## Monday, January 29, 2024

### January 2023 Algebra 2, Part II

This exam was adminstered in January 2023.

More Regents problems.

### 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

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 ax + 1 = ∛(a2) for x algebraically

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

ax + 1 = ∛(a2)
a3x + 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).

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.

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.

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

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

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.

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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