Question 1

Select the curve represented by the parametric equations.​ $$\begin{array} { l } 
x = 5 \cos \theta \
y = 6 \sin \theta
\end{array}$$ ​

Accepted Answer

A) ​   
B) ​   
C) ​   
D) ​   
E) ​   
A) ​   
B) ​   
C) ​   
D) ​   
E) ​

Question 2

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ $$x = t - 1$$ $$y = 5 t + 1$$ ​

Accepted Answer

A)  $y = 5 x - 6$ 
B)  $y = 5 x + 6$ 
C)  $y = x + 5$ 
D)  $y = x + 6$ 
E)  $y = 5 x + 1$ 
A)  $y = 5 x - 6$ 
B)  $y = 5 x + 6$ 
C)  $y = x + 5$ 
D)  $y = x + 6$ 
E)  $y = 5 x + 1$

Question 3

Find a set of parametric equations for the rectangular equation.​ $t = 2 - x$ $y = \frac { 1 } { 3 x }$ ​

Accepted Answer

A)  $x = - 2 - t , y = \frac { 1 } { ( 6 - 6 t ) }$ 
B)  $x = - 2 - t , y = \frac { 1 } { ( 6 - 3 t ) }$ 
C)  $x = 2 - t , y = \frac { 1 } { ( 3 - 3 t ) }$ 
D)  $x = - 2 + t , y = \frac { 1 } { ( 6 - 3 t ) }$ 
E)  $x = 2 - t , y = \frac { 1 } { ( 6 - 3 t ) }$ 
A)  $x = - 2 - t , y = \frac { 1 } { ( 6 - 6 t ) }$ 
B)  $x = - 2 - t , y = \frac { 1 } { ( 6 - 3 t ) }$ 
C)  $x = 2 - t , y = \frac { 1 } { ( 3 - 3 t ) }$ 
D)  $x = - 2 + t , y = \frac { 1 } { ( 6 - 3 t ) }$ 
E)  $x = 2 - t , y = \frac { 1 } { ( 6 - 3 t ) }$

Question 4

A projectile is launched at a height of h feet above the ground at an angle of $\theta$ with the horizontal.The initial velocity is $v _ { 0 }$ feet per second,and the path of the projectile is modeled by the parametric equations $$x = \left( v _ { 0 } \cos \theta \right) t \text { and } y = h + \left( v _ { 0 } \sin \theta \right) t - 16 t ^ { 2 }$$ . Select the correct graph of the path of a projectile launched from ground level at the value of $$\theta$$ and $v _ { 0 }$ .​ $\theta = 60 ^ { \circ }$ , $v _ { 0 } = 93$ feet per second
​

Accepted Answer

A) ​   
B) ​   
C) ​   
D) ​   
E) ​   ​ 
A) ​   
B) ​   
C) ​   
D) ​   
E) ​   ​

Question 5

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ $x = t - 2$ $y = \frac { t } { t - 2 }$ ​

Accepted Answer

A)  $y = x + 2$ 
B)  $y = \frac { x + 2 } { x }$ 
C)  $y = x - 2$ 
D)  $y = \frac { x + 2 } { 3 x }$ 
E)  $y = \frac { x } { x + 2 }$ 
A)  $y = x + 2$ 
B)  $y = \frac { x + 2 } { x }$ 
C)  $y = x - 2$ 
D)  $y = \frac { x + 2 } { 3 x }$ 
E)  $y = \frac { x } { x + 2 }$

Question 6

Select the curve represented by the parametric equations.​ $x = 1 + \cos \theta$ $y = 1 + 2 \sin \theta$ ​

Accepted Answer

A) ​   
B) ​   
C) ​   ​ 
D) ​   
E) ​   
A) ​   
B) ​   
C) ​   ​ 
D) ​   
E) ​

Question 7

Find a set of parametric equations for the rectangular equation.​ $t = 4 - x$ $y = \frac { 1 } { x }$ ​

