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 I: Each correct answer will receive 2 credits.


16. The area of triangle ABC is 42. If AB = 8 and m∠B = 61, the length of BC is approximately

1) 5.1
2) 9.2
3) 12.0
4) 21.7

Answer: 3) 12.0

The area of the triangle is given by A = 1/2 ab sin C, the product of the length of two sides times the sin of the included angle.

42 = 1/2 (8) (x) sin 61
x = 2 * 42 / (8 sin 61) = 12.005...

17. When factored completely, the expression 3x³ - 5x² - 48x + 80 is equivalent to

1) (x² - 16)(3x - 5)
2) (x² + 16)(3x - 5)(3x - 5)
3) (x + 4)(x - 4)(3x - 5)
4) (x + 4)(x - 4)(3x - 5)(3x - 5)

Answer: 3) (x + 4)(x - 4)(3x - 5)

It should be obvious that CHoices (2) and (4) would result in polynomials of order 4. That is, the first tem will be x⁴, not x³. Also, Choice (1) is NOT factored completely, so the only possible answer is Choice (3).

Factor by grouping:

3x³ - 5x² - 48x + 80
= 3x³ - 48x - 5x² + 80
= 3x(x² - 16) - 5(x² - 16)
= (3x - 5)(x² - 16)
This is Choice (1), but it is not factored completely.
= (3x - 5)(x + 4)(x - 4)

18. The value of sin (180 + x) is equivalent to

1) -sin x
2) -sin (90 - x)
3) sin x
4) sin (90 - x)

Answer: 1) -sin x

When a point was rotated 180 degrees about the origin, the point (x, y) becomes (-x, -y). Since any point (x, y) on the unit circle is (cos x, sin x), then adding 180 to sin x would change it to -sinx.

19. The sum of CBRT(6a⁴b²) and CBRT(162a⁴b²) expressed in simplest radical form, is

1) SixthRoot(6a⁴b²)
2) 2a²b CBRT(21a²b)
3) 4a CBRT(6ab²)
4) 10a²b CBRT(8)

Answer: 3) 4a CBRT(6ab²)

The two expressions cannot be added unless they are like terms. Simplify the radicals by finding cube roots.

CBRT(6a⁴b²) + CBRT(162a⁴b²)
= CBRT(6a³a b²) + CBRT(2 * 3⁴ a³a b²)
= CBRT(6a³a b²) + CBRT(2 * 3 3³ a³a b²)
= a * CBRT(6 a b²) + 3a * CBRT(6a b²)
= 4a * CBRT(6 a b²)

Choice (1) with its Sixth root was ridiculous and should have been immediately eliminated.

20. Which equation is represented by the graph below?

1) y = 2 cos 3x
2) y = 2 sin 3
3) y = 2 cos 2π/3 x
4) y = 2 cos 2π/3 x

Answer: 1) y = 2 cos 3x

Cos (0) = 1 and 2 Cos (0) = 2. However 2 Sin (0) = 0, so eliminate choices (3) and (4).

The 3 in front of the x increases the frequency by a factor of 3, meaning that waves will be compressed and each cycle will be smaller, not bigger.

A frequency of 2π/3 would stretch out the graph. And the coefficient of x does not correspond to the number on the x-axis.

The correct choice is (1).

More to come. Comments and questions welcome.

More Regents problems.

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