JEFFERSON MATH PROJECT REGENTS BY STANDARD · 5. Given: ΔABT CBTD CD, , and AB⊥ Write an...

JEFFERSON MATH PROJECTREGENTS BY STANDARDAll 578 NY Math B Regents Exam Questions

from June 2001 to August 2006 Sorted by Standard

www.jmap.org

Dear Sir

I have to acknolege the reciept of your favor of May 14. in which you mention that you have finished the6. first books of Euclid, plane trigonometry, surveying & algebra and ask whether I think a furtherpursuit of that branch of science would be useful to you. there are some propositions in the latter books ofEuclid, & some of Archimedes, which are useful, & I have no doubt you have been made acquainted withthem. trigonometry, so far as this, is most valuable to every man, there is scarcely a day in which he will notresort to it for some of the purposes of common life. the science of calculation also is indispensible as far asthe extraction of the square & cube roots; Algebra as far as the quadratic equation & the use of logarithmsare often of value in ordinary cases: but all beyond these is but a luxury; a delicious luxury indeed; butnot to be indulged in by one who is to have a profession to follow for his subsistence. in this light I view theconic sections, curves of the higher orders, perhaps even spherical trigonometry, Algebraical operationsbeyond the 2d dimension, and fluxions. Letter from Thomas Jefferson to William G. Munford, Monticello, June 18, 1799.

NEW YORK STATE MATHEMATICSPERFORMANCE STANDARDS

# of ?s

MATHEMATICAL REASONINGStudents use mathematical reasoning to analyze mathematical situations, make conjectures,gather evidence, and construct an argument.

25

NUMBER AND NUMERATIONStudents use number sense and numeration to develop an understanding of the multiple useof numbers in the real world, the use of numbers to communicate mathematically, and theuse of numbers in the development of mathematical ideas.

52

OPERATIONSStudents use mathematical operations and relationships among them to understandmathematics.

63

MODELING/MULTIPLE REPRESENTATIONStudents use mathematical modeling/multiple representation to provide a means ofpresenting, interpreting, communicating, and connecting mathematical information andrelationships.

127

MEASUREMENTStudents use measurement in both metric and English measure to provide a major linkbetween the abstractions of mathematics and the real world in order to describe and compareobjects and data.

113

UNCERTAINTYStudents use the ideas of uncertainty to illustrate that mathematics involves more thanexactness when dealing with everyday situations.

67

PATTERNS/FUNCTIONSStudents use patterns and functions to develop mathematical power, appreciate the truebeauty of mathematics, and construct generalizations that describe patterns simply andefficiently.

131

Total 578

Math B Regents Exam Questions by Standard - Mathematical Reasoning PAGE 1www.jmap.org

1. Tom scored 23 points in a basketball game.He attempted 15 field goals and 6 freethrows. If each successful field goal is 2points and each successful free throw is 1point, is it possible he successfully made all6 of his free throws? Justify your answer.

2. A ball is thrown straight up at an initialvelocity of 54 feet per second. The heightof the ball t seconds after it is thrown isgiven by the formula h t t t( ) .= −54 12 2

How many seconds after the ball is thrownwill it return to the ground?

[A] 9.2 [B] 6 [C] 4.5 [D] 4

3. In ΔABC, D is a point on AC such thatBD is a median. Which statement must betrue?

[A] Δ ΔABD CBD≅

[B] ∠ ≅ ∠ABD CBD

[C] AD CD≅ [D] BD AC⊥

4. In the accompanying diagram, ΔABC is notisosceles. Prove that if altitude BD weredrawn, it would not bisect AC.

5. Given: ΔABT CBTD CD, , and AB⊥

Write an indirect proof to show that AT isnot perpendicular to CD.

6. In the accompanying diagram of ΔABC,

AB AC≅ , BD BA=13

, and CE CA=13

.

Triangle EBC can be proved congruent totriangle DCB by

[A] HL ≅ HL [B] SSS ≅ SSS

[C] SAS ≅ SAS [D] ASA ≅ ASA

7. In the accompanying diagram, CA AB⊥ ,

ED DF⊥ , ED AB, CE BF≅ , AB ED≅

and m CAB m FDE∠ = ∠ = 90.

Which statement would not be used to proveΔ ΔABC DEF≅ ?[A] HL ≅ HL [B] SSS ≅ SSS

[C] AAS ≅ AAS [D] SAS ≅ SAS


8. In the accompanying diagram ofparallelogram ABCD, DE BF≅ .

Triangle EGC can be proved congruent totriangle FGA by

[A] AAS ≅ AAS [B] AAA ≅ AAA

[C] HL ≅ HL [D] SSA ≅ SSA

9. In the accompanying diagram, HK bisectsIL and ∠ ≅ ∠H K.

What is the most direct method of proof thatcould be used to prove Δ ΔHIJ KLJ≅ ?

[A] SAS ≅ SAS [B] AAS ≅ AAS

[C] HL ≅ HL [D] ASA ≅ ASA

10. Which condition does not prove that twotriangles are congruent?

[A] SSS ≅ SSS [B] SSA ≅ SSA

[C] ASA ≅ ASA [D] SAS ≅ SAS

11. Which statements could be used to provethat ΔABC and Δ ′ ′ ′A B C are congruent?

[A] AB A B≅ ′ ′, BC B C≅ ′ ′, and∠ ≅ ∠ ′A A

[B] ∠ ≅ ∠ ′A A , AC A C≅ ′ ′, andBC B C≅ ′ ′

[C] ∠ ≅ ∠ ′A A , ∠ ≅ ∠ ′B B , and ∠ ≅ ∠ ′C C

[D] AB A B≅ ′ ′, ∠ ≅ ∠ ′A A , and ∠ ≅ ∠ ′C C

12. Given: parallelogram FLSH, diagonalFGAS , LG FS⊥ , HA FS⊥

Prove: Δ ΔLGS HAF≅

13.

14. Prove that the diagonals of a parallelogrambisect each other.


15. Given: parallelogram ABCD, diagonal AC,and ABE

Prove: m m∠ > ∠1 2

16. In the accompanying diagram of ABCD,where a b≠ , prove ABCD is an isoscelestrapezoid.

17. Quadrilateral KATE has vertices K(1,5),A(4,7), T(7,3), and E(1,-1).a Prove that KATE is a trapezoid. [The useof the grid is optional.]b Prove that KATE is not an isoscelestrapezoid.

18. The coordinates of quadrilateral ABCD areA(-1,-5), B(8,2), C(11,13), and D(2,6).Using coordinate geometry, prove thatquadrilateral ABCD is a rhombus. [The useof the grid is optional.]

19. The coordinates of quadrilateral JKLM areJ(1,-2), K(13,4), L(6,8), and M(-2,4). Provethat quadrilateral JKLM is a trapezoid butnot an isosceles trapezoid. [The use of thegrid is optional.]


20. Jim is experimenting with a new drawingprogram on his computer. He createdquadrilateral TEAM with coordinates T(-2,3), E(-5,-4), A(2,-1), and M(5,6). Jimbelieves that he has created a rhombus butnot a square. Prove that Jim is correct.[The use of the grid is optional.]

21. Given: A(-2,2), B(6,5), C(4,0), D(-4,-3)Prove: ABCD is a parallelogram but not arectangle. [The use of the grid is optional.]

22. In the accompanying diagram of circle O,PA is drawn tangent to the circle at A.Place B on PA anywhere between P and Aand draw OA, OP, and OB. Prove thatOB is not perpendicular to PA.

23. Given: chords AB and CD of circle Ointersect at E, an interior point of circle O;chords AD and CB are drawn.

Prove: (AE)(EB) = (CE)(ED)


24. In the accompanying diagram of circle O,diameter AOB is drawn, tangent CB isdrawn to the circle at B, E is a point on thecircle, and BE ADC.

Prove: Δ ΔABE CAB≅

25. Twenty high school students took anexamination and received the followingscores:70, 60, 75, 68, 85, 86, 78, 72, 82, 88, 88, 73,74, 79, 86, 82, 90, 92, 93, 73Determine what percent of the studentsscored within one standard deviation of themean. Do the results of the examinationapproximate a normal distribution? Justifyyour answer.

Math B Regents Exam Questions by Standard - Number and Numeration PAGE 1www.jmap.org

1. For which equation is the sum of the rootsequal to the product of the roots?

[A] x x2 4 4 0− + = [B] x x2 1 0+ + =

[C] x x2 8 4 0− − = [D] x x2 3 6 0+ − =

2. Which expression is equivalent to 43 2+

?

[A] 12 4 211− [B] 12 4 2

7+

[C] 12 4 211+ [D] 12 4 2

7−

3. Which expression is equal to 2 32 3+−

?

[A] 1 4 37

− [B] 7 4 37

+

[C] 1 4 3− [D] 7 4 3+

4. The expression 72 3−

is equivalent to

[A] 14 7 3+ [B] 2 37+

[C] 14 7 3− [D] 14 37+


is equivalent to

[A] − +3 52

[B] 3 52+

[C] 3 52− [D] − −3 5

2


is equivalent to

[A] 21 27+ [B] 3 2

7+

[C] 3 2+ [D] 3 2−


is equivalent to

[A]12

135−+ [B]

8135

−+

[C]12

135 + [D]8

135 +

8. Which expression represents the sum of13

12

+ ?

[A] 25

[B] 2 3 3 26+

[C] 3 23+ [D] 3 2

2+


is equivalent to

[A] 5 5 54− [B] 5 5 5

4+

[C] 5 5 56− [D] 5

4

10. The expression 123 3+

is equivalent to

[A] 2 3+ [B] 12 3−

[C] 4 2 3− [D] 6 2 3−

11. Which statement is true for all real numbervalues of x?

[A] |x - 1| > 0 [B] |x - 1| > (x - 1)

[C] x x2 = [D] x x2 =

12. If the equation x kx2 36 0− − = has x = 12as one root, what is the value of k?

[A] 9 [B] 3 [C] -3 [D] -9


13. The new corporate logo created by thedesign engineers at Magic Motors is shownin the accompanying diagram.

If chords BA and BC are congruent andmBC = 140, what is m B∠ ?

[A] 40 [B] 140 [C] 280 [D] 80

14. What is the multiplicative inverse of 3i?

[A] −i3

[B] − 3i [C] 13

[D] − 3

15. Express − + + + −48 35 25 27. insimplest a + bi form.

16. What is the sum of 2 4− − and− + −3 16 expressed in simplest a + biform?

[A] − +1 2i [B] − +1 12i

[C] − +1 20i [D] − +14 i

17. If f x x x( ) ,= −3 22 then f i( ) is equivalentto

[A] 2 - i [B] 2 + i

[C] -2 + i [D] -2 - i

18. The expression 23++

ii

is equivalent to

[A] 6 58+ i [B] 7 5

10− i

[C] 68+ i [D] 7

10+ i

19. Melissa and Joe are playing a game withcomplex numbers. If Melissa has a score of5 4− i and Joe has a score of 3 2+ i, what istheir total score?

[A] 8 + 6i [B] 8 - 2i

[C] 8 - 6i [D] 8 + 2i

20. Show that the product of a + bi and itsconjugate is a real number.

21. If the sum of the roots of x x2 3 5+ − isadded to the product of its roots, the result is

[A] -2 [B] 15 [C] -15 [D] -8

22. Jacob is solving a quadratic equation. Heexecutes a program on his graphingcalculator and sees that the roots are real,rational, and unequal. This informationindicates to Jacob that the discriminant is

[A] a perfect square [B] negative

[C] zero [D] not a perfect square

23. Which equation has imaginary roots?

[A] x2 1 0− = [B] x x2 1 0− − =

[C] x x2 1 0+ + = [D] x2 2 0− =

24. The roots of the equation ax x2 4 2+ = − arereal, rational, and equal when a has a valueof

[A] 3 [B] 4 [C] 1 [D] 2

25. In the equation ax x2 6 9 0+ − = , imaginaryroots will be generated if

[A] -1 < a < 1 [B] a < 1, only

[C] a < -1 [D] a > -1, only

26. The equation 2 8 02x x n+ + = hasimaginary roots when n is equal to

[A] 10 [B] 8 [C] 6 [D] 4

27. The roots of the equation 2 8 4 02x x− − =are

[A] real, rational, and equal

[B] real, rational, and unequal

[C] real, irrational, and unequal

[D] imaginary


28. The roots of the equation 2 42x x− = are

[A] real, rational, and equal

[B] real and irrational

[C] real, rational, and unequal

[D] imaginary

29. Which statement must be true if a parabolarepresented by the equation y ax bx c= + +2

does not intersect the x-axis?

[A] b ac2 4 0− > , and b ac2 4− is a perfectsquare.

[B] b ac2 4 0− > , and b ac2 4− is not aperfect square.

[C] b ac2 4 0− < [D] b ac2 4 0− =

30. If the roots of ax bx c2 0+ + = are real,rational, and equal, what is true about thegraph of the function y ax bx c= + +2 ?

[A] It intersects the x-axis in two distinctpoints.

[B] It lies entirely above the x-axis.

[C] It is tangent to the x-axis.

[D] It lies entirely below the x-axis.

31. Find all values of k such that the equation3 2 02x x k− + = has imaginary roots.

32. If 2 + 3i is one root of a quadratic equationwith real coefficients, what is the sum of theroots of the equation?

33. Which equation has imaginary roots?

[A] x(5 + x) = 8 [B] x(x + 6) = -10

[C] x(5 - x) = -3 [D] (2x + l)(x - 3) = 7

34. The roots of the equation 2 5 02x − = are

[A] real, rational, and unequal

[B] real, rational, and equal

[C] imaginary [D] real and irrational

35. Written in simplest form, the expressionx y

xy

2 2 93

−−

is equivalent to

[A] 13+ xy

[B] −1

[C] − +( )3 xy [D] 3+ xy

36. Express the following rational expression insimplest form:

910 28 6

2

2

−− −

xx x

37. For all values of x for which the expression

is defined, 25 6

2

2

x xx x

++ +

is equivalent to

[A] 13x +

[B] xx + 3

[C] 12x +

[D] xx + 2

38. Written in simplest form, the expressionx x

x x

2

2

945 5

−−

is equivalent to

[A] 5 [B] −15

[C] 15

[D] −5

39. The expression 3 124

2

2 3

y yy y−−

is equivalent to

[A] −94

[B] 34

122−

y

[C] 3y

[D] −3y

40. The expression

ab

ba

a b

−

+1 1 is equivalent to

[A] ab [B] a b+

[C] a bab− [D] a b−


41. The fraction

xy

x

y

+

+1 1

is equivalent to

[A] 21

xyy+

[B] x yy

2

1+[C] 2x [D] x

42. In simplest form,

1 1

1 1

2 2x y

y x

−

+ is equal to

[A] y xxy− [B] x y−

[C] y x− [D] x yxy−

43. Express in simplest form:

xx

x

44

1 4

−

−

44. The expression

1 1

1 12 2

x y

x y

+

− is equivalent to

[A] xyy x−

[B] y xxy−

[C] xyx y−

[D] y x−

45. Which expression is equivalent to the

complex fraction

xx

xx

+

−+

21

2

?

