Question ID 91e7ea5e
Assessment Test Domain Skill Difficulty
SAT Math Advanced Math Nonlinear functions
ID: 91e7ea5e
The quadratic function h is defined as shown. In the xy-plane, the graph of
intersects the x-axis at the points and , where t is a constant.
What is the value of t ?
A. 1
B. 2
C. 4
D. 8
ID: 91e7ea5e Answer
Correct Answer: D
Rationale
Choice D is correct. It’s given that the graph of intersects the x-axis at and , where t is a constant. Since
this graph intersects the x-axis when or when , it follows that and . If , then
. Adding 32 to both sides of this equation yields . Dividing both sides of this equation by 2
yields . Taking the square root of both sides of this equation yields . Adding 4 to both sides of this
equation yields . Therefore, the value of t is 8.
Choices A, B, and C are incorrect and may result from calculation errors.
Question Difficulty: Hard
,Question ID fc3d783a
Assessment Test Domain Skill Difficulty
SAT Math Advanced Math Nonlinear equations
in one variable and
systems of
equations in two
variables
ID: fc3d783a
In the -plane, a line with equation intersects a parabola at exactly one point. If the parabola has equation
, where is a positive constant, what is the value of ?
ID: fc3d783a Answer
Correct Answer: 6
Rationale
The correct answer is . It’s given that a line with equation intersects a parabola with equation ,
where is a positive constant, at exactly one point in the xy-plane. It follows that the system of equations consisting of
and has exactly one solution. Dividing both sides of the equation of the line by yields
. Substituting for in the equation of the parabola yields . Adding and subtracting
from both sides of this equation yields . A quadratic equation in the form of ,
where , , and are constants, has exactly one solution when the discriminant, , is equal to zero. Substituting for
and for in the expression and setting this expression equal to yields , or
. Adding to each side of this equation yields . Taking the square root of each side of this equation
yields . It’s given that is positive, so the value of is .
Question Difficulty: Hard
,Question ID 371cbf6b
Assessment Test Domain Skill Difficulty
SAT Math Advanced Math Equivalent
expressions
ID: 371cbf6b
The equation above is true for all x, where a and b are
constants. What is the value of ab ?
A. 18
B. 20
C. 24
D. 40
ID: 371cbf6b Answer
Correct Answer: C
Rationale
Choice C is correct. If the equation is true for all x, then the expressions on both sides of the equation will be equivalent.
Multiplying the polynomials on the left-hand side of the equation gives . On the right-
hand side of the equation, the only -term is . Since the expressions on both sides of the equation are equivalent, it
follows that , which can be rewritten as . Therefore, , which
gives .
Choice A is incorrect. If , then the coefficient of on the left-hand side of the equation would be ,
which doesn’t equal the coefficient of , , on the right-hand side. Choice B is incorrect. If , then the coefficient
of on the left-hand side of the equation would be , which doesn’t equal the coefficient of , , on the
right-hand side. Choice D is incorrect. If , then the coefficient of on the left-hand side of the equation would be
, which doesn’t equal the coefficient of , , on the right-hand side.
Question Difficulty: Hard
, Question ID 34847f8a
Assessment Test Domain Skill Difficulty
SAT Math Advanced Math Equivalent
expressions
ID: 34847f8a
The equation above is true for all , where r and t are positive
constants. What is the value of rt ?
A.
B. 15
C. 20
D. 60
ID: 34847f8a Answer
Correct Answer: C
Rationale
Choice C is correct. To express the sum of the two rational expressions on the left-hand side of the equation as the single
rational expression on the right-hand side of the equation, the expressions on the left-hand side must have the same
denominator. Multiplying the first expression by results in , and multiplying the second expression by
results in , so the given equation can be rewritten as
, or . Since the two
rational expressions on the left-hand side of the equation have the same denominator as the rational expression on the right-
hand side of the equation, it follows that . Combining like terms on the left-hand side yields
, so it follows that and . Therefore, the value of is .
Choice A is incorrect and may result from an error when determining the sign of either r or t. Choice B is incorrect and may
result from not distributing the 2 and 3 to their respective terms in .
Choice D is incorrect and may result from a calculation error.
