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 2012
Part I: Each correct answer will receive 2 credits.
6. What is the product of (x/4 - 1/3) and (x/4 + 1/3)?
1) x2/8 - 1/9
2) x2/16 - 1/9
3) x2/8 - x/6 - 1/9
4) x2/8 - x/6 - 1/9
Answer: 2) x2/16 - 1/9
When two conjugates are multiplied, the result is the Difference of Two Perfect Sqaures, even if those squares are fractions.
(x/4)(x/4) = x2/16
(-1/3)(1/3) = - 1/9
The middle terms create a Zero Pair: (x/4)(1/3) and (-1/3)(x/4).
7. Which is a graph of y = cot x?
Answer: 3) [See Image]
The cotangent graph is always decending, unlike the tangent graph which is always rising. The cotangent graph, which is cos/sin, has an asymptote at 0, where sin x = 0 and the fraction would be undefined. This is Choice (3).
Choice (1) shows a cosine graph, which even if you weren't sure if it was cosine or sine, you should still have known it was incorrect.
Choice (2) is the tangent graph, and the only other choice (other than the answer) that you should have been looking at if you just didn't know right off the bat.
Choice (4) is a cosecant graph, which you might've thought was a secant graph for all that it mattered. You should've known immediately that it's wrong.
8. Which expression always equals 1?
1) cos2 x - sin2 x
2) cos2 x + sin2 x
3) cos x - sin x
4) cos x + sin x
Answer: 2) cos2 x + sin2 x
THis is an importnat trigonometric identity that you must know.
If you need help remembering it, think of the Unit Circle. Then think of either the equation of that circle x2 + y2 = 12. Or think of the Pytagorean Theorem, which is the same formula.
Now replace x with cos x, and y with sin x. You will get the second equation listed above.
The other equations are not identities is any way.
9. What are the sum and product of the roots of the equation
6x2 - 4x - 12 = 0?
1) sum = -2/3; product = -2
2) sum = 2/3; product = -2
3) sum = -2; product = 2/3
4) sum = -2; product = -2/3
Answer: 2) sum = 2/3; product = -2
For any quadratic equation, the sum of the roots is -b/a and the product of the roots is c/a. You don't have to find the roots to answer the question.
In this equation, a = 6, b = -4, and c = -12
The sum of the roots is -b/a = -(-4)/6 = 2/3.
The product of the roots is c/a = -12/6 = -2
10. Given triangle ABC with a = 9, b = 10, and m∠B = 70, what type of
triangle can be drawn?
1) an acute triangle, only
2) an obtuse triangle, only
3) both an acute and an obtuse triangle
4) neither an acute triangle nor an obtuse triangle
Answer: 1) an acute triangle, only
Use the Law of Sines to find possible values for angle A.
sin A / a = sin B / b
sin A / 9 = sin 70 / 10
sin A = 9 * sin 70 / 10 = 0.8457
A = 57.75 degrees, B = 70, C = 180 - 70 - 57.75 = 52.25 degrees. This is an acute triangle.
sin-1(0.8457) could also be 122.25. However, 122.25 + 70 is greater than 180 and cannot make a triangle.
More to come. Comments and questions welcome.
More Regents problems.
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