[A] x2

[B] 2x

[C] 22

xx +

[D] 242

xx +

46. When simplified, the complex fraction

1 1

1 0+

−≠x

xx

x, , is equivalent to

[A] -1 [B]x−1

1 [C] 1 [D]1

1−x

47. Express in simplest form:

1 1

12

2

r srs

−

−

48. Simplify completely:

1

1

−

−

mm

mm

49. Simplify for all values of a for which the

expression is defined: 1 2

4 12

−

−

a

a

50. In a science experiment, when resistor Aand resistor B are connected in a parallel

circuit, the total resistance is 11 1A B+

. This

complex fraction is equivalent to

[A] 1 [B] AB

[C] ABA B+

[D] A + B

51. A rectangle is said to have a golden ratio

when wh

hw h

=−

, where w represents width

and h represents height. When w = 3,between which two consecutive integerswill h lie?

52. The expression seccsc

θθ

is equivalent to

[A] cosθ [B] sincos

θθ

[C] cossin

θθ

[D] sinθ

Math B Regents Exam Questions by Standard - Operations PAGE 1www.jmap.org

1. On a trip, a student drove 40 miles per hourfor 2 hours and then drove 30 miles per hourfor 3 hours. What is the student's averagerate of speed, in miles per hour, for thewhole trip?

[A] 37 [B] 34 [C] 35 [D] 36

2. The volume of any spherical balloon can be

found by using the formula V r=43

3π .

Write an equation for r in terms of V and π .

3. Which quadratic function is shown in theaccompanying graph?

[A] y x= −12

2 [B] y x= 2 2

[C] y x= −2 2 [D] y x=12

2

4. The product of ( )5ab and ( )−2 2 3a b is

[A] − 30 7 4a b [B] − 40 6 4a b

[C] − 30 6 4a b [D] − 40 7 4a b

5. The expression ( )bb b

n

n n

2 1 3

4 3

+

+⋅ is equivalent to

[A] bn

2[B] bn [C] b n−3 [D] b n− +3 1

6. What is the axis of symmetry of the graphof the equation x y= 2 ?

[A] line y = -x [B] line y = x

[C] x-axis [D] y-axis

7. Which transformation is a direct isometry?

[A] D2 [B] T2 5, [C] ry axis− [D] D−2

8. Which transformation is an example of anopposite isometry?

[A] (x,y) → (3x,3y)

[B] (x,y) → (x + 3,y - 6)

[C] (x,y) → (y,-x) [D] (x,y) → (y,x)

9. What are the coordinates of point P, theimage of point (3,-4) after a reflection in theline y = x?

[A] (3,4) [B] (-3,4)

[C] (-4,3) [D] (4,-3)

10. The image of the origin under a certaintranslation is the point (2,-6). The image ofpoint ( , )− −3 2 under the same translation isthe point

[A] (-1,-8) [B] (-6,12)

[C] ( , )−32

13

[D] (-5,4)


11. Point ′P is the image of point P(-3,4) aftera translation defined by T( , ) .7 1− Which othertransformation on P would also produce′P ?

[A] ry axis− [B] ry x=−

[C] R− °90 [D] R90°

12. If the coordinates of point A are (-2,3), whatis the image of A under r Dy axis− 3 ?

[A] (6,9) [B] (5,6)

[C] (9,-6) [D] (-6,-9)

13. What is the image of point (1,1) underr Rx axis− °0 90, ?

[A] (1,-1) [B] (-1,-1)

[C] (1,1) [D] (-1,1)

14. If f(x) = cos x, which graph represents f(x)under the composition r ry axis x axis− − ?

[A]

[B]

[C]

[D]

15. The accompanying graph represents thefigure .

Which graph represents after atransformation defined by r Ry x= °90 ?

[A] [B]

[C] [D]

16. Given point A(-2,3). State the coordinatesof the image of A under the compositionT rx axis− − −3 4, . [The use of the accompanyinggrid is optional.]


17. The graph of the function g(x) is shown onthe accompanying set of axes. On the sameset of axes, sketch the image of g(x) underthe transformation D2 .

18. A box contains one 2-inch rod, one 3-inchrod, one 4-inch rod, and one 5-inch rod.What is the maximum number of differenttriangles that can be made using these rodsas sides?

[A] 1 [B] 4 [C] 3 [D] 2

19. At Mogul's Ski Resort, the beginner's slopeis inclined at an angle of 12.3°, while theadvanced slope is inclined at an angle of26.4°. If Rudy skis 1,000 meters down theadvanced slope while Valerie skis the samedistance on the beginner's slope, how muchlonger was the horizontal distance thatValerie covered?

[A] 895.7 m [B] 977.0 m

[C] 81.3 m [D] 231.6 m

20. A regular hexagon is inscribed in a circle.What is the ratio of the length of a side ofthe hexagon to the minor arc that itintercepts?

[A] π6

[B] 6π

[C] 3π

[D] 36

21. In the accompanying diagram, cabins B andG are located on the shore of a circular lake,and cabin L is located near the lake. PointD is a dock on the lake shore and iscollinear with cabins B and L. The roadbetween cabins G and L is 8 miles long andis tangent to the lake. The path betweencabin L and dock D is 4 miles long.

What is the length, in miles, of BD ?

[A] 24 [B] 4 [C] 8 [D] 12

22. If f ( )x x= −5 12 and g( ) ,x x= −3 1 findg(f(1)).

23. If f ( )x x= +2 42 and g( ) ,x x= − 3 whichnumber satisfies f f g( ) ( )( ) ?x x=

[A] 34

[B] 32

[C] 5 [D] 4


24. The temperature generated by an electricalcircuit is represented by t m m= =f ( ) . ,0 3 2

where m is the number of moving parts.The resistance of the same circuit isrepresented by r t t= = +g( ) ,150 5 where tis the temperature. What is the resistance ina circuit that has four moving parts?

[A] 8,670 [B] 174 [C] 156 [D] 51

25. The revenue, R(x), from selling x units of aproduct is represented by the equationR x x( ) ,= 35 while the total cost, C(x), ofmaking x units of the product is representedby the equation C x x( ) .= +20 500 The totalprofit, P(x), is represented by the equationP x R x C x( ) ( ) ( ).= − For the values of R(x)and C(x) given above, what is P(x)?

[A] 15x + 500 [B] 15x

[C] 15x - 500 [D] 10x + 100

26. Which equation states that the temperature,t, in a room is less than 3° from 68°?

[A] |68 - t| < 3 [B] |68 + t| < 3

[C] |3 - t| < 68 [D] |3 + t| < 68

27. The solution set of 3 2 1x + < contains

[A] only positive real numbers

[B] no real numbers

[C] both positive and negative real numbers

[D] only negative real numbers

28. The graph of y x= −( )3 2 is shifted left 4units and down 2 units. What is the axis ofsymmetry of the transformed graph?

[A] x = -2 [B] x = 1

[C] x = 7 [D] x = -1

29. If the point (a, b) lies on the graphy f x= ( ), the graph of y f x= −1( ) mustcontain point

[A] (b,a) [B] (a,0)

[C] (0,b) [D] (-a,-b)

30. Fractal geometry uses the complex numberplane to draw diagrams, such as the oneshown in the accompanying graph.

Which number is not included in the shadedarea?

[A] -0.9 [B] -0.5 - 0.5i

[C] -0.9 - 0.9i [D] -0.5i


31. Two complex numbers are graphed below.

What is the sum of w and u, expressed instandard complex number form?

[A] 5 + 7i [B] -5 + 3i

[C] 3 + 7i [D] 7 + 3i

32. Expressed in simplest form,i i i i16 6 5 132+ − +

[A] −1 [B] i [C] − i [D] 1

33. When simplified, 27 34i i+ is equal to

[A] i61 [B] i-1 [C] i [D] -i-1

34. What is the value of i i99 3− ?

[A] 1 [B] i96 [C] 0 [D] − i

35. What is the sum of − 2 and −18 ?

[A] 6i [B] 4 2i

[C] 2 5i [D] 5 2i

36. The expression i i i i i0 1 2 3 4⋅ ⋅ ⋅ ⋅ is equal to

[A] i [B] -1 [C] 1 [D] -i

37. The expression ii

16

3 is equivalent to

[A] -i [B] -1 [C] 1 [D] i

38. Mrs. Donahue made up a game to help herclass learn about imaginary numbers. Thewinner will be the student whose expressionis equivalent to -i. Which expression willwin the game?

[A] i48 [B] i46 [C] i47 [D] i49

39. What is the product of 5 36+ − and1 49− − , expressed in simplest a + biform?

[A] 47 - 29i [B] -37 + 41i

[C] 5 - 71i [D] 47 + 41i

40. When expressed as a monomial in terms ofi, 2 32 5 8− − − is equivalent to

[A] 218i [B] 22i

[C] i22 [D] 22i−

41. The expression ( )− +1 3i is equivalent to

[A] -3i [B] 2 + 2i

[C] -1 - i [D] -2 - 2i

42. The complex number c di+ is equal to( ) .2 2+ i What is the value of c?


43. In an electrical circuit, the voltage, E, involts, the current, I, in amps, and theopposition to the flow of current, calledimpedance, Z, in ohms, are related by theequation E = IZ. A circuit has a current of(3 + i) amps and an impedance of (-2 + i)ohms. Determine the voltage in a + biform.

44. The relationship between voltage, E,current, I, and resistance, Z, is given by theequation E IZ= . If a circuit has a currentI i= +3 2 and a resistance Z i= −2 , what isthe voltage of this circuit?

[A] 4 + i [B] 4 - i

[C] 8 + i [D] 8 + 7i

45. Impedance measures the opposition of anelectrical circuit to the flow of electricity.The total impedance in a particular circuit is

given by the formula Z Z ZZ ZT = +

1 2

1 2

. What is

the total impedance of a circuit, ZT , ifZ i1 1 2= + and Z i2 1 2= − ?

[A] 1 [B] 52

[C] 0 [D] −32

46. Express, in simplest a + bi form, the rootsof the equation x x2 5 4+ = .

47. In physics class, Taras discovers that thebehavior of electrical power, x, in aparticular circuit can be represented by thefunction f ( ) .x x x= + +2 2 7 If f ( ) ,x = 0solve the equation and express your answerin simplest a + bi form.

48. The value of ( )3

27

0

23

1− is

[A] 9 [B] 19

[C] −19

[D] − 9

49. The expression 4 212 3⋅ is equal to

[A] 832 [B] 4

32 [C] 4 [D] 16

50. The expression 3

3

13

23


[A] 3 [B] 1 [C] 3 [D] 133

51. If x is a positive integer, 412x is equivalent

to

[A] 4 1x

[B] 2x [C] 2x

[D] 4 x

52. The expression 16 6 44 a b is equivalent to

[A] 2 2a b [B] 4 2a b

[C] 232a b [D] 4

32a b

53. When simplified, the expression

( )( )m m4312


[A] 6 5m [B] 3 2−m

[C] 5 4−m [D] 4 3m


54. Find the value of ( ) ( )x x+ + +−

2 1023 when

x = 7.

55. If ( ) ,aa

x23

2

1= what is the value of x?

[A] -3 [B] 2 [C] 1 [D] -1

56. If f ( ) ,x x=−

32 then f ( )1

4 is equal to

[A] 8 [B] − 4 [C] −18

[D] − 2

57. If log 5 = a, then log 250 can be expressedas

[A] 50a [B] 2a + 1

[C] 10 + 2a [D] 25a

58. Which function is symmetrical with respectto the origin?

[A] y x= 5 [B] yx

= −5

[C] y x= +5 [D] y x= −5

59. Express in simplest form:4 8

12

3 154

2 8 10

2

2

xx

xx

xx x

++

⋅−−

÷−

− −

60. What is the sum of 33x −

and xx3−

?

[A] xx+−

33

[B] −1 [C] 1 [D] 0

61. What is the sum of ( ) ?yy

− ++

5 32

[A]272

+−

yy [B] 5−y

[C]22

+−

yy [D]

2732

+−−

yyy

62. Express in simplest form: 1 13x x

++

63. Which type of symmetry does the equationy x= cos have?

[A] point symmetry with respect to theorigin

[B] line symmetry with respect to y = x

[C] line symmetry with respect to the x-axis

[D] point symmetry with respect to ( , )π2

0

Math B Regents Exam Questions by Standard - Modeling/Multiple Representation PAGE 1www.jmap.org

1. A store advertises that during its Labor Daysale $15 will be deducted from everypurchase over $100. In addition, after thededuction is taken, the store offers an early-bird discount of 20% to any person whomakes a purchase before 10 a.m. If Hakeemmakes a purchase of x dollars, x>100, at 8a.m., what, in terms of x, is the cost ofHakeem's purchase?

[A] 0.85x - 20 [B] 0.80x - 12

[C] 0.20x - 15 [D] 0.20x - 3

2. If Jamar can run 35

of a mile in 2 minutes

30 seconds, what is his rate in miles perminute?

[A] 4 16

[B] 3 110

[C] 45

[D] 625

3. At the local video rental store, José rentstwo movies and three games for a total of$15.50. At the same time, Meg rents threemovies and one game for a total of $12.05.How much money is needed to rent acombination of one game and one movie?

4. The cost of a long-distance telephone call isdetermined by a flat fee for the first 5minutes and a fixed amount for eachadditional minute. If a 15-minute telephonecall costs $3.25 and a 23-minute call costs$5.17, find the cost of a 30-minute call.

5. Island Rent-a-Car charges a car rental fee of$40 plus $5 per hour or fraction of an hour.Wayne's Wheels charges a car rental fee of$25 plus $7.50 per hour or fraction of anhour. Under what conditions does it costless to rent from Island Rent-a-Car? [Theuse of the accompanying grid is optional.]

6. What is the total number of points ofintersection for the graphs of the equationsy x= 2 and y x= − 2 ?

[A] 1 [B] 0 [C] 3 [D] 2


7. A pelican flying in the air over water dropsa crab from a height of 30 feet. Thedistance the crab is from the water as it fallscan be represented by the functionh t t( ) ,= − +16 302 where t is time, inseconds. To catch the crab as it falls, a gullflies along a path represented by thefunction g t t( ) .= − +8 15 Can the gull catchthe crab before the crab hits the water?Justify your answer. [The use of theaccompanying grid is optional.]

8. The price of a stock, A(x), over a 12-monthperiod decreased and then increasedaccording to the equationA x x x( ) . ,= − +0 75 6 202 where x equals the

number of months. The price of anotherstock, B(x), increased according to theequation B x x( ) . .= +2 75 150 over the same12-month period. Graph and label bothequations on the accompanying grid. Stateall prices, to the nearest dollar, when bothstock values were the same.

9. What is the equation of a parabola that goesthrough points (0,1), (-1,6), and (2,3)?

[A] y x x= − +2 3 1

[B] y x x= − +2 3 12

[C] y x= +2 1 [D] y x= +2 12


10. A small rocket is launched from a height of72 feet. The height of the rocket in feet, h,is represented by the equationh( ) ,t t t= − + +16 64 722 where t = time, inseconds. Graph this equation on theaccompanying grid.Use your graph to determine the number ofseconds that the rocket will remain at orabove 100 feet from the ground. [Only agraphic solution can receive full credit.]

11. For which quadratic equation is the axis ofsymmetry x = 3?

[A] y x x= − + +2 3 5

[B] y x x= + +2 6 3

[C] y x x= + +2 3 [D] y x x= − + +2 6 2

12. An acorn falls from the branch of a tree tothe ground 25 feet below. The distance, S,the acorn is from the ground as it falls isrepresented by the equationS t t( ) ,= − +16 252 where t represents time,in seconds. Sketch a graph of this situationon the accompanying grid.Calculate, to the nearest hundredth of asecond, the time the acorn will take to reachthe ground.

13. Two objects are 2 4 1020. × centimetersapart. A message from one object travels tothe other at a rate of 12 105. × centimetersper second. How many seconds does it takethe message to travel from one object to theother?

