19.
An equation of the line passing through (–4, 2) having slope 4/3 is
(A) 4x + 3y = –10
(3)
(B) 4x + 3y = 10
.12
(C) 4x – 3y = 10
.09
(D) 4x – 3y = –22
(E) 4x – 3y = 22
.36 *
.25
.16
F-IF-8. Functions: Interpreting functions: Analyze functions using different representations
8. Write a function defined by an expression in different but equivalent forms to reveal and explain different
properties of the function.
24.
If f (x) = 3x -1 and g(x) = x 3 , then g( f (x)) =
(A) 3x3 – 1
(28)
(B) 27x3 – 1
.10
(C) x3(3x – 1)
.08
(D) x3 + 3x – 1
.07
(E) (3x – 1)3
.73 *
.01
F-BF-1c. Functions: Building functions: Build a function that models a relationship between two quantities
1c. (+) Compose functions.
1.
What is the radian measure of an angle whose degree measure is 150º ?
11
(A)
(29)
6
π
.05
(B)
6
π(C)
5
.06
7
π(D)
6
5
π
6
(E)
1
6
π
.75 *
.13
.00
F-TF-1. Functions: Trigonometric functions: Extend the domain of trigonometric functions using unit circle
1. Understand radian measure of an angle as the length of the arc on the unit circle
27.
æ1ö
log 4 ç ÷ =
è 16 ø
(A) 4
(2)
.29
(B) 2
.06
(C) 1/2
(D) –2
.16
(E) –4
.31 *
.18
F-BF-5. Functions: Building functions: Building new functions from existing functions
5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to
solve problems involving exponents and logarithms.
25.
If f (x) =
(A) 4
(21)
3+ x
then for which value of x does f (x) = 7 ?
x
(B) 10/7
.02
(C) 1(D) 1/2
.11
(E) none of the above
.55 *
.03
.28
F-BF-4a. Functions: Building functions: Build new functions from existing functions
4a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression
for the inverse.
12.
For which values of x in the interval 0º ≤ x ≤ 180º is sin(3x) = 1 ?
(A) 30º and 150º
(B) 60º and 180º
.38 *
(4)
(C) 0º and 120º
.09
(D) 90º
.14
(E) none of these
.21
.16
A-REI-1. Algebra: Reasoning with equations and inequalities: Understand solving
equations as a process of reasoning and explain the reasoning
1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the
previous step, starting from the assumption that the original equation has a solution. Construct a viable
argument to justify a solution method.
29.
The polynomial equation (x + 5)(x2 – 4)(x2 + 9) = 0 has how many real roots?
(A) one only
(10)
.05
(B) two only
(C) three only
.43 *
.24
(D) four only
(E) five
.11
.14
A-APR-3. Algebra: Arithmetic with polynomials and rational expressions: Understand the relationship
between zeros and factors of polynomials
3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a
rough graph of the function defined by the polynomial.
11.
For which values of x in the interval 0 ≤ x ≤ 2π does (cos x – 4)(cos x – 1) = 0?
(A)
(19)
p
2
.19
and
3p
2
(B) 1 and 4
.14
(C) π
.08
(D) 0 and 2π
.52 *
(E)
p
2
.06
A-REI-1. Algebra: Reasoning with equations and inequalities: Understand solving
equations as a process of reasoning and explain the reasoning
1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the
previous step, starting from the assumption that the original equation has a solution. Construct a viable
argument to justify a solution method.
6.
æ 5p ö
tan ç ÷ =
è 6 ø
(A) - 3
(13)
(B) -
1
3
.47 *
.21
1
3
(C)
(D) 1
.20
(E)
.03
3
.07
F-TF-3. Functions: Trigonometric functions: Extend the domain of trigonometric functions using unit circle
3. (+) Use special triangles to determine geometrically the values of sine, cosine, and tangent for π, π/2,
π/3,
π/4, π/6, and 0, and use the unit circle to express the values of sine, cosine, and tangent for π-x, π+x, and
2π-x in terms of their values for x, where x is any real number.
4.
csc(60º) =
(A) 2
(15)
.07
(B) √3
.12
(C) √2
.09
3
2
(D)
(E)
2
3
.48 *
.22
G-SRT-6. Geometry: Similarity, right triangles, and trigonometry: Define trigonometric ratios and solve
problems involving right triangles
6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle,
leading to definitions of trigonometric ratios for acute angles.
23.
(1)
The graph of the parabola y = x 2 - 36x is symmetric with respect to the line
(A) x = –6
(B) x = 6
.30
.25
(C) x = 18
.19 *
(D) y = –6
(E) y = 6
.14
.13
F-IF-8. Functions: Interpreting functions: Analyze functions using different representations
8a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme
values, and symmetry of the graph, and interpret these in terms of a context.
28.
The graphs of the two equations y = 6x – 13 and y = x2 – 4 intersect in how many distinct points?
(A) none (B) one only
(11)
.04
.44 *
(C) two only
.47
(D) three only
0
(E) Insufficient information given
.04
A-REI-7. Algebra: Reasoning with equations and inequalities: Solve systems of equations
7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables
algebraically and graphically.
14.
secq cot q sin2 q =
(A) cot q
(17)
.11
(B) sinq
(C) cosq
.48 *
(E) cscq
(D) sin 2 q cos2 q
.12
.18
.10
F-TF. Functions: Trigonometric functions: Prove and apply trigonometric identities
7.
sin ( 90 - q ) =
(A) cos q
(14)
.47 *
(B) - cos q
(C) sin q
(D) - sin q
.18
.05
.19
(E) 1- sin q
.10
F-TF-3. Functions: Trigonometric functions: Extend the domain of trigonometric functions using unit circle
3. (+) Use special triangles to determine geometrically the values of sine, cosine, and tangent for π, π/2,
π/3,
π/4, π/6, and 0, and use the unit circle to express the values of sine, cosine, and tangent for π-x, π+x, and
2π-x in terms of their values for x, where x is any real number.
or
F-TF-9. Functions: Trigonometric functions: Prove and apply trigonometric identities
9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve
problems.
