Friday, October 22, 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

Part I: Each correct answer will receive 2 credits.

16. Which expression is equivalent to (n°m°p)(x), given m(x) = sin x, n(x) = 3x, and p(x) = x2?

1) sin(3x)2
2) 3 sin x2
3) sin2(3x)
4) 3 sin2x

Answer: 2) 3 sin x2

As with compositions of transformations, when you have a composition of functions, you work from right to left. You want to find n OF m OF p OF x, or more explicitly, n(m(p(x))).

p(x) = x2

m(x2) = sin x2

n(sin x2) = 3 sin x2

The other choices will occur if you do operations in the wrong order.

17. The value of csc 138°23' rounded to four decimal places is

1) -1.3376
2) -1.3408
3) 1.5012
4) 1.5057

Answer: 4) 1.5057

If you can't calculate that directly in your calculator, remember that CSC is 1/SIN.

Since Sin is positive between 0 and 180°, you can eliminate Choices (1) and (2). (Without checking, my guess is that both of these are related to Cosine.)

23/60 = .383333333

1/SIN(138.38333333) = 1/0.664143715 = 1.50569821 = 1.5057

18. Which function is one-to-one?

1) k(x) = x2 + 2
2) g(x) = x3 + 2
3) f(x) = |x| + 2
4) j(x) = x4 + 2

Answer: 2) g(x) = x3 + 2

For a relation to be a function, every x can have only 1 value associated with it. It must pass the Vertical Line Test.

For a function to be one-to-one, every y value must be associate with exactly 1 x value. It must pass a horizontal line test as well.

Choices (1), (3) and (4) are all U or V shaped. They come down from infinity, hit a minimum point and turn back up toward infinity. Every y value, except the minimum, will be associated with both a positive AND a negative x value.

19. The conjugate of the complex expression -5x + 4i is

1) 5x - 4i
2) 5x + 4i
3) -5x - 4i
4) -5x + 4i

Answer: 3) -5x - 4i

When it came to binomials, and the Difference of Squares rule, (x + y)(x - y) were conjugates. The two terms were the same but the sign in the middle was change from + to -.

With complex numbers, it is the same thing: (-5x + 4i) and (-5x - 4i) are conjugates. The sign in the middle -- the sign before the imaginary part of the complex number -- changes. The sign of the Real portion does not change.

20. What is a positive value of tan 1/2 x, when sin x = 0.8?

1) 0.5
2) 0.4
3) 0.33
4) 0.25

Answer: 1) 0.5

The value of of sin x = 0.8 when x = 51.130... degrees, as in a 6-8-10 right triangle.

Half of 51.130... is 26.565..

Tan 25.565 = 0.478..., which rounds to .5.

More to come. Comments and questions welcome.

More Regents problems.

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