[A] 2 0 1015. × [B] 2 88 1025. ×

[C] 12 1015. × [D] 2 0 104. ×

14. The path of a rocket is represented by theequation y x= −25 2 . The path of amissile designed to intersect the path of therocket is represented by the equation

x y=32

. The value of x at the point of

intersection is 3. What is the correspondingvalue of y?

[A] -2 [B] 4 [C] 2 [D] -4


15. Which transformation best describes therelationship between the functions

f ( )x x= 2 and g( ) ( ) ?x x=12

[A] reflection in the origin

[B] reflection in the x-axis

[C] reflection in the y-axis

[D] reflection in the line y = x

16. What are the coordinates of point A', theimage of point A(-4,l) after the compositetransformation R ry x90° = where the origin isthe center of rotation?

[A] (-1,-4) [B] (1,4)

[C] (4,1) [D] (-4,-1)

17. a On the accompanying grid, graph theequation 2 2 42y x= − in the interval− ≤ ≤3 3x and label it a.b On the same grid, sketch the image of aunder T rx axis5 2,− − and label it b.

18. Ileana buys a large circular pizza that isdivided into eight equal slices. Shemeasures along the outer edge of the crust

from one piece and finds it to be 5 12

inches.

What is the diameter of the pizza to thenearest inch?

[A] 8 [B] 7 [C] 4 [D] 14

19. Two tow trucks try to pull a car out of aditch. One tow truck applies a force of1,500 pounds while the other truck applies aforce of 2,000 pounds. The resultant forceis 3,000 pounds. Find the angle between thetwo applied forces, rounded to the nearestdegree.

20. One force of 20 pounds and one force of 15pounds act on a body at the same point sothat the resultant force is 19 pounds. Find,to the nearest degree, the angle between thetwo original forces.

21. Two equal forces act on a body at an angleof 80°. If the resultant force is 100newtons, find the value of one of the twoequal forces, to the nearest hundredth of anewton.

22. The accompanying diagram shows atriangular plot of land that is part of Fran'sgarden. She needs to change thedimensions of this part of the garden, butshe wants the area to stay the same. Sheincreases the length of side AC to 22.5 feet.If angle A remains the same, by how manyfeet should side AB be decreased to makethe area of the new triangular plot of landthe same as the current one?


23. The accompanying diagram shows asemicircular arch over a street that has aradius of 14 feet. A banner is attached tothe arch at points A and B, such that AE =EB = 5 feet. How many feet above theground are these points of attachment forthe banner?

24. If f x x x( ) ( ) ,= + −4 40 1 what is the value off ( ) ?4

[A] −12 [B] 1 116

[C] 4 116

[D] 0

25. Which relation is not a function?

[A] y x= +2 4 [B] x y x= + −2 2 3

[C] x y= −3 2 [D] y x x= − +2 4 3

26. A company calculates its profit by findingthe difference between revenue and cost.The cost function of producing x hammersis C x x( ) .= +4 170 If each hammer is soldfor $10, the revenue function for selling xhammers is R x x( ) .= 10How many hammers must be sold to make aprofit?How many hammers must be sold to make aprofit of $100?

27. Which equation represents the parabolashown in the accompanying graph?

[A] f x x( ) ( )= − + +3 12

[B] f x x( ) ( )= + −1 32

[C] f x x( ) ( )= − − −3 32

[D] f x x( ) ( )= − − +3 12

28. An archer shoots an arrow into the air suchthat its height at any time, t, is given by thefunction h t t kt( ) .= − + +16 32 If themaximum height of the arrow occurs at timet = 4 , what is the value of k?

[A] 4 [B] 64 [C] 8 [D] 128

29. What is the turning point, or vertex, of theparabola whose equation is

?163 2 −+= xxy

[A] (3,44) [B] (-l,-4)

[C] (1,8) [D] (-3,8)

30. What is the minimum point of the graph ofthe equation y x x= + +2 8 92 ?

[A] (-2,-15) [B] (2,17)

[C] (2,33) [D] (-2,1)


31. The height of an object, h(t), is determinedby the formula h t t t( ) = − +16 2562 , where tis time, in seconds. Will the object reach amaximum or a minimum? Explain or showyour reasoning.

32. Vanessa throws a tennis ball in the air. Thefunction h t t t( ) = − + +16 45 72 representsthe distance, in feet, that the ball is from theground at any time t. At what time, to thenearest tenth of a second, is the ball at itsmaximum height?

33. The height, h, in feet, a ball will reach whenthrown in the air is a function of time, t, inseconds, given by the equationh t t t( ) = − + +16 30 62 . Find, to the nearesttenth, the maximum height, in feet, the ballwill reach.

34. When a current, I, flows through a givenelectrical circuit, the power, W, of thecircuit can be determined by the formulaW I I= −120 12 2 . What amount of current,I, supplies the maximum power, W?

35. The equation W I I= −120 12 2 representsthe power (W), in watts, of a 120-volt circuithaving a resistance of 12 ohms when acurrent (I) is flowing through the circuit.What is the maximum power, in watts, thatcan be delivered in this circuit?

36. A baseball player throws a ball from theoutfield toward home plate. The ball'sheight above the ground is modeled by theequation y x x= − + +16 48 62 where yrepresents height, in feet, and x representstime, in seconds. The ball is initiallythrown from a height of 6 feet.How many seconds after the ball is thrownwill it again be 6 feet above the ground?What is the maximum height, in feet, thatthe ball reaches? [The use of theaccompanying grid is optional.]

37. When a baseball is hit by a batter, the heightof the ball, h(t), at time t, t ≥ 0, isdetermined by the equationh t t t( ) .= − + +16 64 42 For which intervalof time is the height of the ball greater thanor equal to 52 feet?


38. The profit, P, for manufacturing a wirelessdevice is given by the equationP x x= − + −10 750 9 0002 , , where x is theselling price, in dollars, for each wirelessdevice. What range of selling prices allowsthe manufacturer to make a profit on thiswireless device? [The use of the grid isoptional.]

39. Which graph represents the solution set ofthe inequality x x2 4 5 0− − < ?

[A]

[B]

[C]

[D]

40. Find the sum of -2 + 3i and -1 - 2i.Graph the resultant on the accompanying setof axes.

41. Which graph represents a quadratic functionwith a negative discriminant?

[A]

[B]

[C]

[D]


42. The expression b b−

>32 0, , is equivalent to

[A] − ( )b 3 [B] 23 )( b

[C]3)(

1b

[D]23 )(

1b

43. Meteorologists can determine how long astorm lasts by using the function

t d d( ) . ,= 0 0732 where d is the diameter of

the storm, in miles, and t is the time, inhours. If the storm lasts 4.75 hours, find itsdiameter, to the nearest tenth of a mile.

44. As shown in the accompanying diagram, acircular target with a radius of 9 inches hasa bull's-eye that has a radius of 3 inches. Iffive arrows randomly hit the target, what isthe probability that at least four hit thebull's-eye?

45. Which equation models the data in theaccompanying table?

[A] y x= +2 5 [B] y x= 2

[C] y x= 5 2( ) [D] y x= 2

46. What is the domain of f x x( ) = 2 ?

[A] all real numbers [B] x ≤ 0

[C] x ≥ 0 [D] all integers

47. The strength of a medication over time isrepresented by the equation y x= −200 15( . ) ,where x represents the number of hourssince the medication was taken and yrepresents the number of micrograms permillimeter left in the blood. Which graphbest represents this relationship?

[A] [B]

[C] [D]

48. On January 1, 1999, the price of gasolinewas $1.39 per gallon. If the price ofgasoline increased by 0.5% per month, whatwas the cost of one gallon of gasoline, to thenearest cent, on January 1 one year later?

49. The Franklins inherited $3,500, which theywant to invest for their child's future collegeexpenses. If they invest it at 8.25% withinterest compounded monthly, determinethe value of the account, in dollars, after 5

years. Use the formula A P rn

nt= +( ) ,1

where A = value of the investment after tyears, P = principal invested, r = annualinterest rate, and n = number of timescompounded per year.


50. On the accompanying grid, solve thefollowing system of equations graphically:

y x x= − + +2 2 1y x= 2

51. On the accompanying grid, sketch thegraphs of y x= 2 and 3 7 3y x= + over theinterval − ≤ ≤3 4x . Identify and state thecoordinates of all points of intersection.

52. The graphs of the equations y x= 2 andy x a= − +2 intersect in Quadrant I forwhich values of a?

[A] a ≥ 1 [B] 0 1< <a[C] a > 1 [D] a < 1

53. If log ,b x y= then x equals

[A] by [B] yb [C] yb

[D] y b⋅

54. The function y x= 2 is equivalent to

[A] x y= log 2 [B] y x= log 2

[C] x y= log2 [D] y x= log2

55. For which value of x is y = log x undefined?

[A] 110

[B] 1483. [C] π [D] 0

56. A hotel finds that its total annual revenueand the number of rooms occupied dailyby guests can best be modeled by thefunction R n n n= + >3 10 02log( ), , where Ris the total annual revenue, in millions ofdollars, and n is the number of roomsoccupied daily by guests. The hotel needsan annual revenue of $12 million to beprofitable. Graph the function on theaccompanying grid over the interval0 100< ≤n .Calculate the minimum number of roomsthat must be occupied daily to be profitable.


57. Which expression is not equivalent tolog ?b 36

[A] 6 2logb [B] log logb b72 2−

[C] 2 6logb [D] log logb b9 4+

58. If log a = 2 and log b = 3, what is the

numerical value of log ?ab3

[A] 8 [B] 25 [C] -25 [D] -8

59. If log a x= and log ,b y= what islog ?a b

[A] x y+

2[B] 2 2x y+

[C] x y+ 2 [D] x y+2

60. The speed of sound, v, at temperature T, indegrees Kelvin, is represented by the

equation v T= 1087

273. Which expression

is equivalent to log v?

[A] log log( )1087 2 273+ +T

[B] 1087 12

273+ −log logT

[C] 1087 12

12

273( log log )T −

[D] log log log1087 12

12

273+ −T

61. If log log log ,k c v p k= + equals

[A] v pc [B] v pc +

[C] cv p+ [D] ( )vp c

62. Solve for x: log ( ) log ( )42

43 5 1x x x+ − + =

63. In the equation log log ,x x4 9 2+ = x isequal to

[A] 18 [B] 13 [C] 65. [D] 6

64. If log ,5 2x = what is the value of x ?

[A] 25 [B] 225 [C] 5 [D] 5

65. The relationship between the relative size ofan earthquake, S, and the measure of theearthquake on the Richter scale, R, is givenby the equation log S = R. If an earthquakemeasured 3.2 on the Richter scale, what wasits relative size to the nearest hundredth?

66. The magnitude (R) of an earthquake is

related to its intensity (I) by R IT

= log( ),

where T is the threshold below which theearthquake is not noticed. If the intensity isdoubled, its magnitude can be representedby

[A] log I - log T [B] 2(log I - log T)

[C] 2 log I - log T

[D] log 2 + log I - log T

67. The scientists in a laboratory company raiseamebas to sell to schools for use in biologyclasses. They know that one ameba dividesinto two amebas every hour and that theformula t N= log2 can be used todetermine how long in hours, t, it takes toproduce a certain number of amebas, N.Determine, to the nearest tenth of an hour,how long it takes to produce 10,000 amebasif they start with one ameba.

68. Solve for x: log ( )2 1 3x + =

69. Solve for x: log log logb b b x36 2− =


70. What is the value of x in the equation81 272 5 4x x+ += ?

[A] 411

[B] −211

[C] −411

[D] −32

71. The growth of bacteria in a dish is modeled

by the function f tt

( ) .= 23 For which valueof t is f t( ) ?= 32

[A] 15 [B] 16 [C] 8 [D] 2

72. Depreciation (the decline in cash value) ona car can be determined by the formulaV C r t= −( )1 , where V is the value of thecar after t years, C is the original cost, and ris the rate of depreciation. If a car's cost,when new, is $15,000, the rate ofdepreciation is 30%, and the value of the carnow is $3,000, how old is the car to thenearest tenth of a year?

73. Growth of a certain strain of bacteria ismodeled by the equation G A t= ( . ) .2 7 0 584 ,where:G = final number of bacteriaA = initial number of bacteriat = time (in hours)In approximately how many hours will 4bacteria first increase to 2,500 bacteria?Round your answer to the nearest hour.

74. The equation for radioactive decay is

ptH= ( . )05 , where p is the part of a

substance with half-life H remainingradioactive after a period of time, t.A given substance has a half-life of 6,000years. After t years, one-fifth of the originalsample remains radioactive. Find t, to thenearest thousand years.

75. An archaeologist can determine theapproximate age of certain ancientspecimens by measuring the amount ofcarbon-14, a radioactive substance,contained in the specimen. The formulaused to determine the age of a specimen is

A At

=−

057602 , where A is the amount of

carbon-14 that a specimen contains, A0 isthe original amount of carbon-14, t is time,in years, and 5760 is the half-life of carbon-14.A specimen that originally contained 120milligrams of carbon-14 now contains 100milligrams of this substance. What is theage of the specimen, to the nearest hundredyears?

76. An amount of P dollars is deposited in anaccount paying an annual interest rate r (asa decimal) compounded n times per year.After t years, the amount of money in theaccount, in dollars, is given by the equation

A P rn

nt= +( ) .1

Rachel deposited $1,000 at 2.8% annualinterest, compounded monthly. In howmany years, to the nearest tenth of a year,will she have $2,500 in the account? [Theuse of the grid is optional.]


77. The current population of Little Pond, NewYork, is 20,000. The population isdecreasing, as represented by the formulaP A t= −( . ) ,.13 0 234 where P = finalpopulation, t = time, in years, and A = initialpopulation.What will the population be 3 years fromnow? Round your answer to the nearesthundred people.To the nearest tenth of a year, how manyyears will it take for the population to reachhalf the present population? [The use of thegrid is optional.]

78. Explain how a person can determine if a setof data represents inverse variation and givean example using a table of values.

79. If R varies inversely as S, when S isdoubled, R is multiplied by

[A] 2 [B] 4 [C] 14

[D] 12

80. To balance a seesaw, the distance, in feet, aperson is from the fulcrum is inverselyproportional to the person's weight, inpounds. Bill, who weighs 150 pounds, issitting 4 feet away from the fulcrum. If Danweighs 120 pounds, how far from thefulcrum should he sit to balance the seesaw?

[A] 4.5 ft [B] 3 ft [C] 5 ft [D] 3.5 ft

81. For a rectangular garden with a fixed area,the length of the garden varies inverselywith the width. Which equation representsthis situation for an area of 36 square units?

[A] yx

=36 [B] y x= 36

[C] x y+ = 36 [D] x y− = 36

82. When air is pumped into an automobile tire,the pressure is inversely proportional to thevolume. If the pressure is 35 pounds whenthe volume is 120 cubic inches, what is thepressure, in pounds, when the volume is 140cubic inches?

83. A pulley that has a diameter of 8 inches isbelted to a pulley that has a diameter of 12inches. The 8-inch-diameter pulley isrunning at 1,548 revolutions per minute. Ifthe speeds of the pulleys vary inversely totheir diameters, how many revolutions perminute does the larger pulley make?

84. Boyle's Law states that the pressure ofcompressed gas is inversely proportional toits volume. The pressure of a certainsample of a gas is 16 kilopascals when itsvolume is 1,800 liters. What is the pressure,in kilopascals, when its volume is 900liters?

