Thursday, October 21, 2021

Algebra 2 Problems of the Day (Algebra 2 Regents, June 2012)

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, June 2012

11. Which graph represents the function log 2 x = y? position?

The log2 x function is the inverse of the exponential function 2x, a reflection across the line y = x.

When x = 1 then y = 0 because when 0 is the exponent of 2 then the result is 1.

12. A circle is drawn to represent a pizza with a 12 inch diameter. The circle is cut into eight congruent pieces. What is the length of the outer edge of any one piece of this circle?

1) 3π/4
2) π
3) 3π/2
4) 3π

The circumference of the pizza is 12π. One eighth of that is 12π/8, which simplifies to 3π/2.

13. What is the solution set for the equation √(5x + 29) = x + 3?

1) {4}
2) {-5}
3) {4, 5}
4) {-5, 4}

You can work this out (see below) or you can check if 4, -5 or 5 work.

√(5x + 29) ?= x + 3
√(5(4) + 29) ?= (4) + 3
√(49) ?= 7
True, x = 4 is a solution. Eliminate Choice (2).

√(5x + 29) ?= x + 3
√(5(5) + 29) ?= (5) + 3
√(54) ?= 8
False, x = 5 is NOT a solution. Eliminate Choice (3).

√(5x + 29) ?= x + 3
√(5(-5) + 29) ?= (-5) + 3
√(4) ?= -2
False, x = 2 is NOT a solution. Eliminate Choice (4).

Choice (1) is the solution.

Alternatively, you could solve the equation:

√(5x + 29) = x + 3
5x + 29 = (x + 3)2
5x + 29 = x2 + 6x + 9
x2 + x - 20 = 0
(x + 5)(x - 4) = 0
x + 5 = 0 or x - 4 = 0
x = -5 or x = 4

Next check for extraneous values. As shown above x = -5 is NOT a solution. Discard it. Then x = 4 is the only solution.

14. When factored completely, x3 + 3x2 - 4x - 12 equals

1) (x + 2)(x - 2)(x - 3)
2) (x + 2)(x - 2)(x + 3)
3) (x2 - 4)(x + 3)
4) (x2 - 4)(x - 3)

Answer: 2) (x + 2)(x - 2)(x + 3)

It says "factored completely" and (x2 - 4) is not factored completely. It can be broken down further. Eliminate Choices (3) and (4).

If you factor the polynomial by grouping:

x3 + 3x2 - 4x - 12
= x3 - 4x + 3x2 - 12
= x(x2 - 4) + 3(x - 4)
= (x2 - 4)(x + 3)
= (x + 2)(x - 2)(x + 3)

15. What is the middle term in the expansion of (x/2 - 2y)6?

1) 20x3y3
2) -15/4 x4y2
3) -20x3y3
4) 15/4 x4y2

The expansion has 7 terms, starting with the x6y0 term and ending with the x0y6 term. The middle term would be the fourth one and it would be in the form x3y3, so eliminate Choices (2) and (4).

The coefficient of the fourth term would be 6C3 * (1/2)3 * (-2)3 = (20)(1/8)(-1/8) = -20.

More to come. Comments and questions welcome.

More Regents problems.

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