Assessment Test Domain Skill Difficulty
SAT Math Advanced Math Nonlinear functions
ID: 91e7ea5e
The quadratic function h is defined as shown. In the xy-plane, the graph of
intersects the x-axis at the points and , where t is a constant.
What is the value of t ?
A. 1
B. 2
C. 4
D. 8
ID: 91e7ea5e Answer
Correct Answer: D
Rationale
Choice D is correct. It’s given that the graph of intersects the x-axis at and , where t is a constant. Since
this graph intersects the x-axis when or when , it follows that and . If , then
. Adding 32 to both sides of this equation yields . Dividing both sides of this equation by 2
yields . Taking the square root of both sides of this equation yields . Adding 4 to both sides of this
equation yields . Therefore, the value of t is 8.
Choices A, B, and C are incorrect and may result from calculation errors.
Question Difficulty: Hard
,Question ID fc3d783a
Assessment Test Domain Skill Difficulty
SAT Math Advanced Math Nonlinear equations
in one variable and
systems of
equations in two
variables
ID: fc3d783a
In the -plane, a line with equation intersects a parabola at exactly one point. If the parabola has equation
, where is a positive constant, what is the value of ?
ID: fc3d783a Answer
Correct Answer: 6
Rationale
The correct answer is . It’s given that a line with equation intersects a parabola with equation ,
where is a positive constant, at exactly one point in the xy-plane. It follows that the system of equations consisting of
and has exactly one solution. Dividing both sides of the equation of the line by yields
. Substituting for in the equation of the parabola yields . Adding and subtracting
from both sides of this equation yields . A quadratic equation in the form of ,
where , , and are constants, has exactly one solution when the discriminant, , is equal to zero. Substituting for
and for in the expression and setting this expression equal to yields , or
. Adding to each side of this equation yields . Taking the square root of each side of this equation
yields . It’s given that is positive, so the value of is .
Question Difficulty: Hard
,Question ID 371cbf6b
Assessment Test Domain Skill Difficulty
SAT Math Advanced Math Equivalent
expressions
ID: 371cbf6b
The equation above is true for all x, where a and b are
constants. What is the value of ab ?
A. 18
B. 20
C. 24
D. 40
ID: 371cbf6b Answer
Correct Answer: C
Rationale
Choice C is correct. If the equation is true for all x, then the expressions on both sides of the equation will be equivalent.
Multiplying the polynomials on the left-hand side of the equation gives . On the right-
hand side of the equation, the only -term is . Since the expressions on both sides of the equation are equivalent, it
follows that , which can be rewritten as . Therefore, , which
gives .
Choice A is incorrect. If , then the coefficient of on the left-hand side of the equation would be ,
which doesn’t equal the coefficient of , , on the right-hand side. Choice B is incorrect. If , then the coefficient
of on the left-hand side of the equation would be , which doesn’t equal the coefficient of , , on the
right-hand side. Choice D is incorrect. If , then the coefficient of on the left-hand side of the equation would be
, which doesn’t equal the coefficient of , , on the right-hand side.
Question Difficulty: Hard
, Question ID 34847f8a
Assessment Test Domain Skill Difficulty
SAT Math Advanced Math Equivalent
expressions
ID: 34847f8a
The equation above is true for all , where r and t are positive
constants. What is the value of rt ?
A.
B. 15
C. 20
D. 60
ID: 34847f8a Answer
Correct Answer: C
Rationale
Choice C is correct. To express the sum of the two rational expressions on the left-hand side of the equation as the single
rational expression on the right-hand side of the equation, the expressions on the left-hand side must have the same
denominator. Multiplying the first expression by results in , and multiplying the second expression by
results in , so the given equation can be rewritten as
, or . Since the two
rational expressions on the left-hand side of the equation have the same denominator as the rational expression on the right-
hand side of the equation, it follows that . Combining like terms on the left-hand side yields
, so it follows that and . Therefore, the value of is .
Choice A is incorrect and may result from an error when determining the sign of either r or t. Choice B is incorrect and may
result from not distributing the 2 and 3 to their respective terms in .
Choice D is incorrect and may result from a calculation error.