85. The time it takes to travel to a locationvaries inversely to the speed traveled. Ittakes 4 hours driving at an average speed of55 miles per hour to reach a location. Tothe nearest tenth of an hour, how long willit take to reach the same location driving atan average speed of 50 miles per hour?


86. The speed of a laundry truck variesinversely with the time it takes to reach itsdestination. If the truck takes 3 hours toreach its destination traveling at a constantspeed of 50 miles per hour, how long will ittake to reach the same location when ittravels at a constant speed of 60 miles perhour?

[A] 2 hours [B] 2 12

hours

[C] 2 23

hours [D] 2 13

hours

87. In a given rectangle, the length variesinversely as the width. If the length isdoubled, the width will

[A] increase by 2 [B] be divided by 2

[C] remain the same

[D] be multiplied by 2

88. Camisha is paying a band $330 to play ather graduation party. The amount eachmember earns, d, varies inversely as thenumber of members who play, n. The graphof the equation that represents therelationship between d and n is an exampleof

[A] an ellipse [B] a parabola

[C] a hyperbola [D] a line

89. According to Boyle's Law, the pressure, p,of a compressed gas is inverselyproportional to the volume, v. If a pressureof 20 pounds per square inch exists whenthe volume of the gas is 500 cubic inches,what is the pressure when the gas iscompressed to 400 cubic inches?

[A] 16 lb / in2 [B] 50 lb / in2

[C] 25 lb / in2 [D] 40 lb / in2

90. The accompanying graph shows therelationship between a person's weight andthe distance that the person must sit fromthe center of a seesaw to make it balanced.

Which equation best represents this graph?

[A] y x= −120 [B] y x= 2 log

[C] yx

=120 [D] y x= 12 2

91. Which equation represents a hyperbola?

[A] y x2 216= − [B] y x= 16 2

[C] yx

=16 [D] y x= −16 2

92. Which graph represents an inverse variationbetween stream velocity and the distancefrom the center of the stream?

[A] [B]

[C] [D]


93. Which graph shows that soil permeabilityvaries inversely to runoff?

[A]

[B]

[C]

[D]

94. In the accompanying diagram, point P(0.6, -0.8) is on unit circle O. What is the value ofθ , to the nearest degree?

95. If sinθ is negative and cosθ is negative, inwhich quadrant does the terminal side of θlie?

[A] III [B] II [C] IV [D] I

96. Expressed as a function of a positive acuteangle, sin (-230°) is equal to

[A] sin 50° [B] -sin 50°

[C] -cos 50° [D] cos 50°

97. In the accompanying diagram, PR istangent to circle O at R, QS OR⊥ , andPR OR⊥ .

Which measure represents sin ?θ

[A] PR [B] QS [C] RO [D] SO

98. In the accompanying diagram of a unit

circle, the ordered pair ( , )− −3

212

represents the point where the terminal sideof θ intersects the unit circle.

What is m∠θ ?

[A] 240 [B] 225 [C] 233 [D] 210


99. If θ is an angle in standard position andP( , )−3 4 is a point on the terminal side ofθ , what is the value of sin θ ?

[A] 45

[B] −35

[C] 35

[D] −45

100. The accompanying diagram shows unitcircle O, with radius OB = 1.

Which line segment has a length equivalentto cos ?θ

[A] OA [B] OC [C] AB [D] CD

101. Through how many radians does the minutehand of a clock turn in 24 minutes?

[A] 0 2. π [B] 08. π

[C] 0 6. π [D] 0 4. π

102. A wedge-shaped piece is cut from a circularpizza. The radius of the pizza is 6 inches.The rounded edge of the crust of the piecemeasures 4.2 inches. To the nearest tenth,the angle of the pointed end of the piece ofpizza, in radians, is

[A] 7.0 [B] 1.4 [C] 0.7 [D] 25.2

103. A student attaches one end of a rope to awall at a fixed point 3 feet above theground, as shown in the accompanyingdiagram, and moves the other end of therope up and down, producing a wavedescribed by the equation y a bx c= +sin .The range of the rope's height above theground is between 1 and 5 feet. The periodof the wave is 4π . Write the equation thatrepresents this wave.

104. The shaded portion of the accompanyingmap indicates areas of night, and theunshaded portion indicates areas of daylightat a particular moment in time.

Which type of function best represents thecurve that divides the area of night from thearea of daylight?

[A] logarithmic [B] quadratic

[C] cosine [D] tangent


105. A building's temperature, T, varies withtime of day, t, during the course of 1 day, asfollows:

T t= +8 78cos

The air-conditioning operates whenT F≥ °80 . Graph this function for6 17≤ <t and determine, to the nearesttenth of an hour, the amount of time in 1day that the air-conditioning is on in thebuilding.

106. The tide at a boat dock can be modeled by

the equation y t= − +26

8cos( ) ,π where t is

the number of hours past noon and y is theheight of the tide, in feet. For how manyhours between t = 0 and t = 12 is the tide atleast 7 feet? [The use of the grid isoptional.]

107. On the accompanying set of axes, graph theequations y x= 4cos and y = 2 in thedomain − ≤ ≤π πx .Express, in terms of π , the interval forwhich 4 2cos .x ≥


108. The times of average monthly sunrise, asshown in the accompanying diagram, overthe course of a 12-month interval can bemodeled by the equation y A Bx D= +cos( ) .Determine the values of A, B, and D, andexplain how you arrived at your values.

109. A ship at sea is 70 miles from one radiotransmitter and 130 miles from another.The angle between the signals sent to theship by the transmitters is 117.4°. Find thedistance between the two transmitters, to thenearest mile.

110. A wooden frame is to be constructed in theform of an isosceles trapezoid, withdiagonals acting as braces to strengthen theframe. The sides of the frame each measure5.30 feet, and the longer base measures12.70 feet. If the angles between the sidesand the longer base each measure 68.4°,find the length of one brace to the nearesttenth of a foot.

111. A ship captain at sea uses a sextant to sightan angle of elevation of 37° to the top of alighthouse. After the ship travels 250 feetdirectly toward the lighthouse, anothersighting is made, and the new angle ofelevation is 50°. The ship's charts show thatthere are dangerous rocks 100 feet from thebase of the lighthouse. Find, to the nearestfoot, how close to the rocks the ship is at thetime of the second sighting.

112. A ski lift begins at ground level 0.75 milefrom the base of a mountain whose face hasa 50° angle of elevation, as shown in theaccompanying diagram. The ski lift ascendsin a straight line at an angle of 20°. Find thelength of the ski lift from the beginning ofthe ski lift to the top of the mountain, to thenearest hundredth of a mile.

113. A ship at sea heads directly toward a cliffon the shoreline. The accompanyingdiagram shows the top of the cliff, D,sighted from two locations, A and B,separated by distance S. If m DAC∠ = 30,m DBC∠ = 45, and S = 30 feet, what is the

height of the cliff, to the nearest foot?

114. The center of a circular sunflower with adiameter of 4 centimeters is (-2,1). Whichequation represents the sunflower?

[A] ( ) ( )x y+ + − =2 1 22 2

[B] ( ) ( )x y− + + =2 1 22 2

[C] ( ) ( )x y− + − =2 1 42 2

[D] ( ) ( )x y+ + − =2 1 42 2


115. What is the equation of a circle with center(-3, 1) and radius 7?

[A] 49)1()3( 22 =−++ yx

[B] 7)1()3( 22 =++− yx

[C] 7)1()3( 22 =−++ yx

[D] 49)1()3( 22 =++− yx

116. A circle has the equation( ) ( ) .x y+ + − =1 3 162 2 What are thecoordinates of its center and the length of itsradius?

[A] (1,-3) and 16 [B] (1,-3) and 4

[C] (-1,3) and 16 [D] (-1,3) and 4

117. What are the coordinates of the center of thecircle represented by the equation( ) ( ) ?x y+ + − =3 4 252 2

[A] (-3,4) [B] (3,-4)

[C] (-3,-4) [D] (3,4)

118. The center of a circle represented by theequation ( ) ( )x y− + + =2 3 1002 2 is locatedin Quadrant

[A] IV [B] I [C] II [D] III

119. A commercial artist plans to include anellipse in a design and wants the length ofthe horizontal axis to equal 10 and thelength of the vertical axis to equal 6. Whichequation could represent this ellipse?

[A] 3 20 2y x= [B] 9 25 2252 2x y+ =

[C] x y2 2 100+ = [D] 9 25 2252 2x y− =

120. Which equation, when graphed on aCartesian coordinate plane, would bestrepresent an elliptical racetrack?

[A] 3 10 288 000x y+ = ,

[B] 3 10 288 0002 2x y− = ,

[C] 30 288 000xy = ,

[D] 3 10 288 0002 2x y+ = ,

121. An object orbiting a planet travels in a pathrepresented by the equation3 1 5 4 152 2( ) ( ) .y x+ + + = In which type ofpattern does the object travel?

[A] ellipse [B] circle

[C] parabola [D] hyperbola

122. The accompanying diagram represents theelliptical path of a ride at an amusementpark.

Which equation represents this path?

[A] x y2

2

2

2150 501+ =

[B] y x x= + +2 100 300

[C] x y2

2

2

2150 501− = [D] x y2 2 300+ =


123. An architect is designing a building toinclude an arch in the shape of a semi-ellipse (half an ellipse), such that the widthof the arch is 20 feet and the height of thearch is 8 feet, as shown in theaccompanying diagram.

Which equation models this arch?

[A] x y2 2

100 641+ = [B] x y2 2

64 1001+ =

[C] x y2 2

400 641+ = [D] x y2 2

64 4001+ =

124. The accompanying diagram shows theelliptical orbit of a planet. The foci of theelliptical orbit are F1 and F2 .

If a, b, and c are all positive and a b c≠ ≠ ,which equation could represent the path ofthe planet?

[A] ax by c2 2 2+ = [B] x y c2 2 2+ =

[C] y ax c= +2 2 [D] ax by c2 2 2− =

125. The accompanying diagram shows theconstruction of a model of an elliptical orbitof a planet traveling around a star. Point Pand the center of the star represent the fociof the orbit.

Which equation could represent the relationshown?

[A] x y2 2

15 91+ = [B] x y2 2

81 2251+ =

[C] x y2 2

15 91− = [D] x y2 2

225 811+ =

126. A designer who is planning to install anelliptical mirror is laying out the design on acoordinate grid. Which equation couldrepresent the elliptical mirror?

[A] x y2 24 144+ =

[B] x y2 2 144+ =

[C] y y= +4 1442 [D] x y2 2144 36= +

127. Solve the following system of equationsalgebraically:

9 93 3

2 2x yx y

+ =− =

Math B Regents Exam Questions by Standard - Measurement PAGE 1www.jmap.org

1. A bug travels up a tree, from the ground,over a 30-second interval. It travels fast atfirst and then slows down. It stops for 10seconds, then proceeds slowly, speeding upas it goes. Which sketch best illustrates thebug's distance (d) from the ground over the30-second interval (t)?

[A] [B]

[C] [D]

2. On her first trip, Sari biked 24 miles in Thours. The following week Sari biked 32miles in T hours. Determine the ratio of heraverage speed on her second trip to heraverage speed on her first trip.

[A] 34

[B] 43

[C] 23

[D] 32

3. In a mathematics class of ten students, theteacher wanted to determine how ahomework grade influenced a student'sperformance on the subsequent test. Thehomework grade and subsequent test gradefor each student are given in theaccompanying table.

a Give the equation of the linear regressionline for this set of data.b A new student comes to the class andearns a homework grade of 78. Based onthe equation in part a, what grade would theteacher predict the student would receive onthe subsequent test, to the nearest integer?


4. The graph of f(x) is shown in theaccompanying diagram.

Which graph represents f x r rx axis y axis( ) ?

− −

[A] [B]

[C] [D]

5. The accompanying diagram shows twocables of equal length supporting a pole.Both cables are 14 meters long, and they areanchored to points in the ground that are 14meters apart.

What is the exact height of the pole, inmeters?

[A] 7 [B] 27 [C] 37 [D] 14

6. An architect commissions a contractor toproduce a triangular window. The architectdescribes the window as ΔABC , wherem A∠ = 50, BC = 10 inches, andAB = 12 inches. How many distinct

triangles can the contractor construct usingthese dimensions?

[A] 2 [B] 1 [C] more than 2 [D] 0

7. Sam is designing a triangular piece for ametal sculpture. He tells Martha that two ofthe sides of the piece are 40 inches and 15inches, and the angle opposite the 40-inchside measures 120°. Martha decides tosketch the piece that Sam described. Howmany different triangles can she sketch thatmatch Sam's description?

[A] 0 [B] 3 [C] 2 [D] 1

8. How many distinct triangles can be formedif m A∠ = 30, side b = 12, and side a = 8?

[A] 1 [B] 2 [C] 0 [D] 3

9. A landscape designer is designing atriangular garden with two sides that are 4feet and 6 feet, respectively. The angleopposite the 4-foot side is 30°. How manydistinct triangular gardens can the designermake using these measurements?

10. Main Street and Central Avenue intersect,making an angle measuring 34°. Angelalives at the intersection of the two roads,and Caitlin lives on Central Avenue 10miles from the intersection. If Leticia lives7 miles from Caitlin, which conclusion isvalid?

[A] Leticia can live at one of two locationson Main Street.

[B] Leticia cannot live on Main Street.

[C] Leticia can live at one of three locationson Main Street.

[D] Leticia can live at only one location onMain Street.

11. In ΔABC, if AC = 12, BC = 11,andm A∠ = 30, angle C could be

[A] a right angle, only

[B] either an obtuse angle or an acute angle

[C] an acute angle, only

[D] an obtuse angle, only


12. What is the total number of distincttriangles that can be constructed if AC = 13,BC = 8, and m A∠ = 36?

[A] 0 [B] 2 [C] 1 [D] 3

13. Sam needs to cut a triangle out of a sheet ofpaper. The only requirements that Sammust follow are that one of the angles mustbe 60°, the side opposite the 60° angle mustbe 40 centimeters, and one of the other sidesmust be 15 centimeters. How manydifferent triangles can Sam make?

[A] 0 [B] 3 [C] 2 [D] 1

14. A homeowner wants to increase the size ofa rectangular deck that now measures 15feet by 20 feet, but building code laws statethat a homeowner cannot have a deck largerthan 900 square feet. If the length and thewidth are to be increased by the sameamount, find, to the nearest tenth, themaximum number of feet that the length ofthe deck may be increased in size legally.

15. A garden in the shape of an equilateraltriangle has sides whose lengths are 10meters. What is the area of the garden?

[A] 50 3 2m [B] 25 2m

[C] 25 3 2m [D] 50 2m

16. If the perimeter of an equilateral triangle is18, the length of the altitude of this triangleis

[A] 6 [B] 3 3 [C] 3 [D] 6 3

17. A picnic table in the shape of a regularoctagon is shown in the accompanyingdiagram. If the length of AE is 6 feet, findthe length of one side of the table to thenearest tenth of a foot, and find the area ofthe table's surface to the nearest tenth of asquare foot.

18. Every time the pedals go through a 360°rotation on a certain bicycle, the tires rotatethree times. If the tires are 24 inches indiameter, what is the minimum number ofcomplete rotations of the pedals needed forthe bicycle to travel at least 1 mile?

[A] 281 [B] 5,280 [C] 561 [D] 12

19. A ball is rolling in a circular path that has aradius of 10 inches, as shown in theaccompanying diagram. What distance hasthe ball rolled when the subtended arc is54°? Express your answer to the nearesthundredth of an inch.


