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

Question 1

What is the value of the algebraic expression (1 + ω - ω²)³?

Accepted Answer

-8

Answer

8

Answer

-1

Answer

1

Question 2

If $y = \sec(3x)$, then the derivative $\frac{dy}{dx}$ is equal to:

Accepted Answer

A) $3\sec(3x) 	an(3x)$

Answer

B) $\sec(3x) 	an(3x)$

Answer

C) $-3\csc(3x) \cot(3x)$

Answer

D) $3\sec^2(3x)$

Question 3

The derivative of $f(x) = 5^{x^2}$ with respect to $x$ is:

Accepted Answer

A) $2x \cdot 5^{x^2}$

Answer

B) $2x \cdot 5^{x^2} \cdot \ln 5$

Answer

C) $5^{x^2} / \ln 5$

Answer

D) $x^2 \cdot 5^{x^2 - 1}$

Question 4

If $x = \sin t$ and $y = \cos t$, then the second derivative $\frac{d^2y}{dx^2}$ is equal to:

Accepted Answer

A) $-\cos t$

Answer

B) $-\sec^3 t$

Answer

C) $-	an t$

Answer

D) $\csc^2 t$

Question 5

The slope of the normal to the curve $y = e^{2x}$ at the point where $x = 0$ is:

Accepted Answer

A) 2

Answer

B) -2

Answer

C) 1/2

Answer

D) -1/2

Question 6

The n-th term of an arithmetic sequence with first term a and common difference d is given by:

Accepted Answer

T_n = a + (n - 1) * d

Answer

T_n = a * r^(n - 1)

Answer

T_n = a + n * d

Answer

T_n = 1/2 n (2a + d)

Question 7

In the arithmetic sequence (3, 7, 11, 15, ...), the common difference d and the 20th term (T_20) are respectively:

Accepted Answer

d = 4 and T_20 = 79

Answer

d = 4 and T_20 = 83

Answer

d = 3 and T_20 = 60

Answer

d = 5 and T_20 = 98

Question 8

The sum of the first n terms of an arithmetic sequence (S_n) is formulated as:

Accepted Answer

S_n = n / 2 * [ 2a + (n - 1) * d ] = n / 2 * [ a + l ]

Answer

S_n = a * (1 - r^n) / (1 - r)

Answer

S_n = n * (a + d)

Answer

S_n = a / (1 - r)

Question 9

The common ratio r of the geometric sequence (64, 32, 16, 8, ...) is:

Accepted Answer

r = 1/2

Answer

r = 2

Answer

r = -1/2

Answer

r = 4

Question 10

The sum to infinity (S_infinity) of a geometric series exists and converges if and only if the common ratio satisfies:

Accepted Answer

|r| < 1 (i.e. -1 < r < 1), and S_infinity = a / (1 - r)

Answer

r > 1

Answer

r = 1

Answer

|r| >= 2

Point of comparison	The novel	The short story
Life presented	A whole life of one character, or the lives of several	One situation or one concentrated moment
Characters	Many	Limited
Events	Numerous and branching	Few
Time	Stretches over years	Usually short in range
Place	Places are many	Limited
Language	Expansion, and comprehensive precise description to imitate reality	The utmost conciseness: every description is intended and every phrase carries a sense
Deletion	Some scenes or dialogues can be cut without damage	No sentence can be cut without the build being affected

First: the short story and how it differs from the novel

01First: the short story and how it differs from the novel

The concept

Comparing the novel and the short story

The aim of the short story

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