Cho số phức z thỏa mãn ( 3 − i ) ( z + 1 ) + ( 2 − i ) ( z ‾ + 3 i ) = 1 − i \left(3-i\right)\left(z+1\right)+\left(2-i\right)\left(\overline{z}+3i\right)=1-i ( 3 − i ) ( z + 1 ) + ( 2 − i ) ( z + 3 i ) = 1 − i . Tính mô-đun của số phức w = i − z 1 + z w=\frac{i-z}{1+z} w = 1 + z i − z

82 8 \frac{\sqrt[]{82}}{8} 8 8 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

82 4 \frac{\sqrt[]{82}}{4} 4 8 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

2 82 9 \frac{2\sqrt[]{82}}{9} 9 2 8 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

3 82 5 \frac{3\sqrt[]{82}}{5} 5 3 8 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

Tìm số phức liên hợp của số phức z = i ( 3 i + 1 ) z=i\left(3i+1\right) z = i ( 3 i + 1 )

z ‾ = − 3 − i \ \overline{z}=-3-i z = − 3 − i

z ‾ = 3 − i \ \overline{z}=3-i z = 3 − i

z ‾ = − 3 + i \ \overline{z}=-3+i z = − 3 + i

z ‾ = 3 + i \ \overline{z}=3+i z = 3 + i

Hàm số y = |3cosx – 4sinx + 8| với x ∈ [ 0 ; 2 π ] x\in\left[0;2\pi\right] x ∈ [ 0 ; 2 π ] . Gọi M, m lần lượt là giá trị lớn nhất, giá trị nhỏ nhất của hàm số. Khi đó tổng M + m bằng bao nhiêu?

8 2 8\sqrt[]{2} 8 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

7 3 7\sqrt[]{3} 7 3 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

8 3 8\sqrt[]{3} 8 3 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

CAO THỦ CASIO

Authored by Quy Kim

Mathematics

9th - 12th Grade

CCSS covered

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20 questions

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MULTIPLE CHOICE QUESTION

2 mins • 1 pt

Tìm nghiệm của phương trình sau: $3^{x-1}\ =\ 27$

x = 9

x = 3

x = 4

x = 10

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

Phương trình $\log_2^2x\ -5\log_2x\ +\ 4\ =0$ có 2 nghiệm $x_1$ ; $x_2$ khi đó tích $x_1\cdot x_2$ bằng:

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

Cho hàm số $y\ =\ 3\sin x\ -4\sin^2x$ . Giá trị lớn nhất của hàm số trên khoảng $\left(-\frac{\pi}{2};\frac{\pi}{2}\right)$ bằng:

-1

Cho $\int_0^1\frac{dx}{e^{x\ }+\ 1}=\ a\ +\ b\ln\frac{1\ +e}{2}$ , với a,b là các số hữu tỉ. Tính $S=a^3+b^3$

S = 0

S = 2

S = -2

S = 1

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

Tìm tập xác định D của hàm số $y=\left(x^{4\ }-3x^2\ -\ 4\right)^{\sqrt[]{2}}$

$D=\left(-\infty;-1\right)\cup\left(4;+\infty\right)$

$D=\left(-\infty;-2\right)\cup\left(2;+\infty\right)$

$D=\left(-\infty;-2\right]\cup\left[2;+\infty\right)$

$D=\left(-\infty;+\infty\right)$

MULTIPLE CHOICE QUESTION

5 mins • 1 pt

Tìm tất cả các giá trị của m để hàm số $y=2^{x^3-x^2+mx\ }$ đồng biến trên [1;2]

$m>\frac{1}{3}$

$m\ge\frac{1}{3}$

$m\ge1$

$m\ \ge-1$

MULTIPLE CHOICE QUESTION

10 mins • 1 pt

Cho số phức z thỏa mãn $\left(3-i\right)\left(z+1\right)+\left(2-i\right)\left(\overline{z}+3i\right)=1-i$ . Tính mô-đun của số phức $w=\frac{i-z}{1+z}$