20. Cities H and K are located on the same lineof longitude and the difference in thelatitude of these cities is 9°, as shown in theaccompanying diagram. If Earth's radius is3,954 miles, how many miles north of cityK is city H along arc HK? Round youranswer to the nearest tenth of a mile.

21. Kathy and Tami are at point A on a circulartrack that has a radius of 150 feet, as shownin the accompanying diagram. They runcounterclockwise along the track from A toS, a distance of 247 feet. Find, to thenearest degree, the measure of minor arcAS.

22. The accompanying diagram shows the pathof a cart traveling on a circular track ofradius 2.40 meters. The cart starts at pointA and stops at point B, moving in acounterclockwise direction. What is thelength of minor arc AB, over which the carttraveled, to the nearest tenth of a meter?

23. The circumference of a circular plot of landis increased by 10%. What is the bestestimate of the total percentage that the areaof the plot increased?

[A] 21% [B] 25%

[C] 10% [D] 31%

24. The accompanying diagram shows a 24-footladder leaning against a building. A steelbrace extends from the ladder to the pointwhere the building meets the ground. Thebrace forms a right angle with the ladder.

If the steel brace is connected to the ladderat a point that is 10 feet from the foot of theladder, which equation can be used to findthe length, x, of the steel brace?

[A] 1024xx

= [B] 10 242 2 2+ =x

[C] 1014xx

= [D] 10 142 2 2+ =x

25. Gregory wants to build a garden in theshape of an isosceles triangle with one ofthe congruent sides equal to 12 yards. If thearea of his garden will be 55 square yards,find, to the nearest tenth of a degree, thethree angles of the triangle.


26. The accompanying diagram shows the floorplan for a kitchen. The owners plan tocarpet all of the kitchen except the "workspace," which is represented by scalenetriangle ABC. Find the area of this workspace to the nearest tenth of a square foot.

27. Two sides of a triangular-shaped poolmeasure 16 feet and 21 feet, and theincluded angle measures 58°. What is thearea, to the nearest tenth of a square foot, ofa nylon cover that would exactly cover thesurface of the pool?

28. The triangular top of a table has two sidesof 14 inches and 16 inches, and the anglebetween the sides is 30°. Find the area ofthe tabletop, in square inches.

29. A landscape architect is designing atriangular garden to fit in the corner of a lot.The corner of the lot forms an angle of 70°,and the sides of the garden including thisangle are to be 11 feet and 13 feet,respectively. Find, to the nearest integer,the number of square feet in the area of thegarden.

30. Kimi wants to determine the radius of acircular pool without getting wet. She islocated at point K, which is 4 feet from thepool and 12 feet from the point of tangency,as shown in the accompanying diagram.

What is the radius of the pool?

[A] 32 ft [B] 16 ft

[C] 20 ft [D] 4 10 ft

31. An overhead view of a revolving door isshown in the accompanying diagram. Eachpanel is 1.5 meters wide.

What is the approximate width of d, theopening from B to C?

[A] 1.73 m [B] 3.00 m

[C] 1.50 m [D] 2.12 m


32. In the accompanying diagram of circle O,chord AY is parallel to diameter DOE ,AD is drawn, and 40.mAD =

What is m DAY∠ ?

[A] 130 [B] 110 [C] 150 [D] 90

33. A toy truck is located within a circular playarea. Alex and Dominic are sitting onopposite endpoints of a chord that containsthe truck. Alex is 4 feet from the truck, andDominic is 3 feet from the truck. Meira andTamara are sitting on opposite endpoints ofanother chord containing the truck. Meira is8 feet from the truck. How many feet, tothe nearest tenth of a foot, is Tamara fromthe truck? Draw a diagram to support youranswer.

34. In the accompanying diagram, the length of

ABC is 32π radians.

What is m ABC∠ ?

[A] 45 [B] 36 [C] 53 [D] 72

35. The accompanying diagram shows a child'sspin toy that is constructed from two chordsintersecting in a circle. The curved edge ofthe larger shaded section is one-quarter ofthe circumference of the circle, and thecurved edge of the smaller shaded section isone-fifth of the circumference of the circle.

What is the measure of angle x?

[A] 108° [B] 72° [C] 81° [D] 40°

36. The accompanying diagram representscircular pond O with docks located at pointsA and B. From a cabin located at C, twosightings are taken that determine an angleof 30° for tangents CA and .CB

What is m CAB∠ ?

[A] 60 [B] 75 [C] 30 [D] 150


37. An architect is designing a park with anentrance represented by point C and acircular garden with center O, as shown inthe accompanying diagram. The architectplans to connect three points on thecircumference of the garden, A, B, and D, tothe park entrance, C, with walkways so thatwalkways CA and CB are tangent to thegarden, walkway DOEC is a path throughthe center of the garden,

: 3 : 2,mADB mAEB = BC = 60 meters, andEC = 43.6 meters.Find the measure of the angle betweenwalkways CA and CB.Find the diameter of the circular garden, tothe nearest meter.

38. A small fragment of something brittle, suchas pottery, is called a shard. Theaccompanying diagram represents theoutline of a shard from a small round platethat was found at an archaeological dig.

If BC is a tangent to AC at B andm ABC∠ = 45, what is the measure of AC ,the outside edge of the shard?

[A] 45° [B] 90°

[C] 225° [D] 135°

39. Point P lies outside circle O, which has adiameter of AOC . The angle formed bytangent PA and secant PBC measures 30°.Sketch the conditions given above and findthe number of degrees in the measure ofminor arc CB.

40. The accompanying diagram shows acircular machine part that has rods PT andPAR attached at points T, A, and R, whichare located on the circle;

: : 1: 3 : 5;mTA mAR RTm = RA = 12centimeters; and PA = 5 centimeters.

Find the measure of ∠P, in degrees, andfind the length of rod PT , to the nearesttenth of a centimeter.

41. In the accompanying diagram, PA istangent to circle O at A, secant PBC isdrawn, PB = 4, and BC = 12. Find PA.

42. Given circle O with diameter GOAL;secants HUG and HTAM intersect at pointH; : : 7 :3 : 2;mGM mML LTm = and chord

GU ≅ chord UT. Find the ratio ofm UGL∠ to m H∠ .


43. A machine part consists of a circular wheelwith an inscribed triangular plate, as shownin the accompanying diagram. If SE EA≅ ,SE = 10, and 140,mSE = find the length of

SA to the nearest tenth.

44. In the accompanying diagram, circle O hasradius OD, diameter BOHF , secant CBA,and chords DHG and BD; CE is tangentto circle O at D; 80;mDF = and : : 3 : 2 :1.mBA mAG mGF =Find ,mGF and m BHD∠ , m BDG∠ ,m GDE∠ , m C∠ , and m BOD∠ .

45. Which graph represents the inverse of f(x) ={(0,1),(1,4),(2,3)}?

[A]

[B]

[C]

[D]

46. Draw f x x( ) = 2 2 and f x−1( ) in theinterval 0 2≤ ≤x on the accompanying setof axes. State the coordinates of the pointsof intersection.


47. The table below, created in 1996, shows ahistory of transit fares from 1955 to 1995.On the accompanying grid, construct ascatter plot where the independent variableis years. State the exponential regressionequation with the coefficient and baserounded to the nearest thousandth. Usingthis equation, determine the prediction thatshould have been made for the year 1998, tothe nearest cent.

48. The breaking strength, y, in tons, of steelcable with diameter d, in inches, is given inthe table below.

On the accompanying grid, make a scatterplot of these data. Write the exponentialregression equation, expressing theregression coefficients to the nearest tenth.

49. Determine the value of x and y if 2 8y x=and 3 3 4y x= + .

[A] x = 2, y = 6 [B] x = 6, y = 2

[C] x = y [D] x = -2, y = -6

50. The amount A, in milligrams, of a 10-milligram dose of a drug remaining in thebody after t hours is given by the formulaA t= 10 08( . ) . Find, to the nearest tenth of

an hour, how long it takes for half of thedrug dose to be left in the body.


51. The price per person to rent a limousine fora prom varies inversely as the number ofpassengers. If five people rent thelimousine, the cost is $70 each. How manypeople are renting the limousine when thecost per couple is $87.50?

52. A rectangular prism has a length of2 2 24

4

2

2

x xx x+ −

+, a width of x x

x

2 64

+ −+

, and a

height of 8 29

2

2

x xx+−

. For all values of x for

which it is defined, express, in terms of x,the volume of the prism in simplest form.

53. If the length of a rectangular garden is

represented by x xx x

2

2

22 15+

+ − and its width is

represented by 2 62 4

xx−+

, which expression

represents the area of the garden?

[A] xx +5

[B] x xx

2 22 5

++( )

[C] x [D] x + 5

54. Working by herself, Mary requires 16minutes more than Antoine to solve amathematics problem. Working together,Mary and Antoine can solve the problem in6 minutes. If this situation is represented by

the equation 6 616

1t t+

+= , where t

represents the number of minutes Antoineworks alone to solve the problem, howmany minutes will it take Antoine to solvethe problem if he works by himself?

55. If the sine of an angle is 35

and the angle is

not in Quadrant I, what is the value of thecosine of the angle?

56. If θ is an angle in standard position and itsterminal side passes through the point

( , )12

32

on a unit circle, a possible value of

θ is

[A] 30° [B] 120° [C] 60° [D] 150°

57. If sinθ > 0 and sec ,θ < 0 in which quadrantdoes the terminal side of angle θ lie?

[A] II [B] IV [C] III [D] I

58. If the tangent of an angle is negative and itssecant is positive, in which quadrant doesthe angle terminate?

[A] I [B] II [C] IV [D] III

59. Which angle is coterminal with an angle of125°?

[A] -235° [B] 425°

[C] 235° [D] -125°

60. If θ is an obtuse angle and sin ,θ = b then itcan be concluded that

[A] tanθ > b [B] cosθ > b

[C] sin 2θ < b [D] cos2θ > b

61. Two straight roads intersect at an anglewhose measure is 125°. Which expressionis equivalent to the cosine of this angle?

[A] -cos 55° [B] cos 55°

[C] -cos 35° [D] cos 35°

62. If tan .θ = 2 7 and csc ,θ < 0 in whichquadrant does θ lie?

[A] II [B] I [C] III [D] IV

63. If x is a positive acute angle and

cos ,x = 34

what is the exact value of

sin ?x

[A] 134

[B] 45

[C] 35

[D] 35


64. An art student wants to make a stringcollage by connecting six equally spacedpoints on the circumference of a circle to itscenter with string. What would be theradian measure of the angle between twoadjacent pieces of string, in simplest form?

65. An arc of a circle that is 6 centimeters inlength intercepts a central angle of 1.5radians. Find the number of centimeters inthe radius of the circle.

66. What is the radian measure of the angleformed by the hands of a clock at 2:00 p.m.?

[A] π6

[B] π3

[C] π4

[D] π2

67. A dog has a 20-foot leash attached to thecorner where a garage and a fence meet, asshown in the accompanying diagram. Whenthe dog pulls the leash tight and walks fromthe fence to the garage, the arc the leashmakes is 55.8 feet.

What is the measure of angle θ between thegarage and the fence, in radians?

[A] 160 [B] 0.36 [C] 3.14 [D] 2.79

68. What is the number of degrees in an angle

whose radian measure is 712π ?

69. Kristine is riding in car 4 of the Ferris wheelrepresented in the accompanying diagram.The Ferris wheel is rotating in the directionindicated by the arrows. The eight cars areequally spaced around the circular wheel.Express, in radians, the measure of thesmallest angle through which she will travelto reach the bottom of the Ferris wheel.

70. The pendulum of a clock swings through anangle of 2.5 radians as its tip travels throughan arc of 50 centimeters. Find the length ofthe pendulum, in centimeters.

71. The brightness of the star MIRA over timeis given by the equation

y x = + 24

6sin π , where x represents

time and y represents brightness. What isthe period of this function, in radianmeasure?

72. A monitor displays the graph y x= 3 5sin . What will be the amplitude after a dilationof 2?

[A] 10 [B] 6 [C] 7 [D] 5

73. What is the amplitude of the function

y x=23

4sin ?

[A] 4 [B] π2

[C] 23

[D] 3π


74. The path traveled by a roller coaster ismodeled by the equation y x= +27 13 30sin .What is the maximum altitude of the rollercoaster?

[A] 57 [B] 30 [C] 27 [D] 13

75. A certain radio wave travels in a pathrepresented by the equation y x= 5 2sin .What is the period of this wave?

[A] 5 [B] 2 [C] π [D] 2π

76. A sound wave is modeled by the curvey x= 3 4sin . What is the period of thiscurve?

[A] π2

[B] 3 [C] π [D] 4

77. What is the period of the graph of the

equation y x= 2 13

sin ?

[A] 23π [B] 3

2π [C] 6π [D] 2π

78. A radio transmitter sends a radio wave fromthe top of a 50-foot tower. The wave isrepresented by the accompanying graph.

What is the equation of this radio wave?

[A] y x= sin [B] y x= 15. sin

[C] y x= sin .15 [D] y x= 2sin

79. The accompanying diagram shows a sectionof a sound wave as displayed on anoscilloscope.

Which equation could represent this graph?

[A] y x=12 2

sin π [B] y x=12

2cos

[C] y x= 2

2sin [D] y x

= 22

cos

80. If sin ,A =45

tan ,B =5

12 and angles A and

B are in Quadrant I, what is the value ofsin( ) ?A B+

[A] 3365

[B] −6365

[C] −3365

[D] 6365

81. If A is a positive acute angle and

sin ,A =5

3 what is cos 2A?

[A] −19

[B] 19

[C] 13

[D] −13

82. If sin ,x = 1213

cos ,y = 35

and x and y are

acute angles, the value of cos( )x y− is

[A] −3365

[B] −1465

[C] 2165

[D] 6365


83. If x is an acute angle and sin ,x = 1213

then

cos2x equals

[A] 25169

[B] 119169

[C] −25

169[D] −

119169

84. If A and B are positive acute angles,

sin ,A =5

13 and cos ,B =

45

what is the

value of sin( ) ?A B+

[A] 5665

[B] 6365

[C] 3365

[D] −1665

85. The expression cos 40° cos 10° + sin 40° sin10° is equivalent to

[A] sin 50° [B] cos 30°

[C] cos 50° [D] sin 30°

86. If θ is a positive acute angle and

sin ,2 32

θ = then (cos sin )θ θ+ 2 equals

[A] 30° [B] 60° [C] 1 32

+ [D] 1

87. What value of x in the interval 0 180°≤ ≤ °xsatisfies the equation 3 1 0tan ?x + =

[A] -30° [B] 150° [C] 30° [D] 60°

88. Two straight roads, Elm Street and PineStreet, intersect creating a 40° angle, asshown in the accompanying diagram.John's house (J) is on Elm Street and is 3.2miles from the point of intersection. Mary'shouse (M) is on Pine Street and is 5.6 milesfrom the intersection. Find, to the nearesttenth of a mile, the direct distance betweenthe two houses.

89. Kieran is traveling from city A to city B. Asthe accompanying map indicates, Kierancould drive directly from A to B alongCounty Route 21 at an average speed of 55miles per hour or travel on the interstates,45 miles along I-85 and 20 miles along I-64.The two interstates intersect at an angle of150° at C and have a speed limit of 65 milesper hour. How much time will Kieran saveby traveling along the interstates at anaverage speed of 65 miles per hour?

