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 II: Each correct answer will receive 2 credits. Partial credit is possible.


28. Determine the sum of the first twenty terms of the sequence whose first five terms are 5, 14, 23, 32, and 41.

Answer:

The sequence is Arithmetic because it has a common difference of 9: 14 - 5 = 23 - 14 = 32 - 23 ... etc.

From the back of the booklet, we know that the formula for summing terms in an arithmetic sequence is:

S_n = n(a₁ + a_n) / 2

So we need to know the 20th term in the sequence. The formula for the sequence is

a_n = 9(n - 1) + 5
so for n = 20
a₂₀ = 9(20 - 1) + 5 = 176

S₂₀ = 20(a₁ + a_n) / 2 = 20(5 + 176) / 2 = 1810

29. Determine the sum and the product of the roots of 3x² = 11x - 6

Answer:

A quadratic in standard form is ax² + bx + c. When a = 1, then -b is the sum of the roots and c is the product of the roots.

If a > 1, then the sum of the roots is -b/a, and the product is c/a.

Rewrite the equation in standard form:

3x² = 11x - 6
3x² - 11x + 6 = 0

Sum of the Roots = c/a = 6/3 = 2

Product of the Roots = -b/a = 11/3

If you didn't know the formulas, you could find the roots of the equation.

What are the factors of 18 that add up to add up to -11? -9 and -2. Factor by grouping.

3x² - 9x - 2x + 6 = 0
3x(x - 3) - 2(x - 3) = 0
(3x - 2)(x - 3) = 0
3x - 2 = 0 or x - 3 = 0
x = 2/3 or x = 3

Product of (2/3)(3) = 2

Sum of (2/3) + (3) = 3 2/3 or 11/3

30. If sec (a + 15)° = csc (2a)°, find the smallest positive value of a, in degrees.

Answer:

Sec = 1 / cos and csc = 1 / sin.

sec (a + 15)° = csc (2a)°
1 / cos (a + 15)° = 1 / sin (2a)°
cos (a + 15)° = sin (2a)°
sin (90 - (a + 15))° = sin (2a)°
2a = 90° - a - 15°
3a = 75°
a = 25°

31. The heights, in inches, of 10 high school varsity basketball players are 78, 79, 79, 72, 75, 71, 74, 74, 83, and 71. Find the interquartile range of this data set.

Answer:

If you put the data into your calculator, it will spit out the Five-Number Summary. Then calculate IQR = Q3 - Q1

Without the calculator, you need to sort the data first:

71, 71, 72, 74, 74, 75, 78, 79, 79, 83
Min = 71
Max = 83
Med = (74+75)/2 = 74.5
Q1 = 72
Q3 = 79
IQR = Q3 - Q1 = 79 - 72 = 7

More to come. Comments and questions welcome.

More Regents problems.

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