$\frac{\sqrt[]{82}}{4}$

$\frac{\sqrt[]{82}}{8}$

$\frac{2\sqrt[]{82}}{9}$

$\frac{3\sqrt[]{82}}{5}$

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

Tìm số phức liên hợp của số phức $z=i\left(3i+1\right)$

$\ \overline{z}=3-i$

$\ \overline{z}=-3+i$

$\ \overline{z}=3+i$

$\ \overline{z}=-3-i$

Tính tích phân $\int_1^ex\ln xdx$

$I\ =\frac{1}{2}$

$I\ =\frac{e^2-2}{2}$

$I\ =\frac{e^2+1}{4}$

$I\ =\frac{e^2-1}{4}$

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CAO THỦ CASIO

Tìm nghiệm của phương trình sau: 3x−1 = 273^{x-1}\ =\ 273x−1 = 27

Phương trình log⁡22x −5log⁡2x + 4 =0\log_2^2x\ -5\log_2x\ +\ 4\ =0log22​x −5log2​x + 4 =0 có 2 nghiệm x1x_1x1​; x2x_2x2​khi đó tích x1⋅x2x_1\cdot x_2x1​⋅x2​bằng:

Cho hàm số y = 3sin⁡x −4sin⁡2xy\ =\ 3\sin x\ -4\sin^2xy = 3sinx −4sin2x. Giá trị lớn nhất của hàm số trên khoảng (−π2;π2)\left(-\frac{\pi}{2};\frac{\pi}{2}\right)(−2π​;2π​)bằng:

Cho ∫01dxex + 1= a + bln⁡1 +e2\int_0^1\frac{dx}{e^{x\ }+\ 1}=\ a\ +\ b\ln\frac{1\ +e}{2}∫01​ex + 1dx​= a + bln21 +e​, với a,b là các số hữu tỉ. Tính S=a3+b3S=a^3+b^3S=a3+b3

Tìm tập xác định D của hàm số y=(x4 −3x2 − 4)2y=\left(x^{4\ }-3x^2\ -\ 4\right)^{\sqrt[]{2}}y=(x4 −3x2 − 4)2​

Tìm tất cả các giá trị của m để hàm số y=2x3−x2+mx y=2^{x^3-x^2+mx\ }y=2x3−x2+mx đồng biến trên [1;2]

Cho số phức z thỏa mãn (3−i)(z+1)+(2−i)(z‾+3i)=1−i\left(3-i\right)\left(z+1\right)+\left(2-i\right)\left(\overline{z}+3i\right)=1-i(3−i)(z+1)+(2−i)(z+3i)=1−i. Tính mô-đun của số phức w=i−z1+zw=\frac{i-z}{1+z}w=1+zi−z​

Trong không gian với hệ trục tọa độ Oxyz, tìm tọa độ hình chiếu vuông góc của điểm A (0;1;2) trên mặt phẳng (P): x+y+z=0

Tìm số phức liên hợp của số phức z=i(3i+1)z=i\left(3i+1\right)z=i(3i+1)

Tính tích phân ∫1exln⁡xdx\int_1^ex\ln xdx∫1e​xlnxdx

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Tìm nghiệm của phương trình sau: $3^{x-1}\ =\ 27$

Phương trình $\log_2^2x\ -5\log_2x\ +\ 4\ =0$ có 2 nghiệm $x_1$ ; $x_2$ khi đó tích $x_1\cdot x_2$ bằng:

Cho hàm số $y\ =\ 3\sin x\ -4\sin^2x$ . Giá trị lớn nhất của hàm số trên khoảng $\left(-\frac{\pi}{2};\frac{\pi}{2}\right)$ bằng:

Cho $\int_0^1\frac{dx}{e^{x\ }+\ 1}=\ a\ +\ b\ln\frac{1\ +e}{2}$ , với a,b là các số hữu tỉ. Tính $S=a^3+b^3$

Tìm tập xác định D của hàm số $y=\left(x^{4\ }-3x^2\ -\ 4\right)^{\sqrt[]{2}}$

Tìm tất cả các giá trị của m để hàm số $y=2^{x^3-x^2+mx\ }$ đồng biến trên [1;2]

Cho số phức z thỏa mãn $\left(3-i\right)\left(z+1\right)+\left(2-i\right)\left(\overline{z}+3i\right)=1-i$ . Tính mô-đun của số phức $w=\frac{i-z}{1+z}$

Tìm số phức liên hợp của số phức $z=i\left(3i+1\right)$

Tính tích phân $\int_1^ex\ln xdx$