1.Find the exact value of ∣z−w∣.
∣z−w∣=x$\sqrt{y-\sqrt{z}}$; x=  , y=  , z=  .
2.1.Find the exact perimeter of triangle AOB.
perimeter=x+∣z−w∣; x=  .
2.2.Find the exact area of triangle AOB.
area=$\frac{x}{y}$ ; x=  , y=  .

参考答案:     查看本题详细解析

标记此题
4#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  A circle is drawn on d42n dj69*pna(8 onx cwkr5tgan Argand diagram as shown below. The tangent to the circle from the point B(0,9) 5 x ojwgcn(n8nt6arddp *4k 29meets the circle at the point A as shown. Let w= OA$^{→}$.

1.Show that ∣w∣=3$\sqrt{3}$.

2.Find arg w.w=$\frac{\pi}{x}$,x=  .

3.Hence write w in the form a+bi where a,b∈R.
w=$\frac{3\sqrt{3}}{x}+\frac{y}{2}i$,x=  , y=  .

参考答案:     查看本题详细解析

标记此题
5#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Let $z_1$ =2$\sqrt{3} cis( $\frac{7π}{12}$), $z_3$=2cisθ, and $z_2$=$z_1$+$z_3$ be represented by the points
A, B and C on an Argand diagram as shown below.

The shape OABC is a rectangle.

1.Show that θ= $\frac{π}{12}$.

2.Find arg($z_1$-$z_2$)=-$\frac{xπ}{y}$, x=  ,y=  .

3.Express $z_2$ in modulus-argument form.$z_2$=xcis($\frac{yΠ}{z}$),x=  ,y=  ,z=  .

参考答案:     查看本题详细解析

标记此题
6#
 
问答题 ( 1.0 分) 切至整卷模式 搜藏此题  
On an Argand diagram, the complex numbers ak lp/a24 82uen3evgs.dy la7$z_1$ =2+2$\sqrt{3}$i, $z_2$=1−i and $z_3$=$z_1$$z_1$ are represented by the vertices of a triangle.

Find the area of the triangle.
参考答案:    

标记此题
7#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Points A and B represent h t+ pik44z-2rs xywk:pvg)+83sjfp vthe complex numbers $z_1$ = 3​ −i and $z_2$=-3-3i as shown on an Argand diagram below.

1.Find the angle AOB=$\frac{x\pi}{y}$ ; x=  ,y=  .

2.Find the argument of $z_1$$z_2$
arg$z_1$$z_2$=-$frac{x\pi}{y}$ ; x=  ,y=  .

3.Given that the real powers of p$z_1$$z_2$ , for p>0, all lie on a unit circle centred at the origin, find the exact value of p.
p=$\frac{\sqrt{x}}{y}$ ; x=  ,y=  .

参考答案:     查看本题详细解析

标记此题
8#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Let z=r$e^{i\frac{\pi}{3}}$ where r∈R$^+$.
1.For r= $sqrt{2}$,

1.1.express $z^2$ and $z^3$ in the form a+bi where a,b∈R;
$z^2$=-1+$\sqrt{x}$i and $z^3$=-2$\sqrt{2}$+yi; x=  ,y=  .

1.2.draw $z^2$ and $z^3$ on the following Argand diagram.

2.Given that the integer powers of w=(3−3i)z lie on a unit circle centred at the origin, find the value of r.
r=$\frac{\sqrt{x}}{y}$ ; x=  .y=  .

参考答案:     查看本题详细解析

标记此题
9#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Let z=r$e^{i\frac{\pi}{3}}$ where r∈R$^+$.
1.For r= $sqrt{3}$,

1.1.express $z^2$ and $z^3$ in the form a+bi where a,b∈R;
$z^2$=$\frac{3}{2}$+$\frac{3\sqrt{x}}{y}$i and $z^3$=0+$3sqrt{z}$i; x=  ,y=  ,z=  .

1.2.draw $z^2$ and $z^3$ on the following Argand diagram.

2.Given that the integer powers of w=$\frac{z}{6+2i}$ lie on a unit circle centred at the origin, find the value of r.
r=$x\sqrt{y}$ ; x=  .y=  .

参考答案:     查看本题详细解析

标记此题
10#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Let z=$\sqrt{2}cis(\frac{3\pi}{8})$ and w=2cis($\frac{n\pi}{24}$),where n∈$Z^+$.

