Tuesday, September 28, 2021

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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