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	First Cairo Revolt	Second Cairo Revolt
Date	October 1798	After the convention of El-Arish (1800)
Leadership	Al-Azhar and its ulema	The people of Cairo
Immediate cause	The French occupation and the killing of the French governor of the city	The suffering of the Egyptians, exploiting the French preoccupation with fighting the Ottomans
French response	Repression, terror, entering al-Azhar on horseback	Quelling the revolt in Lower Egypt and allying with Murad Bey against Upper Egypt

The Second Cairo Revolt and the killing of Kleber

07The Second Cairo Revolt and the killing of Kleber

The Second Cairo Revolt

The killing of Kleber

Comparison between the two revolts

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