Accepted Answer

A)  $x = 4 - t , y = \frac { - 1 } { ( t + 4 ) }$ 
B)  $x = 4 - t , y = \frac { - 1 } { ( - t - 4 ) }$ 
C)  $x = 4 - t , y = \frac { - 1 } { ( t - 4 ) }$ 
D)  $x = 4 + t , y = \frac { - 1 } { ( t - 8 ) }$ 
E)  $x = 8 - t , y = \frac { - 1 } { ( t - 4 ) }$ 
A)  $x = 4 - t , y = \frac { - 1 } { ( t + 4 ) }$ 
B)  $x = 4 - t , y = \frac { - 1 } { ( - t - 4 ) }$ 
C)  $x = 4 - t , y = \frac { - 1 } { ( t - 4 ) }$ 
D)  $x = 4 + t , y = \frac { - 1 } { ( t - 8 ) }$ 
E)  $x = 8 - t , y = \frac { - 1 } { ( t - 4 ) }$

Question 8

Using following result find a set of parametric equation of conic. ​
Hyperbola: $x = h + a \sec \theta , y = k + b \tan \theta$ ​
Hyperbola: vertices: $( \pm 15,0 )$ ;foci: $( \pm 17,0 )$ ​

Accepted Answer

A)  $x = 8 \sec \theta , y = 15 \tan \theta$ 
B)  $x = 17 - 15 \sec \theta , y = 17 + 8 \tan \theta$ 
C)  $x = 15 \sec \theta , y = 8 \tan \theta$ 
D)  $x = - 8 + 15 \sec \theta , y = 17 - 8 \tan \theta$ 
E)  $x = 17 \sec \theta , y = 17 \tan \theta$ 
A)  $x = 8 \sec \theta , y = 15 \tan \theta$ 
B)  $x = 17 - 15 \sec \theta , y = 17 + 8 \tan \theta$ 
C)  $x = 15 \sec \theta , y = 8 \tan \theta$ 
D)  $x = - 8 + 15 \sec \theta , y = 17 - 8 \tan \theta$ 
E)  $x = 17 \sec \theta , y = 17 \tan \theta$

Question 9

Find a set of parametric equations for the rectangular equation.​ $t = 2 - x$ $y = 4 x - 2$ ​

Accepted Answer

A)  $x = - t + 4 y = - 4 t + 2$ 
B)  $x = - t + 6 , y = - 4 t + 6$ 
C)  $x = - t + 2 , y = - 6 t + 4$ 
D)  $x = t + 2 , y = - 4 t + 6$ 
E)  $x = - t + 2 , y = - 4 t + 6$ 
A)  $x = - t + 4 y = - 4 t + 2$ 
B)  $x = - t + 6 , y = - 4 t + 6$ 
C)  $x = - t + 2 , y = - 6 t + 4$ 
D)  $x = t + 2 , y = - 4 t + 6$ 
E)  $x = - t + 2 , y = - 4 t + 6$

Question 10

A projectile is launched at a height of h feet above the ground at an angle of $\theta$ with the horizontal.The initial velocity is $v _ { 0 }$ feet per second,and the path of the projectile is modeled by the parametric equations $x = \left( v _ { 0 } \cos \theta \right) t \text { and } y = h + \left( v _ { 0 } \sin \theta \right) t - 16 t ^ { 2 }$ . Select the correct graph of the path of a projectile launched from ground level at the value of $\theta$ and $v _ { 0 }$ .​ $\theta = 45 ^ { \circ } , v _ { 0 } = 114$ feet per second
​

