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Mathematics

Unit 1

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Unit 2

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Unit 3

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Unit 4

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Unit 5

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Unit 6

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Unit 7

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Quiz

Lesson 1 Quiz: Complex Numbers in Trigonometric and Exponential Form and De Moivres Theorem

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  • Questions count: 20
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1.
If a, b, c form a geometric sequence of positive real numbers, the geometric mean b equals:
b = a * c
b = a2 + c2
b = (a + c) / 2
b = + sqrt(a * c)
2.
STEM Integration Problem 28: Evaluate the definite integral ∫[0 to 2] (3x2 - 4x + 1) dx.
2
0
4
6
3.
STEM Solid Geometry Problem 42: What is the equation of a sphere in 3D Cartesian coordinates with center C(2, -1, 3) and radius r = 5?
(x + 2)2 + (y - 1)2 + (z + 3)2 = 25
(x - 2)2 + (y + 1)2 + (z - 3)2 = 25
x2 + y2 + z2 - 4x + 2y - 6z = 25
(x - 2)2 + (y - 1)2 + (z - 3)2 = 5
4.
The sum of all coefficients in the expansion of \((3x - 2y)^7\) is equal to:
A) 1
D) 128
C) 0
B) \(5^7\)
5.
Using the Quotient Rule, the derivative of y = (x + 3) / (x - 2) for x != 2 is:
-5 / (x - 2)2
5 / (x - 2)2
1 / (x - 2)2
(2x + 1) / (x - 2)2
6.
STEM Algebra Problem 38: In the expansion of (x + 1/x)10, find the value of the term independent of x (the constant term).
210
120
1
252 (the 6th term T_6 = 10C5)
7.
STEM Statics Problem 18: A body of weight 60 N rests on a rough horizontal plane. If the coefficient of static friction μ_s = √3 / 3, what horizontal force is required to place the body on the verge of motion?
20√3 N (approximately 34.64 N)
10√3 N
30 N
60 N
8.
STEM Algebra Problem 11: In the expansion of (x + 1/x)10, find the value of the term independent of x (the constant term).
210
1
120
252 (the 6th term T_6 = 10C5)
9.
STEM Solid Geometry Problem 6: What is the equation of a sphere in 3D Cartesian coordinates with center C(2, -1, 3) and radius r = 5?
(x + 2)2 + (y - 1)2 + (z + 3)2 = 25
(x - 2)2 + (y + 1)2 + (z - 3)2 = 25
x2 + y2 + z2 - 4x + 2y - 6z = 25
(x - 2)2 + (y - 1)2 + (z - 3)2 = 5
10.
STEM Solid Geometry Problem 29: What is the equation of a sphere in 3D Cartesian coordinates with center C(2, -1, 3) and radius r = 5?
(x + 2)2 + (y - 1)2 + (z + 3)2 = 25
(x - 2)2 + (y + 1)2 + (z - 3)2 = 25
x2 + y2 + z2 - 4x + 2y - 6z = 25
(x - 2)2 + (y - 1)2 + (z - 3)2 = 5
11.
The sum to infinity of the geometric series 12 + 6 + 3 + 1.5 + ... equals:
18
36
Infinity
24
12.
STEM Vector Algebra Problem 17: If vector A = (2, -3, 1) and vector B = (1, 2, -4), calculate the dot product A · B.
-8
6
10
-4
13.
STEM Integration Problem 20: Evaluate the definite integral ∫[0 to 2] (3x2 - 4x + 1) dx.
4
6
2
0
14.
STEM Calculus Problem 17: Find the derivative dy/dx for the function y = sin3(2x).
6 sin2(2x)
3 sin2(2x) cos(2x)
6 sin2(2x) cos(2x)
3 sin2(2x)
15.
STEM Limits Problem 38: Evaluate the limit: lim [x -> 0] (sin(5x) / tan(2x)).
1
Undefined
0
5/2 (or 2.5)
16.
STEM Vector Algebra Problem 14: If vector A = (2, -3, 1) and vector B = (1, 2, -4), calculate the dot product A · B.
6
-4
-8
10
17.
STEM Vector Algebra Problem 25: If vector A = (2, -3, 1) and vector B = (1, 2, -4), calculate the dot product A · B.
-8
10
6
-4
18.
STEM Vector Algebra Problem 6: If vector A = (2, -3, 1) and vector B = (1, 2, -4), calculate the dot product A · B.
10
-4
-8
6
19.
STEM Vector Algebra Problem 34: If vector A = (2, -3, 1) and vector B = (1, 2, -4), calculate the dot product A · B.
6
10
-4
-8
20.
STEM Solid Geometry Problem 32: What is the equation of a sphere in 3D Cartesian coordinates with center C(2, -1, 3) and radius r = 5?
(x - 2)2 + (y + 1)2 + (z - 3)2 = 25
(x - 2)2 + (y - 1)2 + (z - 3)2 = 5
x2 + y2 + z2 - 4x + 2y - 6z = 25
(x + 2)2 + (y - 1)2 + (z + 3)2 = 25
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