1.Find the value of $z^6$ . Give your answer in the form re$^{iθ}$, where r≥0, -Π<θ≤Π.
$z^6$=xe$^{i\frac{Π}{y}}$ . x=  ,y=  .
2.Find the value of $(wz)^4$ for n=5 . Give your answer in the form re$^{iθ}$, where r≥0, -Π<θ≤Π.
$(wz)^4$=xe$^{i\frac{Π}{y}}$ . x=  ,y=  .
3.Find the smallest integer n>9 such that $\frac{z}{w}$ ∈R.
n=  .

参考答案:     查看本题详细解析

标记此题
11#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Two voltage sources are connected t+f1ad0 q6are jo a circuit. At time t milliseconds (ms), the voltage from0fa6 1aeq +drj the first source is $V_1$(t)=12cos(20t) and the voltage from the second source is $V_2$(t)=18cos(20t+5), shere both $V_1$(t) and $V_2$(t) are
easured in volts.
1.Write, in the form V(t)=Acos(ωt+φ), an expression for the total voltage in the circuit at time t ms.
2.Hence write down the highest voltage in the circuit.
Hence the highest voltage in the circuit is    volts


参考答案:     查看本题详细解析

标记此题
12#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  In an unbalanced three-phase ele6yqj6nm gngl: /gd:. y7s6pfx b.im5p6)wk ayctrical circuit, the current at time t ms is given bnyifsk:6l6.. y nag j qm5/g )7py6w md6xgbp:y I(t)=2sin(5t)+5sin(5t-$\frac{5\pi}{4}$),
where I(t) is measured in milliamperes (mA).

1.Write I(t) in the form Acos(ωt+φ).

2.Hence find the highest current flowing through the circuit, and the time it first occurs.
Hence the highest current is   mA and the 1st time it occurs is at   ms

参考答案:    

标记此题
13#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  The revenues of a four seasons hotel can be modelled by the funaqe ai0 :fm)h9ction
R(t)=58.2sin(0.0172t−1.25)+204,
where t is the number of days after midnight on 31 December.
In a similar way, the operating costs of the hotel can be modelled by the function
C(t)=31.4sin(0.0172t+1.14)+85.0.

Both R(t) and C(t) are measured in thousand dollars.

1.Show that the profits of the hotel can be modelled by the function P(t)=83.9sin(0.0172t−1.51)+119.

2.According to the model, find:

2.1.the highest profit the hotel will make;
According to the model, the highest profit is
$P_{max}$≈  thousand.

2.2.the date on which the highest profit will occur.
Hence the highest profit occurs on day    or June   .


参考答案:     查看本题详细解析

标记此题
14#
 
填空题 ( 1.0 分) 切至整卷模式 搜藏此题  
  Ali is swimming in a pi lq-fsn1q yg8r +6so p;9r6lpublic pool with some of his friends. At time t seconds, he lfio 8lsp6grp+ r6n19-qsy;q encounters some waves with height
$h_1$​(t)=0.15sin(3t) from big Bobby jumping into the pool, and waves of height
$h_2$(t)=0.08sin(3t+1.25) from small Suzie jumping into the pool. Both
$h_1$(t) and $h_2$ (t) are measured in metres.

1.Write, in the form h(t)=Asin(ωt+φ), an expression for the total height of the waves Ali encounters at time t seconds.

2.Find the times in the first 5 seconds when Ali isn't affected by any waves.

3.Find the first time when the waves reaching Ali has maximum height.
Hence the first time when the wave reaching Ali has max height is at   s


参考答案:     查看本题详细解析

总分:14分 及格:8.4分 时间:不限时
未答题: 已答题:0 答错题:
当前第 题,此次习题练习共有 14 道题
本平台(https://tiku.one/)支持练习,作业与考试(自动评分)三种模式,群组成绩统计排序导出,题目动态变量实现同卷千人千题
如果您对本系统感兴趣,想加入我们或者想进行任何形式的合作,请加微信 skysky1258

浏览记录|手机版首页|使用帮助|题库网 (https://tiku.one)

GMT+8, 2025-9-18 05:03 , Processed in 0.134698 second(s), 49 queries , Redis On.