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9000064003

Level:

Consider the convergent sequence

(a_{n})_{n = 1}^{\infty} = {(\frac{4 n^{2} + 3 n - 250}{2 n^{2}})}_{n = 1}^{\infty}

and its limit

L

. Find the maximal difference between

L

and the subsequence

(a_{n})_{n = 250}^{\infty}

. (In other words, find the maximal difference between

L

and the terms of the sequence starting at

a_{250}

0.004

0.04

0.504

0.54

9000064008

Level:

Find the limit of the following sequence.

{(\frac{(n^{2} + 2 n + 1)^{n}}{n^{2 n}})}_{n = 1}^{\infty}

Hint: The limit of the sequence

{({(1 + \frac{1}{n})}^{n})}_{n = 1}^{\infty}

is the Euler number

e

e^{2}

2 e

e + 2

\infty

9000064009

Level:

Find the limit of the following sequence.

{({(\frac{\sqrt[n]{2}}{n} + \sqrt[n]{2})}^{n})}_{n = 1}^{\infty}

Hint: The limit of the sequence

{({(1 + \frac{1}{n})}^{n})}_{n = 1}^{\infty}

is the Euler number

e

2 e

e^{2}

e + 2

\infty

9000064010

Level:

Find the limit of the following sequence.

{({(\frac{2 n + 1}{n})}^{n})}_{n = 1}^{\infty}

Hint: The limit of the sequence

{({(1 + \frac{1}{n})}^{n})}_{n = 1}^{\infty}

is the Euler number

e

\infty

2 e

e^{2}

e + 2

9000062406

Level:

Identify an asymptote to the graph of the following function.

f (x) = \frac{x^{2} + 4}{x}

y = x

y = x + 1

y = x - 1

y = - x

9000046507

Level:

Identify the optimal first step convenient to solve the following trigonometric equation. Do not consider the step which is possible but does not help to solve the equation.

\sqrt{3} \cos x = 1 - \sin x

3 \cos^{2} x = (1 - \sin x)^{2}

3 \cos^{2} x = 1 - \sin^{2} x

substitution

1 - \sin x = z

substitution

\cos x = z

9000046508

Level:

Identify the optimal first step convenient to solve the following trigonometric equation. Do not consider the step which is possible but does not help to solve the equation.

\sqrt{3} \sin x = 2 - \cos x

3 \sin^{2} x = 4 - 4 \cos x + \cos^{2} x

substitution

2 - \cos x = z

3 \sin^{2} x = 4 - \cos^{2} x

3 \sin^{2} x = 1 - 2 \cos x + \cos^{2} x

9000039001

Level:

Assuming

x \in R

, find the solution set of the following system.

\begin{aligned} - 4 x & < - 6 - 2 x \\ 2 (x + 5) & \leq 16 \end{aligned}

\emptyset

(3; + \infty)

(- \infty; 3)

{3}

9000039103

Level:

Assuming

x \in R

y \in R

, solve the following equation.

(2 + 5 i) x + (1 - i) y = 13 i + 8

x = 3, y = 2

x = 13, y = 8

x = 8, y = 13

x = 2, y = 3

9000039104

Level:

Assuming

x \in R

y \in R

, solve the following equation.

(3 - 2 i) x + (5 - 7 i) y = 1 + 3 i

x = 2, y = - 1

x = - 1, y = 2

x = - 2, y = - 1

x = - 1, y = - 2

C

9000064003

9000064008

9000064009

9000064010

9000062406

9000046507

9000046508

9000039001

9000039103

9000039104

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