January 2023 Algebra 2, Part II

This exam was adminstered in January 2023.

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Algebra 2 January 2023

Part II: Each correct answer will receive 2 credits. Partial credit can be earned. One mistake (computational or conceptual) will lose 1 point. A second mistake will lose the other point. It is sometimes possible to get 1 point for a correct answer with no correct work shown.


25. Algebraically determine the zeros of the function below.

r(x) = 3x³ + 12x² - 3x - 12

Answer:

Set the expression equal to 0. Then factor by grouping.

3x³ + 12x² - 3x - 12 = 0

3x³ - 3x + 12x² - 12 = 0

3x(x² - 1) + 12(x² - 1) = 0

(3x + 12)(x² - 1) = 0

(3x + 12)(x - 1)(x + 1) = 0

3x + 12 = 0 or x - 1 = 0 or x + 1 = 0

x = -4 or x = 1 or x = -1

26. Given a > 0, solve the equation a^{x + 1} = ∛(a²) for x algebraically

Answer:

Cube both sides of the equation to get rid of the radical and then solve the equation that results from the exponents being equal.

a^{x + 1} = ∛(a²)
a^{3x + 3} = a²
3x + 3 = 2
3x = -1
x = -1/3

27. Given P(A) = 1/3 and P(B) = 5/12, where A and B are independent events, determine P(A ∩ B).

Answer:

The probably of two independent events happening in the probability of one of them happening times the probability of the other.

P(A ∩ B) = (1/3)(5/12) = 5/36.

28. The scores on a collegiate mathematics readiness assessment are approximately normally distributed with a mean of 680 and a standard deviation of 120.

Determine the percentage of scores between 690 and 900, to the nearest percent.

Answer:

Use your graphing calculator. Find the function normalcdf.

Enter the following command: normalcdf(690,900,680,120) for the range minimum and maximum, followed by the median and the standard deviation. The answer will be 43%.

If you estimate it using a standard deviation chart, you won't get an exact answer.

29. Consider the data in the table below.

x	1	2	3	4	5	6
y	3.9	6	11	18.1	28	40.3

State an exponential regression equation to model these data, rounding all values to the nearest thousandth.

Answer:

Put the data into List 1 and List 2 on your graphing calculator. Run a Exponential Regression (ExpReg).

You should get the following output: a = 2.4585... and b = 1.6159...

The equation you want is y = (2.459)(1.616)^x

30. Write the expression A(x) • B(x) - 3C(x) as a polynomial in standard form.

A(x) = x³ + 2x - 1
B(x) = x² + 7
C(x) = x⁴ - 5x

Answer:

Multiply the first two expression A(x) and B(x). Subtract the product of 3 times C(x).

(x³ + 2x - 1)(x² + 7) - 3(x⁴ - 5x)
x⁵ + 7x³ + 2x³ + 14x - x² - 7 - 3x⁴ + 15x
x⁵ + 9x³ - x² + 14x - 7 - 3x⁴ + 15x
x⁵ - 3x⁴ + 9x³ - x² + 29x - 7

31. Over the set of integers, completely factor x⁴ - 5x² + 4

Answer:

The first step is to factor it the way you would factor y² - 5y + 4. Then factor the quadratic expression that result.

x⁴ - 5x² + 4

(x² - 4)(x² - 1)

(x - 2)(x + 2)(x - 1)(x + 1)

32. Natalia’s teacher has given her the following information about angle θ.

• π < θ < 2π • cos θ = &sqrt;(3)/4

Explain how Natalia can determine if the value of tan θ is positive or negative.

Answer:

Cosine is positive in Quadrants I and IV, and π < θ < 2π indicates Quadrants III and IV, so θ must put the angle into quadrant IV.

Tangent is negative in Quadrant IV.

End of Part II

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