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	Mediterranean region (north coast)	Hot desert region (south and the deserts)
Location in Egypt	Coastal strip from Sallum to Rafah	South of Egypt from Asyut southward, plus Western and Eastern Deserts
Temperature, summer and winter	Temperate summer, warm winter under the sea's influence	Very hot summer, very cold nights in winter
Daily and annual range	Very small — the moderating water body	Very large — arid continental climate
Season and amount of rain	Winter cyclonic rain, the heaviest in Egypt (up to 200 mm)	Rare or none; falls as sudden flash floods every few years
Natural vegetation	Mediterranean forest, olive, steppe grass and pastures	Thorny plants — cactus, acacia, artemisia — with deep roots

Seventh: comparison — Egypt's Mediterranean coast vs its hot desert

07 Seventh: comparison — Egypt's Mediterranean coast vs its hot desert

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