90. To measure the distance through a mountainfor a proposed tunnel, surveyors chosepoints A and B at each end of the proposedtunnel and a point C near the mountain.They determined that AC = 3,800 meters,BC = 2,900 meters, and m ACB∠ = 110. Draw a diagram to illustrate this situationand find the length of the tunnel, to thenearest meter.

91. A surveyor is mapping a triangular plot ofland. He measures two of the sides and theangle formed by these two sides and findsthat the lengths are 400 yards and 200 yardsand the included angle is 50°.What is the measure of the third side of theplot of land, to the nearest yard?What is the area of this plot of land, to thenearest square yard?

92. A farmer has determined that a crop ofstrawberries yields a yearly profit of $1.50per square yard. If strawberries are plantedon a triangular piece of land whose sides are50 yards, 75 yards, and 100 yards, howmuch profit, to the nearest hundred dollars,would the farmer expect to make from thispiece of land during the next harvest?


93. The Vietnam Veterans Memorial inWashington, D.C., is made up of two walls,each 246.75 feet long, that meet at an angleof 125.2°. Find, to the nearest foot, thedistance between the ends of the walls thatdo not meet.

94. A triangular plot of land has sides thatmeasure 5 meters, 7 meters, and 10 meters.What is the area of this plot of land, to thenearest tenth of a square meter?

95. Carmen and Jamal are standing 5,280 feetapart on a straight, horizontal road. Theyobserve a hot-air balloon between themdirectly above the road. The angle ofelevation from Carmen is 60° and fromJamal is 75°. Draw a diagram to illustratethis situation and find the height of theballoon to the nearest foot.

96. While sailing a boat offshore, Donna sees alighthouse and calculates that the angle ofelevation to the top of the lighthouse is 3°,as shown in the accompanying diagram.When she sails her boat 700 feet closer tothe lighthouse, she finds that the angle ofelevation is now 5°. How tall, to thenearest tenth of a foot, is the lighthouse?

97. A sign 46 feet high is placed on top of anoffice building. From a point on thesidewalk level with the base of the building,the angle of elevation to the top of the signand the angle of elevation to the bottom ofthe sign are 40° and 32°, respectively.Sketch a diagram to represent the building,the sign, and the two angles, and find theheight of the building to the nearest foot.

98. In ΔABC, m A∠ = 33, a = 12, and b = 15.What is m B∠ to the nearest degree?

[A] 48 [B] 41 [C] 43 [D] 44

99. In the accompanying diagram of ΔABC,m A∠ = 30, m A∠ = 30, and AC = 13.

What is the length of side AB to the nearesttenth?

[A] 10.1 [B] 12.0 [C] 11.5 [D] 6.6

100. In ΔABC, a = 19, c = 10, and m A = 111.∠Which statement can be used to find thevalue of ∠C ?

[A] sin C =1019

[B] sin sinC =°10 69

19

[C] sin sinC =°10 21

19

[D] sin sinC =°19 69

10

101. As shown in the accompanying diagram,two tracking stations, A and B, are on aneast-west line 110 miles apart. A forest fireis located at F, on a bearing 42° northeast ofstation A and 15° northeast of station B.How far, to the nearest mile, is the fire fromstation A?


102. The accompanying diagram shows the plansfor a cell-phone tower that is to be built neara busy highway. Find the height of thetower, to the nearest foot.

103. In ΔABC, m A∠ = 53, m B∠ = 14, anda = 10. Find b to the nearest integer.

104. Which graph represents data used in a linearregression that produces a correlationcoefficient closest to −1?

[A] [B]

[C] [D]

105. On a standardized test, a score of 86 fallsexactly 1.5 standard deviations below themean. If the standard deviation for the testis 2, what is the mean score for this test?

[A] 84.5 [B] 89 [C] 87.5 [D] 84

106. An electronics company produces aheadphone set that can be adjusted toaccommodate different-sized heads.Research into the distance between the topof people's heads and the top of their earsproduced the following data, in inches:4.5, 4.8, 6.2, 5.5, 5.6, 5.4, 5.8, 6.0, 5.8, 6.2,4.6, 5.0, 5.4, 5.8The company decides to design theirheadphones to accommodate three standarddeviations from the mean. Find, to thenearest tenth, the mean, the standarddeviation, and the range of distances thatmust be accommodated.

107. Jean's scores on five mathematics tests were98, 97, 99, 98, and 96. Her scores on fiveEnglish tests were 78, 84, 95, 72, and 79.Which statement is true about the standarddeviations for the scores?

[A] The standard deviation for the Englishscores is greater than the standarddeviation for the math scores.

[B] The standard deviation for the mathscores is greater than the standarddeviation for the English scores.

[C] More information is needed todetermine the relationship between thestandard deviations.

[D] The standard deviations for both sets ofscores are equal.

108. In a certain school district, the ages of allnew teachers hired during the last 5 yearsare normally distributed. Within this curve,95.4% of the ages, centered about the mean,are between 24.6 and 37.4 years. Find themean age and the standard deviation of thedata.

109. Professor Bartrich has 184 students in hermathematics class. The scores on the finalexamination are normally distributed andhave a mean of 72.3 and a standarddeviation of 8.9. How many students in theclass can be expected to receive a scorebetween 82 and 90?

110. The amount of ketchup dispensed from amachine at Hamburger Palace is normallydistributed with a mean of 0.9 ounce and astandard deviation of 0.1 ounce. If themachine is used 500 times, approximatelyhow many times will it be expected todispense 1 or more ounces of ketchup?

[A] 80 [B] 5 [C] 100 [D] 16


111. The amount of juice dispensed from amachine is normally distributed with a meanof 10.50 ounces and a standard deviation of0.75 ounce. Which interval represents theamount of juice dispensed about 68.2% ofthe time?

[A] 9.75-10.50 [B] 9.00-12.00

[C] 10.50-11.25 [D] 9.75-11.25

112. The mean of a normally distributed set ofdata is 56, and the standard deviation is 5.In which interval do approximately 95.4%of all cases lie?

[A] 56-71 [B] 46-66

[C] 46-56 [D] 51-61

113. The national mean for verbal scores on anexam was 428 and the standard deviationwas 113. Approximately what percent ofthose taking this test had verbal scoresbetween 315 and 541?

[A] 26.4% [B] 38.2%

[C] 52.8% [D] 68.2%

Math B Regents Exam Questions by Standard - Uncertainty PAGE 1www.jmap.org

1. What is the mean of the data in theaccompanying table?

[A] 15 [B] 16 [C] 14.5 [D] 11

2. The 1999 win-loss statistics for theAmerican League East baseball teams on aparticular date is shown in theaccompanying chart.

Find the mean for the number of wins, W ,and the mean for the number of losses, L,and determine if the point (W , L ) is a pointon the line of best fit. Justify your answer.

3. A real estate agent plans to compare theprice of a cottage, y, in a town on theseashore to the number of blocks, x, thecottage is from the beach. Theaccompanying table shows a random sampleof sales and location data.Write a linear regression equation thatrelates the price of a cottage to its distancefrom the beach.Use the equation to predict the price of acottage, to the nearest dollar, located threeblocks from the beach.


4. The table below shows the results of anexperiment that relates the height at which aball is dropped, x, to the height of its firstbounce, y.

Find x , the mean of the drop heights.Find y, the mean of the bounce heights.Find the linear regression equation that bestfits the data.Show that ( , )x y is a point on the line ofregression. [The use of the grid is optional.]

5. Two different tests were designed tomeasure understanding of a topic. The twotests were given to ten students with thefollowing results:

Construct a scatter plot for these scores, andthen write an equation for the line of best fit(round slope and intercept to the nearesthundredth).

Find the correlation coefficient.Predict the score, to the nearest integer, ontest y for a student who scored 87 on test x.


6. The accompanying table illustrates thenumber of movie theaters showing apopular film and the film's weekly grossearnings, in millions of dollars.

Write the linear regression equation for thisset of data, rounding values to five decimalplaces.Using this linear regression equation, findthe approximate gross earnings, in millionsof dollars, generated by 610 theaters.Round your answer to two decimal places.Find the minimum number of theaters thatwould generate at least 7.65 million dollarsin gross earnings in one week.

7. Since 1990, fireworks usage nationwide hasgrown, as shown in the accompanying table,where t represents the number of years since1990, and p represents the fireworks usageper year, in millions of pounds.

Find the equation of the linear regressionmodel for this set of data, where t is theindependent variable. Round values to fourdecimal places.Using this equation, determine in what yearfireworks usage would have reached 99million pounds.Based on this linear model, how manymillions of pounds of fireworks would beused in the year 2008? Round your answerto the nearest tenth.

8. A factory is producing and stockpilingmetal sheets to be shipped to an automobilemanufacturing plant. The factory ships onlywhen there is a minimum of 2,050 sheets instock. The accompanying table shows theday, x, and the number of sheets in stock,f(x).

Write the linear regression equation for thisset of data, rounding the coefficients to fourdecimal places.Use this equation to determine the day thesheets will be shipped.

9. A cellular telephone company has twoplans. Plan A charges $11 a month and$0.21 per minute. Plan B charges $20 amonth and $0.10 per minute. After howmuch time, to the nearest minute, will thecost of plan A be equal to the cost of planB?

[A] 1 hr 22 m [B] 1 hr 36 min

[C] 81 hr 48 min [D] 81 hr 8 min

10. What is the domain of the function

f x xx

( ) ?=−

29

2

2

[A] all real numbers except 0

[B] all real numbers

[C] all real numbers except 3 and -3

[D] all real numbers except 3


11. What is the domain of the function

f x xx

( ) ?=−

349

2

2

[A] }0,numbersreal|{ ≠∈ xxx

[B] }7,numbersreal|{ ±≠∈ xxx

[C] }numbersreal|{ ∈xx

[D] }7,numbersreal|{ ≠∈ xxx

12. If f xx

( ) ,=−

12 4

the domain of f x( ) is

[A] x > 2 [B] x < 2[C] x ≥ 2 [D] x = 2

13. The roots of the equation x x2 3 2 0− − = are

[A] real, rational, and unequal

[B] imaginary

[C] real, rational, and equal

[D] real, irrational, and unequal

14. What is the last term in the expansion of( )x y+ 2 5 ?

[A] 2 5y [B] 10 5y

[C] y5 [D] 32 5y

15. What is the middle term in the expansion of( ) ?x y+ 4

[A] 4 2 2x y [B] 2 2 2x y

[C] x y2 2 [D] 6 2 2x y

16. What is the fourth term in the expansion of( ) ?y −1 7

[A] − 35 3y [B] 35 4y

[C] − 35 4y [D] 35 3y

17. What is the third term in the expansion of(cos ) ?x + 3 5

[A] 270 2cos x [B] 90 2cos x

[C] 60 3cos x [D] 90 3cos x

18. At a certain intersection, the light foreastbound traffic is red for 15 seconds,yellow for 5 seconds, and green for 30seconds. Find, to the nearest tenth, theprobability that out of the next eighteastbound cars that arrive randomly at thelight, exactly three will be stopped by a redlight.

19. Team A and team B are playing in a league.They will play each other five times. If the

probability that team A wins a game is 13

,

what is the probability that team A will winat least three of the five games?

20. After studying a couple's family history, adoctor determines that the probability of anychild born to this couple having a gene fordisease X is 1 out of 4. If the couple hasthree children, what is the probability thatexactly two of the children have the gene fordisease X?

21. Which fraction represents the probability ofobtaining exactly eight heads in ten tossesof a fair coin?

[A] 1801 024,

[B] 641 024,

[C] 901 024,

[D] 451 024,

22. The probability that Kyla will score above a

90 on a mathematics test is 45

. What is the

probability that she will score above a 90 onthree of the four tests this quarter?

[A] [B]

[C] [D]

23. On any given day, the probability that theentire Watson family eats dinner together is25

. Find the probability that, during any 7-

day period, the Watsons eat dinner togetherat least six times.


24. When Joe bowls, he can get a strike (knockdown all the pins) 60% of the time. Howmany times more likely is it for Joe to bowlat least three strikes out of four tries as it isfor him to bowl zero strikes out of fourtries? Round your answer to the nearestwhole number.

25. A board game has a spinner on a circle thathas five equal sectors, numbered 1, 2, 3, 4,and 5, respectively. If a player has fourspins, find the probability that the playerspins an even number no more than twotimes on those four spins.

26. The Hiking Club plans to go camping in aState park where the probability of rain onany given day is 0.7. Which expression canbe used to find the probability that it willrain on exactly three of the seven days theyare there?

[A] 7 33 40 7 0 3C ( . ) ( . )

[B] 4 33 40 7 0 7C ( . ) ( . )

[C] 7 33 40 3 0 7C ( . ) ( . ) [D] 4 3

4 30 4 0 3C ( . ) ( . )

27. Tim Parker, a star baseball player, hits onehome run for every ten times he is at bat. IfParker goes to bat five times duringtonight's game, what is the probability thathe will hit at least four home runs?

28. If the probability that it will rain on anygiven day this week is 60%, find theprobability it will rain exactly 3 out of 7days this week.

29. The probability that a planted watermelon

seed will sprout is 34

. If Peyton plants

seven seeds from a slice of watermelon,find, to the nearest ten thousandth, theprobability that at least five will sprout.

30. The Coolidge family's favorite televisionchannels are 3, 6, 7, 10, 11, and 13. If theCoolidge family selects a favorite channel atrandom to view each night, what is theprobability that they choose exactly threeeven-numbered channels in five nights?Express your answer as a fraction or as adecimal rounded to four decimal places.

31. During a recent survey, students atFranconia College were asked if they drinkcoffee in the morning. The results showedthat two-thirds of the students drink coffeein the morning and the remainder do not.What is the probability that of six studentsselected at random, exactly two of themdrink coffee in the morning? Express youranswer as a fraction or as a decimal roundedto four decimal places.

32. Ginger and Mary Anne are planning avacation trip to the island of Capri, wherethe probability of rain on any day is 0.3.What is the probability that during their fivedays on the island, they have no rain onexactly three of the five days?

33. On mornings when school is in session inJanuary, Sara notices that her school bus islate one-third of the time. What is theprobability that during a 5-day school weekin January her bus will be late at least threetimes?

34. A used car was purchased in July 1999 for$11,900. If the car depreciates 13% of itsvalue each year, what is the value of the car,to the nearest hundred dollars, in July2002?

35. A box containing 1,000 coins is shaken, andthe coins are emptied onto a table. Only thecoins that land heads up are returned to thebox, and then the process is repeated. Theaccompanying table shows the number oftrials and the number of coins returned tothe box after each trial.

Write an exponential regression equation,rounding the calculated values to thenearest ten-thousandth.Use the equation to predict how many coinswould be returned to the box after the eighthtrial.


36. The accompanying table shows the averagesalary of baseball players since 1984. Usingthe data in the table, create a scatter plot onthe grid and state the exponential regressionequation with the coefficient and baserounded to the nearest hundredth.Using your written regression equation,estimate the salary of a baseball player inthe year 2005, to the nearest thousanddollars.

37. Jean invested $380 in stocks. Over the next5 years, the value of her investment grew, asshown in the accompanying table.

Write the exponential regression equationfor this set of data, rounding all values totwo decimal places.Using this equation, find the value of herstock, to the nearest dollar, 10 years afterher initial purchase.

38. If sin ,x = 45

where 0° < x < 90°, find the

value of cos (x + 180°).