Accepted Answer

A) ​   ​ 
B) ​   
C) ​   
D) ​   
E) ​   
A) ​   ​ 
B) ​   
C) ​   
D) ​   
E) ​

Question 11

Select the parametric equations matching with the following graph.​   ​

Accepted Answer

A) Involute of circle: $$x = \frac { 1 } { 5 } ( \cos \theta + \Theta \sin \theta ) , y = \frac { 1 } { 5 } ( \sin \theta - \Theta \cos \theta )$$ 
B) Involute of circle: $x = \frac { 1 } { 5 } ( \cos \theta + \Theta \sin \theta ) , y = \frac { 1 } { 5 } ( \sin \theta + \Theta \cos \theta )$ 
C) Involute of circle: $x = \frac { 1 } { 5 } ( \cos \Theta - \Theta \sin \theta ) , y = \frac { 1 } { 5 } ( \sin \theta - \Theta \cos \theta )$ 
D) Involute of circle: $x = \frac { 1 } { 5 } ( \cos \theta - \Theta \sin \theta ) , y = \frac { 1 } { 5 } ( \sin \theta + \Theta \cos \theta )$ 
E) Involute of circle: $x = \frac { 1 } { 5 } ( \cos \theta + \Theta \sin \theta ) , y = \frac { 1 } { 5 } ( \cos \theta - \Theta \sin \theta )$ 
A) Involute of circle: $$x = \frac { 1 } { 5 } ( \cos \theta + \Theta \sin \theta ) , y = \frac { 1 } { 5 } ( \sin \theta - \Theta \cos \theta )$$ 
B) Involute of circle: $x = \frac { 1 } { 5 } ( \cos \theta + \Theta \sin \theta ) , y = \frac { 1 } { 5 } ( \sin \theta + \Theta \cos \theta )$ 
C) Involute of circle: $x = \frac { 1 } { 5 } ( \cos \Theta - \Theta \sin \theta ) , y = \frac { 1 } { 5 } ( \sin \theta - \Theta \cos \theta )$ 
D) Involute of circle: $x = \frac { 1 } { 5 } ( \cos \theta - \Theta \sin \theta ) , y = \frac { 1 } { 5 } ( \sin \theta + \Theta \cos \theta )$ 
E) Involute of circle: $x = \frac { 1 } { 5 } ( \cos \theta + \Theta \sin \theta ) , y = \frac { 1 } { 5 } ( \cos \theta - \Theta \sin \theta )$

Question 12

Select the curve represented by the parametric equations.(indicate the orientation of the curve)​ $x = t - 1$ $y = 4 t + 1$ ​

Accepted Answer

A) ​   
B) ​   
C) ​   
D) ​   
E) ​   
A) ​   
B) ​   
C) ​   
D) ​   
E) ​

Question 13

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ $\begin{array} { l } 
x = 4 \cos \theta \
y = 2 \sin \theta
\end{array}$ ​

Accepted Answer

A) ​ $$\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 16 } = 1$$ 
B)  $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1$ 
C)  $$y = \frac { x } { 4 }$$ 
D)  $y = \frac { x } { 2 }$ 
E) ​ $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1$ 
A) ​ $$\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 16 } = 1$$ 
B)  $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1$ 
C)  $$y = \frac { x } { 4 }$$ 
D)  $y = \frac { x } { 2 }$ 
E) ​ $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1$

Question 14

Select the curve represented by the parametric equations.​ $$\begin{array} { l } 
x = \frac { 2 } { 3 } t \\
y = t ^ { 2 }
\end{array}$$ ​ ​

Accepted Answer

A) ​   
B) ​   
C) ​   
D) ​   
E) ​   
A) ​   
B) ​   
C) ​   
D) ​   
E) ​

Question 15

Select the curve represented by the parametric equations. ​
Cycloid: $$x = 5 ( \Theta - \sin \theta ) , y = 5 ( 1 - \cos \theta )$$ ​

Accepted Answer

A) ​   
B) ​   
C) ​   
D) ​   
E) ​   
A) ​   
B) ​   
C) ​   
D) ​   
E) ​

Question 16

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ $\begin{array} { l } 
x = 5 \sin 2 \Theta \
y = 5 \cos 2 \theta
\end{array}$ ​

Accepted Answer

A)  $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 25 } = 1$ 
B)  $\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 25 } = 1$ 
C)  $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 2 } = 1$ 
D)  $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 2 } = 1$ 
E)  $y = \frac { x } { 2 }$ 
A)  $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 25 } = 1$ 
B)  $\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 25 } = 1$ 
C)  $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 2 } = 1$ 
D)  $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 2 } = 1$ 
E)  $y = \frac { x } { 2 }$

Question 17

Select the curve represented by the parametric equations.​ $x = \sqrt { t }$ $y = 1 - t$ ​

Accepted Answer

A) ​   ​ 
B) ​   ​ 
C) ​   
D) ​   
E) ​   
A) ​   ​ 
B) ​   ​ 
C) ​   
D) ​   
E) ​

Question 18

Find a set of parametric equations for the rectangular equation.​ $\begin{array} { l } 
t = 2 - x \
y = 1 - 6 x ^ { 2 }
\end{array}$ ​​

