Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.
More Regents problems.
Integrated Algebra Regents, January 2012
Part I: Each correct answer will receive 2 credits.
21. Campsite A and campsite B are located directly opposite each other
on the shores of Lake Omega, as shown in the diagram below.
The two campsites form a right triangle with Sam’s position, S.
The distance from campsite B to Sam’s position is 1,300 yards, and
campsite A is 1,700 yards from his position.
What is the distance from campsite A to campsite B, to the nearest yard?
1) 1,095
2) 1,096
3) 2,140
4) 2,141
Answer: 1) 1,095
THe diagram shows a right triangle, along with the length of one leg and the hypotenuse. Use Pythagorean Theorem to find the approximate length of the other leg.
x2 + 13002 = 17002
x2 = 17002 - 13002 = 1200000
x = √(1200000) = 1095.44512..., which gets rounded down.
22. Which set builder notation represents {-2, -1, 0, 1, 2, 3}?
1) {x|-3 < x < 3, where x is an integer}
2) {x|-3 < x < 4, where x is an integer}
3) {x|-2 < x < 3, where x is an integer}
4) {x|-2 < x < 3, where x is an integer}
Answer: 4) {x|-2 < x < 3, where x is an integer}
Look at each of the choices one at a time.
Choice (1) is incorrect because -3 would be a member of the set.
Choice (2) is incorrect because 4 would be a member of the set.
Choice (3) is incorrect because -2 would NOT be a member of the set, not would 3.
Choice (4) is the correct answer because -2 is less than or equal to all the integers in the set and all the integers are less than 4. This isn't how I would write it -- I would want the two symbols to match, which is possible in this case -- but it isn't incorrect.
23. The roots of the equation 3x2 - 27x = 0 are
1) 0 and 9
2) 0 and -9
3) 0 and 3
4) 0 and -3
Answer: 1) 0 and 9
Zero is definitely a root, and it's in all four choices. So you only need to check the threes and nines.
You know that 3x2 has to be positive, so if you subtract something and get zero, then you have to subtract a positive value from it. This means that you can eliminate Choices (2) and (4).
Simplify the equation:
x2 - 9x = 0
x(x - 9) = 0
The roots are x = 0 and x = 9.
24. Which equation is an example of the use of the associative property of addition?
1) x + 7 = 7 + x
2) 3(x + y) = 3x + 3y
3) (x + y) + 3 = x + (y + 3)
4) 3 + (x + y) = (x + y) + 3
Answer: 3) (x + y) + 3 = x + (y + 3)
The Associative Property states that it doesn't matter how you groups the numbers you are adding. Think of an association as a group. It's helpful when you have something like 27 + 18 + 2 -- adding the 18 + 2 makes the calculation much simpler.
Choice (1) shows the Commutative Property. The order of the addends doesn't matter. Your commute at the end of the day is in the opposite direction from your commute in the morning.
Choice (2) is the Distributive Property of Multiplication over Division.
Choice (3) is the Associative Property. They are grouped differently.
Choice (4) is the Commutative Property again. Yes, there are parentheses, but they are still around (x + y) on both sides of the equation. The order has been changed, putting the (x + y) first instead of last.
25. Given:
B = {3, 5, 8, 9}
What is A ∪ B?
1) {5}
2) {5, 8}
3) {2, 3, 4, 7, 9}
4) {2, 3, 4, 5, 7, 8, 9}
Answer: 4) {2, 3, 4, 5, 7, 8, 9}
The Union of the two sets are all the elements that appear in either one or both of the two sets.
Put the two sets together and you {2, 3, 4, 5, 7, 8, 9}.
Choice (2) is the Intersection ( ∩ ) of the two sets, which is only those elements that appear in BOTH sets.
More to come. Comments and questions welcome.
More Regents problems.
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