This exam was adminstered in August 2024 .
August 2024 Algebra 2, Part III
Each correct answer is worth up to 4 credits. Partial credit can be given. Work must be shown or explained.
33. Solve algebraically for x:
Express your answers in simplest a + bi form.
Answer:
Cross-multiply and solve the quadratic equation.
2/x = (2x + 3) / (x - 4)
2(x - 4) = x(2x + 3)
2x - 8 = 2x2 + 3x
2x2 + x + 8 = 0
Use the Quadratic Formula:
x = (-1 + √((1)2 - 4(2)(8)) ) / ( 2(2))
x = (-1 + √(1 - 64) ) / (4)
x = -1/4 + √(-63) / 4
x = -1/4 + i √(63) / 4
x = -1/4 + i √((9)(7)) / 4
x = -1/4 + 3i √(7) / 4
34. A highly selective college reports that the mean score earned by accepted students on the
Mathematics Level 2 subject test is 750 with a standard deviation of 20 and that the scores are
approximately normally distributed.
Given this information, determine the interval representing the middle 95% of student scores. To the nearest whole percent, determine the percentage of accepted students who scored a
760 or less.
Answer:
The middle 95% is the mean minus 2 times the standard deviation to the mean plus 2 times the standard deviation. (Note that I copied this sentence from January 2024, question 34. Similar question, same question number.)
750 - 2(20) = 710, 750 + 2(20) = 790.
The interval representing the middle 95% of student scores is 710 < p < 790.
To answer the second part of the question, you need to use the normalcdf function on your calculator. Use 10-99 as your minimum number and 760 as the maximum (as stated in the question). Then enter the mean of 750 and standard deviation of 20.
normalcdf(10-99, 760, 750, 20) = 0.691
69% of accepted students scored 760 or less.
This particular problem could be solved without using a calculator:
Note that 760 is 750 + 10, which is one-half of a standard deviation of above the mean. The normal distribution curve is no longer on the reference chart. However, if you know the percentages, then you know that 19.1% of the data is one-half of a standard deviation above the mean, and, of course, 50% of the data is below the mean. Therefore, 50% + 19.1% = 69.1% of the data.
35. For c(x) = 3x2 - 4x + 7 and d(x) = x - 2, determine c(x) • d(x) - [d(x)]3 as a polynomial in standard form.
Answer:
There are three steps to complete. Multiply the two functions. Cube the d(x) function. Then subtract the two products.
First step:
c(x) • d(x) = 3x3 - 10x2 + 15x - 14
Second step:
Either multiply (x - 2)(x - 2)(x - 2) or use the fact that (x - a)3 = x3 - 3ax2 + 3a2x - a3.
Using the rule, (x - 2)3 = x3 - 6x2 + 12x - 8. Or you can do it in two steps:
Note: I checked the official response guide and "f(x) + 5" was acceptable as an answer.
Final step:
3x3 - 10x2 + 15x - 14
x3 - 6x2 + 12x - 8
2x3 - 4x2 + 3x - 6
36. Christopher works for a defense contractor and earned $85,000 his first year. For each additional
year he will receive a 2.5% raise.
Write a geometric series formula, Cn, for Christopher’s total earnings over n years.
Use this formula to find Christopher’s total earnings, to the nearest hundred dollars, over his first
10 years of employment.
Answer:
Use the formula for the Reference sheet to answer the first part. Then use that equation, substitute 10 for n, and evaluate.
Cn = (a1 - a1 rn) / (1 - r)
Cn = (850001 - 850001 (1.025)n) / (1 - 1.025)
Substitute n = 10.
Cn = (85000 - 85000(1.025)10) / (1 - 1.025) = 952287.45
To the nearest $100, C10 = $952,300.
End of Part III
How did you do?
Questions, comments and corrections welcome.
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