Friday, September 24, 2021

Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, January 2013)



Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Algebra 2/Trigonometry Regents, January 2013

Part I: Each correct answer will receive 2 credits.


6. Which expression is equivalent to (9x2y6)(-1/2)

1) 1/(3xy3)
2) 3xy3
3) 3/(xy3)
4) xy3/3

Answer: 1) 1/(3xy3)


The exponent (-1/2) is the product of (-1)(1/2). The (-1) tells you to take the reciprocal, and the (1/2) tells you to take the square root. When taking the square root, apply it to the coefficient the normal way and halve all the exponents on the variables.

(9x2y6)(-1/2)
= (9x2y6)((-1)(1/2)
= (3xy3)(-1)
= 1/(3xy3)





7. In a certain high school, a survey revealed the mean amount of bottled water consumed by students each day was 153 bottles with a standard deviation of 22 bottles. Assuming the survey represented a normal distribution, what is the range of the number of bottled waters that approximately 68.2% of the students drink?

1) 131-164
2) 131-175
3) 142-164
4) 142-175

Answer: 2) 131-175


The 68.2% of the population refers to one standard deviation below and above the mean.

153 - 22 = 131 and 153 + 22 = 175.





8. What is the fourth term in the binomial expansion (x - 2)8?

1) 448x5
2) 448x4
3) -448x5
4) -448x4

Answer: 3) -448x5


Refer to the Binomial Theorem in the back of the booklet.

The first term will have x8 and each term after will have the exponent reduced by 1. So the 2nd will be x7, the 3rd will be x6 and the 4th will be x5. Eliminate Choices (2) and (4).

The first term will have (-2)0 -- that is, it's not there because it only has factors of x. The second term with have (-2)1, then (-2)2, then (-2)3. This is (-8), which means that the coefficient will be negative. The answer must be Choice (3).

The last part of the term is nC3. If you use your calculator, or if you can doodle that far along Pascal's triangle, you will see that

8C3 = 8 * 7 * 6 / (3 * 2 * 1) = 56

Multiply (-8)(56) = -448





9. Which value of k satisfies the equation 83k+4 = 42k-1?

1) -1
2) 9/4
3) -2
4) -14/5

Answer: 4) -14/5


8 is 23 and 4 is 22. You can rewrite each expression as a power of 2 and then compare the exponents.

83k+4 = (23)3k + 4 = 29k + 12

42k-1 = (22)2k - 1 = 24k - 2

9k + 12 = 4k - 2
5k = -14
k = -14/5





10. There are eight people in a tennis club. Which expression can be used to find the number of different ways they can place first, second, and third in a tournament?

1) 8P3
2) 8C3
3) 8P5
4) 8C5

Answer: 1) 8P3


Since the order matters, you want a Permutation not a Combination. Eliminate Choices (2) and (4). Also, you should have noticed that (2) and (4) have the SAME value.

There are 8 possibilities for first, then 7 for second and then 6 for third. That is 8P3.




More to come. Comments and questions welcome.

More Regents problems.

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