39. For a carnival game, John is painting twocircles, V and M, on a square dartboard.a On the accompanying grid, draw and labelcircle V, represented by the equationx y2 2 25+ = , and circle M, represented bythe equation ( ) ( )x y− + + =8 6 42 2 .

b A point, (x,y), is randomly selected suchthat − ≤ ≤10 10x and − ≤ ≤10 10y . Whatis the probability that point (x,y) lies outsideboth circle V and circle M?


40. Two social studies classes took the samecurrent events examination that was scoredon the basis of 100 points. Mr. Wong'sclass had a median score of 78 and a rangeof 4 points, while Ms. Rizzo's class had amedian score of 78 and a range of 22 points.Explain how these classes could have thesame median score while having verydifferent ranges.

41. The accompanying graph shows the heartrate, in beats per minute, of a jogger duringa 4-minute interval.

What is the range of the jogger's heart rateduring this interval?

[A] 1-4 [B] 60-110

[C] 0-110 [D] 0-4

42. The effect of pH on the action of a certainenzyme is shown on the accompanyinggraph.

What is the domain of this function?

[A] 4 13≤ ≤x [B] x ≥ 0[C] 4 13≤ ≤y [D] y ≥ 0

43. Which scatter diagram shows the strongestpositive correlation?

[A] [B]

[C] [D]

44. The relationship of a woman's shoe size andlength of a woman's foot, in inches, is givenin the accompanying table.

The linear correlation coefficient for thisrelationship is

[A] 1 [B] -1 [C] 0.5 [D] 0

45. A linear regression equation of best fitbetween a student's attendance and thedegree of success in school is h = 0.5x +68.5. The correlation coefficient, r, forthese data would be

[A] 0 < r < 1 [B] r = 0

[C] -1 < r < 0 [D] r = -1

46. On a nationwide examination, the AdamsSchool had a mean score of 875 and astandard deviation of 12. The BoswellSchool had a mean score of 855 and astandard deviation of 20. In which schoolwas there greater consistency in the scores?Explain how you arrived at your answer.

47. In a New York City high school, a surveyrevealed the mean amount of cola consumedeach week was 12 bottles and the standarddeviation was 2.8 bottles. Assuming thesurvey represents a normal distribution, howmany bottles of cola per week willapproximately 68.2% of the students drink?

[A] 6.4 to 12 [B] 6.4 to 17.6

[C] 9.2 to 14.8 [D] 12 to 20.4


48. Battery lifetime is normally distributed forlarge samples. The mean lifetime is 500days and the standard deviation is 61 days.Approximately what percent of batterieshave lifetimes longer than 561 days?

[A] 68% [B] 34%

[C] 84% [D] 16%

49. On a standardized test, the distribution ofscores is normal, the mean of the scores is75, and the standard deviation is 5.8. If astudent scored 83, the student's score ranks

[A] below the 75th percentile

[B] between the 75th percentile and the84th percentile

[C] between the 84th percentile and the97th percentile

[D] above the 97th percentile

50. The mean score on a normally distributedexam is 42 with a standard deviation of12.1. Which score would be expected tooccur less than 5% of the time?

[A] 32 [B] 25 [C] 67 [D] 60

51. Mrs. Ramírez is a real estate broker. Lastmonth, the sale prices of homes in her areaapproximated a normal distribution with amean of $150,000 and a standard deviationof $25,000.A house had a sale price of $175,000. Whatis the percentile rank of its sale price, to thenearest whole number? Explain what thatpercentile means.Mrs. Ramírez told a customer that most ofthe houses sold last month had selling pricesbetween $125,000 and $175,000. Explainwhy she is correct.

52. A set of normally distributed student testscores has a mean of 80 and a standarddeviation of 4. Determine the probabilitythat a randomly selected score will bebetween 74 and 82.

53. The amount of time that a teenager playsvideo games in any given week is normallydistributed. If a teenager plays video gamesan average of 15 hours per week, with astandard deviation of 3 hours, what is theprobability of a teenager playing videogames between 15 and 18 hours a week?

54. A shoe manufacturer collected dataregarding men's shoe sizes and found thatthe distribution of sizes exactly fits thenormal curve. If the mean shoe size is 11and the standard deviation is 1.5, find:a the probability that a man's shoe size isgreater than or equal to 11b the probability that a man's shoe size isgreater than or equal to 12.5

c P sizeP size( . )

( )≥≥12 5

8

55. What is the value of ( )2 1 1

1

3m m

m+∑ −

=?

[A] 55 [B] 245 [C] 15 [D] 57

56. What is the value of ( ) ?mm

2

2

51−∑

=

[A] 53 [B] 58 [C] 50 [D] 54

57. Evaluate: 2 2 11

5( )n

n−∑

=

58. Evaluate: ( )n nn

2

1

5+∑

=

59. The projected total annual profits, in dollars,for the Nutyme Clothing Company from2002 to 2004 can be approximated by the

model ( , ),13 567 2940

2n

n+∑

= where n is the

year and n = 0 represents 2002. Use thismodel to find the company's projected totalannual profits, in dollars, for the period2002 to 2004.


60. A ball is dropped from a height of 8 feet andallowed to bounce. Each time the ballbounces, it bounces back to half its previousheight. The vertical distance the balltravels, d, is given by the formula

dk

n k

= + ∑=

8 16 121

( ) , where n is the number of

bounces. Based on this formula, what is thetotal vertical distance that the ball hastraveled after four bounces?

[A] 22.0 ft [B] 8.9 ft

[C] 23.0 ft [D] 15.0 ft

61. If n rC represents the number ofcombinations of n items taken r at a time,

what is the value of 4r=1

3Cr∑ ?

[A] 24 [B] 14 [C] 4 [D] 6

62. The value of 52

4Cr

r=∑ is

[A] 5 [B] 45 [C] 10 [D] 25

63. Evaluate: ( cos )3 10

3k

kπ +∑

=

64. Evaluate: ( )( )!−−

∑−

=

12 1

1

1

2 k

k k

65. What is the value of ( ( ) )20

3−∑

=b i

b?

[A] 8-6i [B] 8-5i [C] 2-5i [D] 2-6i

66. What is the value of ( ) ?− +∑=

2 1001

5n

n

[A] 130 [B] 470 [C] 530 [D] 70

67. Jonathan's teacher required him to express

the sum 23

34

45

56

67

+ + + + using sigma

notation. Jonathan proposed four possibleanswers. Which of these four answers is notcorrect?

[A] kkk +

∑= 11

5[B] k

kk

++

∑=

121

5

[C] kkk +

∑= 12

6[D] k

kk

−∑=

13

7

Math B Regents Exam Questions by Standard - Patterns/Functions PAGE 1www.jmap.org

1. What is the solution set of the equationx x x2 2 3 6− = − ?

[A] {±3} [B] {2,3}

[C] {2} [D] {2,±3}

2. The availability of leaded gasoline in NewYork State is decreasing, as shown in theaccompanying table.

Determine a linear relationship for x (years)versus y (gallons available), based on thedata given. The data should be enteredusing the year and gallons available (inthousands), such as (1984,150).If this relationship continues, determine thenumber of gallons of leaded gasolineavailable in New York State in the year2005.If this relationship continues, during whatyear will leaded gasoline first becomeunavailable in New York State?

3. Solve for x: x− =3 2764

4. The solution set of the equation x x+ =6is

[A] {-2} [B] {-2,3} [C] {3} [D] { }

5. What is the solution set of the equationx x= −2 2 3 ?

[A] { } [B] {2,6} [C] {2} [D] {6}

6. Solve algebraically: x x+ + =5 1

7. What is the solution set of the equation9 10x x+ =

[A] {9} [B] {10}

[C] {-1} [D] {10, -1}

8. If 2 1 2 5x − + = , then x is equal to

[A] 4 [B] 2 [C] 5 [D] 1

9. What is the solution of the equation2 3 3 6x − − = ?

[A] 39 [B] 6 [C] 42 [D] 3

10. Solve for all values of q that satisfy theequation 3 7 3q q+ = + .

11. Solve algebraically for x: 3 1 1x x+ + =

12. A wrecking ball suspended from a chain is atype of pendulum. The relationshipbetween the rate of speed of the ball, R, themass of the ball, m, the length of the chain,

L, and the force, F, is R mLF

= 2π .

Determine the force, F, to the nearesthundredth, when L = 12, m = 50, and R =0.6.

13. The lateral surface area of a right circularcone, s, is represented by the equations r r h= +π 2 2 , where r is the radius of thecircular base and h is the height of the cone.If the lateral surface area of a large funnel is236.64 square centimeters and its radius is4.75 centimeters, find its height, to thenearest hundredth of a centimeter.


14. The equation V C= +20 273 relates speedof sound, V, in meters per second, to airtemperature, C, in degrees Celsius. What isthe temperature, in degrees Celsius, whenthe speed of sound is 320 meters persecond? [The use of the accompanying gridis optional.]

15. The number of people, y, involved inrecycling in a community is modeled by thefunction y x= +90 3 400 , where x is thenumber of months the recycling plant hasbeen open.Construct a table of values, sketch thefunction on the grid, and find the number ofpeople involved in recycling exactly 3months after the plant opened.After how many months will 940 people beinvolved in recycling?

16. What is the domain of h x x x( ) ?= − −2 4 5

[A] { }x x x≥ ≤ −1 5or

[B] { }x x− ≤ ≤5 1 [C] { }x x− ≤ ≤1 5

[D] { }x x x≥ ≤ −5 1or

17. Which transformation of the graph ofy x= 2 would result in the graph ofy x= +2 2?

[A] 2D [B] 2=yr [C] 90,0R [D] 2,0T

18. Which transformation is not an isometry?

[A] D2 [B] ry x= [C] R0 90, ° [D] T3 6,

19. Which transformation is not an isometry?

[A] translation [B] line reflection

[C] dilation [D] rotation

20. Which transformation is an oppositeisometry?

[A] rotation of 90° [B] translation

[C] line reflection [D] dilation

21. Which transformation does not preserveorientation?

[A] dilation

[B] reflection in the y-axis

[C] rotation [D] translation


22. In the accompanying diagram of squareABCD, F is the midpoint of AB, G is themidpoint of BC, H is the midpoint of CD,and E is the midpoint of DA.

Find the image of ΔEOA after it is reflectedin line .Is this isometry direct or opposite? Explainyour answer.

23. The graph below represents f(x).

Which graph best represents f(-x)?

[A] [B]

[C] [D]

24. Two parabolic arches are to be built. Theequation of the first arch can be expressedas y x= − +2 9, with a range of 0 9≤ ≤y ,and the second arch is created by thetransformation T7 0, . On the accompanyingset of axes, graph the equations of the twoarches. Graph the line of symmetry formedby the parabola and its transformation andlabel it with the proper equation.

25. Graph and label the following equations, aand b, on the accompanying set ofcoordinate axes.

a y xb y x

:: ( )

=

= − − +

2

24 3Describe the composition of transformationsperformed on a to get b.


26. On the accompanying grid, graph and labelAB, where A is (0,5) and B is (2,0). Under

the transformation r r ABx axis y axis− − ( ), Amaps to ′′A and B maps to ′′B . Graph andlabel ′′ ′′A B . What single transformationwould map AB to ′′ ′′A B ?

27. In which quadrant would the image of point(5,-3) fall after a dilation using a factor of -3?

[A] I [B] III [C] II [D] IV

28. In the accompanying graph, the shadedregion represents set A of all points (x,y)such that x y2 2 1+ ≤ . The transformation Tmaps point (x, y) to point (2x, 4y).

Which graph shows the mapping of set A bythe transformation T?

[A]

[B]

[C] [D]

29. Chad had a garden that was in the shape of arectangle. Its length was twice its width.He decided to make a new garden that was 2feet longer and 2 feet wider than his firstgarden. If x represents the original width ofthe garden, which expression represents thedifference between the area of his newgarden and the area of the original garden?

[A] 2 2x [B] 8

[C] 6 4x + [D] x x2 3 2+ +


30. A small, open-top packing box, similar to ashoebox without a lid, is three times as longas it is wide, and half as high as it is long.Each square inch of the bottom of the boxcosts $0.008 to produce, while each squareinch of any side costs $0.003 to produce.Write a function for the cost of the boxdescribed above.Using this function, determine thedimensions of a box that would cost $0.69to produce.

31. A rectangular piece of cardboard is to beformed into an uncovered box. The piece ofcardboard is 2 centimeters longer than it iswide. A square that measures 3 centimeterson a side is cut from each corner. When thesides are turned up to form the box, itsvolume is 765 cubic centimeters. Find thedimensions, in centimeters, of the originalpiece of cardboard.

32. Given: A(1,6), B(7,9), C(13,6), and D(3,1)Prove: ABCD is a trapezoid. [The use of theaccompanying grid is optional.]

33. Which relation is a function?

[A] x = 4 [B] x y= +2 1

[C] y x= sin [D] x y2 2 16+ =

34. Which equation represents a function?

[A] 4 36 92 2y x= −

[B] y x x= − −2 3 4

[C] x y2 2 4+ = [D] x y x= − +2 6 8

35. Which set of ordered pairs is not a function?

[A] {(1,2), (3,4), (4,5), (5,6)}

[B] {(3,1), (2,1), (1,2), (3,2)}

[C] {(4,1), (5,1), (6,1), (7,1)}

[D] {(0,0), (1,1), (2,2), (3,3)}

36. Which relation is a function?

[A] x y2 2 7− = [B] x y2 2 7+ =

[C] xy = 7 [D] x = 7

37. Which equation does not represent afunction?

[A] y x= [B] y x x= +2 5

[C] y = 4 [D] x = π

38. Which graph is not a function?

[A] [B]

[C] [D]


39. Which graph does not represent a functionof x?

[A]

[B]

[C]

[D]

40. Which diagram represents a relation inwhich each member of the domaincorresponds to only one member of itsrange?

[A]

[B]

[C] [D]

41. On the accompanying diagram, draw amapping of a relation from set A to set Bthat is not a function. Explain why therelationship you drew is not a function.

42. Which diagram represents a one-to-onefunction?

[A]

[B]

[C]

[D]


43. Each graph below represents a possiblerelationship between temperature andpressure. Which graph does not represent afunction?

[A]

[B]

[C]

[D]

44. If f ( )x x= 5 2 and g( ) ,x x= 2 what is thevalue of (f g)(8)?

[A] 16 [B] 80

[C] 8 10 [D] 1 280,

45. If f ( )x x=23 and g( ) ,x x=

−8

12 find

( )( )f g x and (f g)(27).

46. If f ( )x x= −2 1 and g( ) ,x x= −2 1determine the value of (f g)(3).

47. If f ( )x x= − +2 7 and g( ) ,x x= −2 2 thenf(g(3)) is equal to

[A] -3 [B] 7 [C] -1 [D] -7

48. If f and g are two functions defined byf ( )x x= +3 5 and g( ) ,x x= +2 1 theng(f( ))x is

[A] x x2 3 6+ + [B] 9 30 262x x+ +

[C] 3 82x + [D] 9 262x +

49. If f ( )xx

=+2

3 and g( ) ,x

x=

1 then

( )g f)(x is equal to

[A] xx+ 3

2[B] x + 3

2

[C] 21 3

xx+

[D] 1 32+ x

x

50. If f ( )x x= +1 and g( ) ,x x= −2 1 theexpression ( )g f)(x equals 0 when x isequal to

[A] -2, only [B] 1 and -1

[C] 0 and -2 [D] 0, only

51. The accompanying graph is a sketch of thefunction y x= f ( ) over the interval0 7≤ ≤x .

What is the value of ( )( ) ?f f 6

[A] -2 [B] 0 [C] 2 [D] 1


52. A certain drug raises a patient's heart rate,h( ),x in beats per minute, according to thefunction h( ) . ,x x= +70 0 2 where x is thebloodstream drug level, in milligrams. Thelevel of the drug in the patient's bloodstreamis a function of time, t, in hours, accordingto the formula g( ) ( . ) .t t= 300 08 Find thevalue of h(g( )),4 the patient's heart rate inbeats per minute, to the nearest wholenumber.