8.
For all real numbers x, cos2(5x) + sin2(5x) =
(A) cos(10x)
(26)
.08
(B) 1(C) 0(D) 5
.71 *
(E) sin(10x)
.08
.12
.01
F-TF-8. Functions: Trigonometric functions: Prove and apply trigonometric identities
8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given
sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
9.
For all real numbers x, cos(2x) =
æxö
(A) sin ç ÷
è2ø
(8)
.18
æxö
(B) cos ç ÷
è2ø
.03
(C) 2 cos x
.30
(D) 2 sin x cos x
.08
(E) cos2 x – sin2 x
.43 *
F-TF-9. Functions: Trigonometric functions: Prove and apply trigonometric identities
9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve
problems.
5.
In the right triangle shown in the figure, sinq =
(16)
(B) x 2 +1
1- x 2
(A)
.48 *
(C)
.04
1
1- x 2 )
(
2
(D) x
.09
2
(E) x 1- x
.25
.08
G-SRT-8. Geometry: Similarity, right triangles, and trigonometry: Define trigonometric ratios and solve
problems involving right triangles
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*
3.
In the figure shown (not drawn to scale), if sin R = 0.37 and r = 8, then what is q?
(A)
4
5
(B)
8
0.37
10
(C)
(D) 8(0.37)
(E)
Insufficient information given
(25)
.01
.62 *
.04
.14
.18
G-SRT-8. Geometry: Similarity, right triangles, and trigonometry: Define trigonometric ratios and solve
problems involving right triangles
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles inapplied problems.*
18.
If f (x) = 4 x + x 4 , then f (-1) =
(A) 5/4
(23)
.60 *
(B) 3/4
.14
(C) –3/4
(D) –5/4
.07
.03
(E) –3
.15
F-IF-2. Functions: Interpreting functions: Understand the concept of a function and use function notation
2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use
function notation in terms of a context.
26.
Which of the following is the domain of the function f (x) = x 2 - 9 ?
(A) all real numbers
(6)
.19
(B) –3 ≤ x ≤ 3
.19
(C) x ≤ –3 or x ≥ 3
.41 *
(D) x ≤ 3
(E) x ≥ –3
.09
.11
F-IF-5. Functions: Interpreting functions: Interpret functions that arise in terms of the context
5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it
describes.
21.
Of the following, which best represents the graph of y = x - 2 ?
(A)
(B)
(9)
(C)
.14
.11
(D)
.06
(E)
.43 *
.25
F-IF-7b. Functions: Interpreting functions: Analyze functions using different representations
7b. Graph functions expressed symbolically and show key features of the graph … including step functions
and absolute value functions.
30.
Of the following, which best represents an equation of the graph in the figure shown?
(A) y2 = x + 1
(12)
.09
(B) y = ex
.44 *
(C) y = 1 – x2
.12
(D) y = ln x
.30
(E) y = x + 1
.04
F-IF-7e. Functions: Interpreting functions: Analyze functions using different representations
7e. Graph exponential and logarithmic functions, showing intercepts and end behavior.
13.
æ 3ö
æ 3ö
-1
The value of the expression 2 Arc cos ç ÷ (or equivalently, 2 cos ç ÷ ) is
è 2 ø
è 2 ø
(A) 0
(5)
(B)
.03
p
(C)
6
p
(D)
3
.39 *
.24
3
.27
(E) π
.04
F-TF-6. Functions: Trigonometric functions: Model periodic phenomena with trigonometric functions
6. (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or
always decreasing allows its inverse to be constructed.
22.
If t = e x-5 then x =
(A) 5 + ln t
(20)
(B)
.55 *
t+5
e
(C)
.09
t
+5
e
(D) ln(t + 5)
.04
.28
(E) ln t
.03
F-BF-5. Functions: Building functions: Building new functions from existing functions
5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to
solve problems involving exponents and logarithms.
17.
If a rectangle has vertices (–2, 1), (10, 1), (–2, 6), and (10, 6), then the length of a diagonal is
(A) 169
(30)
.04
(B) 37
.08
(C) 13
.76 *
(D) 5
(E)
.06
13
.04
8.G-8. 8th grade Geometry: Understand and apply the Pythagorean Theorem
8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
15.
Recall that for triangle ABC the law of cosines states that
c 2 = a2 + b2 - 2ab cos C . In the triangle
shown in the figure (not drawn to scale), what is cos Q ?
(A) 3/2
(7)
.04
(B) 7/8
.09
(C) 11/16
.42 *
(D) 1/2
.23
(E) 1/4
.21
G-SRT-11. Geometry: Similarity, right triangles, and trigonometry: Apply trigonometry to general triangles
11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in
triangles.
20.
If f (x) = 2x 2 , then f (x + k) =
(A) 2x 3 + 2x 2 k (B) 2x 2 + k
(24)
.03
.15
(C) 2x 2 + 2k 2
(D) 2x 2 + 4xk + 2k 2
.61 *
.18
(E) 4x
.02
F-IF-2. Functions: Interpreting functions: Understand the concept of a function and use function notation
2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use
function notation in terms of a context.
16.
The slope of the line 3x – 4y + 5 = 0 is
(A) 4/3
(27)
.03
(B) 3/4
.71 *
(C) 5/4
.03
(D) –3/4
.17
(E) –4/3
.05
A-SSE-3. Algebra: Seeing structure in expressions: Write expressions in equivalent forms to solve
problems
3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity
represented by the expression.