Sunday, April 28, 2024

January 2024 Algebra 2 Regents, Part I (multiple choice)

This exam was adminstered in January 2024.

More Regents problems.

Part I

9. The equation of the parabola that has its focus at the point (-3, 2) and directrix at y = 0 is

(1) y = 1/4 (x + 3)2 + 1
(2) y = 1/4 (x - 3)2 + 1
(3) y = 1/8 (x + 3)2 + 1
(4) y = 1/8 (x - 3)2 + 1

Answer: (1) y = 1/4 (x + 3)2 + 1

If the focus (which is on the axis of symmetry) is on x = -3, then the sign is flipped in the equation. Eliminate Choices (2) and (4).

The directrix of the parabola is y = k - a = 0, so k = a. And since the vertex is the midpoint between the focus and the directrix, (2 + 0)/2 = 1, so k = a = 1.

The equation of the parabola would be written as

y = (1/4(1))(x + 3)2 + 1

This is Choice (1).

10. The seventh term of the geometric sequence √(6), -2√(3), 2√(6), -4√(3), ... is

(1) 6√(6)
(2) -6√(3)
(3) 8√(6)
(4) -8√(3)

If you divide 2√(6) by -2√(3), you will get a common ratio of -√(2),

There is no reason to use a formula. Just multiply by -√(2) three more times.

4√(6), -8√(3), 8√(6).

11. A company wishes to determine the cooking time for one pound of spaghetti. The company’s technicians cooked one pound of spaghetti and recorded the time needed for the spaghetti to be ready to eat. Repeating this process 35 times resulted in an approximately normal distribution, with a mean of 9.82 minutes and a standard deviation of 1.4 minutes. In which interval should the middle 95% of cooking times fall?

(1) (8.42, 11.22)
(2) (7.02, 12.62)
(3) (9.35, 10.29)
(4) (6.82, 11.32)

The middle 95% interval is 9.82 + 2(1.4), which 9.82 - 2.8 = 7.02 and 9.82 + 2.8 = 12.62.

The correct answer is Choice (2).

If you forgot to multiply by 2, you would've gotten Choice (1).

12. Given f(x) = 2x2 + 7x - 15 and g(x) = 3 - 2x, what is f(x)/g(x) for all defined values?

(1) -x - 5
(2) -x + 5
(3) x - 5
(4) x + 5

Divide 2x2 + 7x - 15 by -2x + 3. It's as simple as factoring.

2x2 + 7x - 15

(-2x + 3)(-x - 5)

Notice that 2x2 divided by -2x is -x, and -15 divided by 3 is -5. You can check -2x * -5 = 10x and 3 * -x = -3x, and 10 - 3x = 7x. Check.

Choice (1) is the correct answer.

13. Which equation is equivalent to P = 210x4/3y7/3

Change the improper fractions into mixed number exponents. Separate the whole number portion from the fraction. The fractions both have denominators of 3 meaning that it is a cube root. Only the variables with fractional exponents will go under the cube root sign.

210x4/3y7/3 = 210 x1 x1/3 y2 y1/3

= 210 xy2 x1/3 y1/3

= 210 xy2 ∛(xy)

This is Choice (3).

14. The average cost of a gallon of milk in the United States between the years of 1995 and 2018 can be modeled by the equation P(t) = -0.0004t3 + 0.0114t2 - 0.0150t + 2.6602, where P(t) represents the cost, in dollars, and t is time in years since January 1995. During this time period, in what year did P(t) reach its maximum?

(1) 1995
(2) 2013
(3) 2014
(4) 2018

Graph the equation and look at the table of values to find the maximum. You'll see it occurs about year 18.

Add 18 years to 1995 to get 2013.

15. The temperature, F, in degrees Fahrenheit, after t hours of a roast put into an oven is given by the equation F = 325 - 185e-0.4t. What was the temperature of the roast when it was put into the oven?

(1) 325
(2) 200
(3) 185
(4) 140

When t = 0, the exponent will be zero, so e0 = 1 and 185e0 = 185.

Subtract F = 325 - 185 = 140, which is Choice (4).

16. The roots of the equation 0 = x2 + 6x + 10 in simplest a + bi form are

(1) -3 + 2i
(2) -6 + i
(3) -3 + i
(4) -3 + i√(2)

Find the discriminant using b2 - 4ac (don't confuse the a's and b's here).

62 - 4(1)(10) = 36 - 40 = -4. And the square root of -4 is 2i.

So the roots are x = (-6 + 2i) / (2(1)), which simplifies to -3 + i.

You could also Complete the Square:

x2 + 6x + 10 = 0

x2 + 6x + 9 = -1

(x + 3)2 = -1

x + 3 = + i

x = -3 + i

The correct answer is Choice (3).

More to come.

More Regents problems.

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