53. The cost (C) of selling x calculators in astore is modeled by the equation

Cx

= +3 200 000 60 000, , , . The store profit

(P) for these sales is modeled by theequation P x= 500 . What is the minimumnumber of calculators that have to be soldfor profit to be greater than cost?

54. The profit a coat manufacturer makes eachday is modeled by the equationP x x x( ) = − + −2 120 2000 , where P is theprofit and x is the price for each coat sold.For what values of x does the companymake a profit? [The use of theaccompanying grid is optional.]

55. What is the solution set of the inequality3 2 4− ≥x ?

[A] { | }x x x≤ ≥72

12

or

[B] { | }x x x≤ − ≥12

72

or

[C] { | }x x72

12

≤ ≤ −

[D] { | }x x− ≤ ≤12

72

56. What is the solution of the inequalityx + ≤3 5?

[A] x or x≤ − ≥2 8

[B] x or x≤ − ≥8 2

[C] − ≤ ≤2 8x [D] − ≤ ≤8 2x

57. The solution of 2 3 5x − < is

[A] x > -1 [B] -1 < x < 4

[C] x < -1 or x > 4 [D] x < 4

58. What is the solution of the inequalityy + >8 3?

[A] -5 < y < 11 [B] -11 < y < -5

[C] y > -5 or y < -11 [D] y > -5

59. Which graph represents the solution set of2 1 7x − < ?

[A]

[B]

[C]

[D]


60. Which graph represents the solution set forthe expression 2 3 7x + > ?

[A]

[B]

[C]

[D]

61. The solution set of which inequality isrepresented by the accompanying graph?

[A] 2 7− < −x [B] x − >2 7

[C] x − <2 7 [D] 2 7− > −x

62. The inequality 15 24 30. C − ≤ representsthe range of monthly average temperatures,C, in degrees Celsius, for Toledo, Ohio.Solve for C.

63. A depth finder shows that the water in acertain place is 620 feet deep. Thedifference between d, the actual depth of thewater, and the reading is d − 620 and mustbe less than or equal to 0.05d. Find theminimum and maximum values of d, to thenearest tenth of a foot.

64. The heights, h, of the students in the chorusat Central Middle School satisfy the

inequality h −≤

57 52

325. . , when h is

measured in inches. Determine the intervalin which these heights lie and express youranswer to the nearest tenth of a foot. [Onlyan algebraic solution can receive fullcredit.]

65. Which equation is represented by theaccompanying graph?

[A] y x= − +3 1 [B] y x= − +( )3 12

[C] y x= − 3 [D] y x= + −3 1

66. The graph below represents f x( ) .

Which graph best represents f x( ) ?

[A] [B]

[C] [D]

67. Given the function y f x= ( ), such that theentire graph of the function lies above the x-axis. Explain why the equation f x( ) = 0has no real solutions.


68. A rock is thrown vertically from the groundwith a velocity of 24 meters per second, andit reaches a height of 2 24 4 9 2+ −t t. after tseconds. How many seconds after the rockis thrown will it reach maximum height, andwhat is the maximum height the rock willreach, in meters? How many seconds afterthe rock is thrown will it hit the ground?Round your answers to the nearesthundredth. [Only an algebraic or graphicsolution will be accepted.]

69. If a function is defined by the equationy x= +3 2 , which equation defines theinverse of this function?

[A] x y= +13

12

[B] y x= − −3 2

[C] y x= −13

23

[D] y x= +13

12

70. A function is defined by the equationy x= −5 5. Which equation defines theinverse of this function?

[A] y x= +5 5 [B] xy

=−1

5 5

[C] yx

=−1

5 5[D] x y= −5 5

71. A function is defined by the equation

y x= −12

32

. Which equation defines the

inverse of this function?

[A] y x= −2 3 [B] y x= +2 3

[C] y x= +2 32

[D] y x= −2 32

72. Given: f x x( ) = 2 and g x x( ) = 2a The inverse of g is a function, but theinverse of f is not a function. Explain whythis statement is true.b Find ))3((1 fg − to the nearest tenth.

73. The accompanying diagram shows thegraph of the line whose equation is

y x= − +13

2.

On the same set of axes, sketch the graph ofthe inverse of this function.State the coordinates of a point on theinverse function.

74. What is the inverse of the functiony x= log ?4

[A] y x4 = [B] 4 x y=

[C] x y4 = [D] 4 y x=

75. The inverse of a function is a logarithmicfunction in the form y xb= log . Whichequation represents the original function?

[A] by x= [B] y bx=

[C] y bx= [D] x b y=


76. The height of a projectile is modeled by theequation y x x= − + +2 38 102 , where x istime, in seconds, and y is height, in feet.During what interval of time, to the nearesttenth of a second, is the projectile at least125 feet above ground? [The use of theaccompanying grid is optional.]

77. Which graph represents the solution set ofx x2 12 0− − < ?

[A]

[B]

[C]

[D]

78. What is the value of x in the equation5 2 3− =x i ?

[A] 4 [B] -2 [C] 7 [D] 1

79. Solve for x in simplest a + bi form:x x2 8 25 0+ + =

80. Barb pulled the plug in her bathtub and itstarted to drain. The amount of water in thebathtub as it drains is represented by theequation where L represents the number ofliters of water in the bathtub and t representsthe amount of time, in minutes, since theplug was pulled.How many liters of water were in thebathtub when Barb pulled the plug? Showyour reasoning.Determine, to the nearest tenth of a minute,the amount of time it takes for all the waterin the bathtub to drain.

81. The roots of a quadratic equation are real,rational, and equal when the discriminant is

[A] 4 [B] 0 [C] 2 [D] -2

82. For which positive value of m will theequation 4 9 02x mx+ + = have roots thatare real, equal, and rational?

[A] 4 [B] 9 [C] 12 [D] 3

83. The height, f(x), of a bouncing ball after xbounces is represented by f ( ) ( . ) .x x= 80 05How many times higher is the first bouncethan the fourth bounce?

[A] 8 [B] 2 [C] 4 [D] 16

84. The accompanying graph represents thevalue of a bond over time.

Which type of function does this graph bestmodel?

[A] logarithmic [B] trigonometric

[C] exponential [D] quadratic


85. The expression log ( )3 8− x is defined for allvalues of x such that

[A] x > 8 [B] x ≥ 8[C] x < 8 [D] x ≤ 8

86. The cells of a particular organism increaselogarithmically. If g represents cell growthand h represents time, in hours, which graphbest represents the growth pattern of thecells of this organism?

[A]

[B]

[C]

[D]

87. If log x = a, log y = b, and log z = c, then

log x yz

2

is equivalent to

[A] 2 12

ab c− [B] 2 12

a b c+ −

[C] a b c2 12

+ − [D] 42 12

a b c+ +

88. The expression log log10 102x x+ − isequivalent to

[A] 1100

[B] − 2 [C] 100 [D] 2

89. The solution set of 2 22 2 1x x+ −= is

[A] {-1} [B] {-1, 1}

[C] { } [D] {1}

90. Solve algebraically for x: 27 92 1 4x x+ =

91. Solve for m: 3 5 221m+ − =

92. Solve algebraically for x: 8 42 6x =

93. After an oven is turned on, its temperature,T, is represented by the equationT m= − −400 350 32 0 1( . ) . where m representsthe number of minutes after the oven isturned on and T represents the temperatureof the oven, in degrees Fahrenheit.How many minutes does it take for theoven's temperature to reach 300°F? Roundyour answer to the nearest minute. [The useof the grid is optional.]

94. Sean invests $10,000 at an annual rate of5% compounded continuously, according tothe formula A Pert= , where A is theamount, P is the principal, e = 2.718, r is therate of interest, and t is time, in years.Determine, to the nearest dollar, the amountof money he will have after 2 years.Determine how many years, to the nearestyear, it will take for his initial investment todouble.


95. What is the solution set of the equationx

x x x x−−

+=

− −41

328

122 ?

[A] { } [B] {4} [C] {4,-6} [D] {-6}

96. Solve for x and express your answer insimplest radical form:

4 31

7x x−

+=

97. Solve for all values of x: 9 92

12x x+

−=

98. Electrical circuits can be connected inseries, one after another, or in parallelcircuits that branch off a main line. Ifcircuits are hooked up in parallel, thereciprocal of the total resistance in the seriesis found by adding the reciprocals of eachresistance, as shown in the accompanyingdiagram.

If R x R x1 2 3= = +, , and the totalresistance, RT , is 2.25 ohms, find thepositive value of R1 to the nearest tenth ofan ohm.

99. Is 12

2sin x the same expression as sin ?x

Justify your answer.

100. A modulated laser heats a diamond. Itsvariable temperature, in degrees Celsius, isgiven by f t T at( ) sin= . What is the periodof the curve?

[A] 2πa

[B] 2aaπ [C] T [D] 1

a

101. What is the period of the functiony x= 5 3sin ?

[A] 5 [B] 3 [C] 25π [D] 2

3π

102. In physics class, Eva noticed the patternshown in the accompanying diagram on anoscilloscope.

Which equation best represents the patternshown on this oscilloscope?

[A] y x= +sin 1 [B] y x= +2 1sin

[C] y x= − +2 12

1sin( )

[D] y x= +sin( )12

1

103. The graphs below show the average annualprecipitation received at different latitudeson Earth. Which graph is a translatedcosine curve?

[A] [B]

[C] [D]


104. A pair of figure skaters graphed part of theirroutine on a grid. The male skater's path isrepresented by the equation

m x x( ) sin= 3 12

, and the female skater's

path is represented by the equationf x x( ) cos= −2 . On the accompanying

grid, sketch both paths and state how manytimes the paths of the skaters intersectbetween x = 0 and x = 4π .

105. An object that weighs 2 pounds issuspended in a liquid. When the object isdepressed 3 feet from its equilibrium point,it will oscillate according to the formulax t= 3 8cos( ), where t is the number ofseconds after the object is released. Howmany seconds are in the period ofoscillation?

[A] π [B] 3 [C] 2π [D] π4

106. The expression (1 + cos x)(1 - cos x) isequivalent to

[A] sin2 x [B] 1

[C] csc2 x [D] sec2 x

107. If θ is a positive acute angle and sin ,θ = awhich expression represents cosθ in termsof a?

[A] 11 2− a

[B] 1 2− a

[C] 1a

[D] a

108. The expression tansec

θθ

is equivalent to

[A] cosθ [B] sinθ

[C] sincos

θθ2 [D] cos

sin

2 θθ

109. The expression 22

cossin

θθ

is equivalent to

[A] sin θ [B] sec θ

[C] cot θ [D] csc θ

110. The expression 1 2

2

− cossin

xx

is equivalent to

[A] 1 [B] sin x [C] −1 [D] cos x

111. A crate weighing w pounds sits on a ramppositioned at an angle of θ with thehorizontal. The forces acting on this crateare modeled by the equationMw wcos sin ,θ θ= where M is the

coefficient of friction. What is anexpression for M in terms of θ ?

[A] M = tanθ [B] M = cotθ

[C] M = cscθ [D] M = secθ

112. Express in simplest terms: 2 2 2− sincos

xx

113. If θ is an acute angle such that sin ,θ =5

13what is the value of sin ?2θ

[A] 120169

[B] 1026

[C] 60169

[D] 1213


114. If sin ,θ =5

3 then cos2θ equals

[A] −19

[B] 13

[C] −13

[D] 19

115. If x is a positive acute angle and sin ,x = 12

what is sin ?2x

[A] −3

2[B] 3

2[C] −

12

[D] 12

116. The expression sinsin

22

θθ

is equivalent to

[A] 2 cosθ [B] 2 tanθ

[C] 2sinθ

[D] 2 cotθ

117. On a monitor, the graphs of two impulsesare recorded on the same screen, where0 360°≤ < °x . The impulses are given bythe following equations: y x= 2 2sin y x= −1 sinFind all values of x, in degrees, for whichthe two impulses meet in the interval0 360°≤ < °x . [Only an algebraic solutionwill be accepted.]

118. If (sec )( sec ) ,x x− − =2 2 1 0 then xterminates in

[A] Quadrants I and IV, only

[B] Quadrants I and II, only

[C] Quadrant I, only

[D] Quadrants I, II, III, and IV

119. If sin 6A = cos 9A, then m A∠ is equal to

[A] 6 [B] 1 12

[C] 36 [D] 54

120. What is a positive value of x for which

9 13

− =cos ?x

[A] 30° [B] 90° [C] 45° [D] 60°

121. A solution set of the equation 5 3 3sinθ + =contains all multiples of

[A] 90° [B] 180° [C] 45° [D] 135°

122. Navigators aboard ships and airplanes usenautical miles to measure distance. Thelength of a nautical mile varies withlatitude. The length of a nautical mile, L, infeet, on the latitude line θ is given by theformula L = −6 077 31 2, cos .θFind, to the nearest degree, the angleθ θ, ,0 90≤ ≤ ° at which the length of anautical mile is approximately 6,076 feet.

123. Solve algebraically for all values of θ in theinterval 0 360°≤ ≤ °θ that satisfy the

equation sincos

.2

11θ

θ+=

124. Solve the following equation algebraicallyfor all values of θ in the interval0 180°≤ ≤ °x .

2 1 0sinθ − =

125. In the interval 0 360°≤ ≤ °A , solve for allvalues of A in the equationcos sin .2 3 1A A= − −

126. Find, to the nearest degree, all values of θin the interval 0° < θ < 360° that satisfy theequation 3 2 1 0cos sin .θ θ+ − =


127. An architect is using a computer program todesign the entrance of a railroad tunnel.The outline of the opening is modeled bythe function f ( ) sin ,x x= +8 2 in theinterval 0 ≤ ≤x π , where x is expressed inradians.Solve algebraically for all values of x in theinterval 0 ≤ ≤x π , where the height of theopening, f(x), is 6. Express your answer interms of π .If the x-axis represents the base of thetunnel, what is the maximum height of theentrance of the tunnel?

128. In the accompanying diagram of ΔABC,m A∠ = 65, m B∠ = 70, and the sideopposite vertex B is 7. Find the length ofthe side opposite vertex A, and find the areaof ΔABC.

129. The number of children of each of the first41 United States presidents is given in theaccompanying table. For this population,determine the mean and the standarddeviation to the nearest tenth.How many of these presidents fall withinone standard deviation of the mean?

130. From 1984 to 1995, the winning scores for agolf tournament were 276, 279, 279, 277,278, 278, 280, 282, 285, 272, 279, and 278.Using the standard deviation for the sample,Sx , find the percent of these winning scoresthat fall within one standard deviation of themean.

131. Beth's scores on the six Earth science testsshe took this semester are 100, 95, 55, 85,75, and 100. For this population, how manyscores are within one standard deviation ofthe mean?

JEFFERSON MATH PROJECT REGENTS BY STANDARD · 5. Given: ΔABT CBTD CD, , and AB⊥ Write an...

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Transcript of JEFFERSON MATH PROJECT REGENTS BY STANDARD · 5. Given: ΔABT CBTD CD, , and AB⊥ Write an...