Accepted Answer

A)  $x = - t + 2 , y = - 6 t ^ { 2 } + 24 t - 23$ 
B)  $x = t + 2 , y = - 6 t ^ { 2 } + 24 t - 23$ 
C)  $x = t - 2 , y = - 6 t ^ { 2 } + 24 t - 23$ 
D)  $x = - t + 2 , y = - 6 t ^ { 2 } + 23 t - 24$ 
E)  $x = t + 2 , y = - 24 t ^ { 2 } + 6 t + 23$ 
A)  $x = - t + 2 , y = - 6 t ^ { 2 } + 24 t - 23$ 
B)  $x = t + 2 , y = - 6 t ^ { 2 } + 24 t - 23$ 
C)  $x = t - 2 , y = - 6 t ^ { 2 } + 24 t - 23$ 
D)  $x = - t + 2 , y = - 6 t ^ { 2 } + 23 t - 24$ 
E)  $x = t + 2 , y = - 24 t ^ { 2 } + 6 t + 23$

Question 19

Select the curve represented by the parametric equations.​ $$\begin{array} { l } 
x = | t - 2 | \
y = t + 6
\end{array}$$ ​

Accepted Answer

A) ​   
B) ​   ​ 
C) ​   
D) ​   
E) ​   
A) ​   
B) ​   ​ 
C) ​   
D) ​   
E) ​

Question 20

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ $x = t + 8$ $y = \frac { t } { t + 8 }$ ​

Accepted Answer

A)  $y = \frac { x - 8 } { 9 x }$ 
B)  $y = \frac { x } { x - 8 }$ 
C)  $y = x - 8$ 
D)  $y = \frac { x - 8 } { x }$ 
E)  $y = x + 8$ 
A)  $y = \frac { x - 8 } { 9 x }$ 
B)  $y = \frac { x } { x - 8 }$ 
C)  $y = x - 8$ 
D)  $y = \frac { x - 8 } { x }$ 
E)  $y = x + 8$

Exam 64: Parametric Equations

Select the curve represented by the parametric equations.​ x=5\theta y=6\theta ​

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ x=t−1x = t - 1x=t−1 y=5t+1y = 5 t + 1y=5t+1 ​

Find a set of parametric equations for the rectangular equation.​ t=2−xt = 2 - xt=2−x y=13xy = \frac { 1 } { 3 x }y=3x1​ ​

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ x=t−2x = t - 2x=t−2 y=tt−2y = \frac { t } { t - 2 }y=t−2t​ ​

Select the curve represented by the parametric equations.​ x=1+cos⁡θx = 1 + \cos \thetax=1+cosθ y=1+2sin⁡θy = 1 + 2 \sin \thetay=1+2sinθ ​

Find a set of parametric equations for the rectangular equation.​ t=4−xt = 4 - xt=4−x y=1xy = \frac { 1 } { x }y=x1​ ​

Using following result find a set of parametric equation of conic. ​ Hyperbola: x=h+asec⁡θ,y=k+btan⁡θx = h + a \sec \theta , y = k + b \tan \thetax=h+asecθ,y=k+btanθ ​ Hyperbola: vertices: (±15,0)( \pm 15,0 )(±15,0) ;foci: (±17,0)( \pm 17,0 )(±17,0) ​

Find a set of parametric equations for the rectangular equation.​ t=2−xt = 2 - xt=2−x y=4x−2y = 4 x - 2y=4x−2 ​

Select the parametric equations matching with the following graph.​ ​

Select the curve represented by the parametric equations.(indicate the orientation of the curve)​ x=t−1x = t - 1x=t−1 y=4t+1y = 4 t + 1y=4t+1 ​

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ x=4\theta y=2\theta ​

Select the curve represented by the parametric equations.​ x=t y= ​ ​

Select the curve represented by the parametric equations. ​ Cycloid: x=5(Θ−sin⁡θ),y=5(1−cos⁡θ)x = 5 ( \Theta - \sin \theta ) , y = 5 ( 1 - \cos \theta )x=5(Θ−sinθ),y=5(1−cosθ) ​

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ x=52\Theta y=52\theta ​

Select the curve represented by the parametric equations.​ x=tx = \sqrt { t }x=t​ y=1−ty = 1 - ty=1−t ​

Find a set of parametric equations for the rectangular equation.​ t=2-x y=1-6 ​​

Select the curve represented by the parametric equations.​ x=|t-2| y=t+6 ​

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.​ x=t+8x = t + 8x=t+8 y=tt+8y = \frac { t } { t + 8 }y=t+8t​ ​

Rectangular Coordinates

Graphs of Equations

Linear Equations in Two Variables

Functions

Analyzing Graphs of Functions

A Library of Parent Functions

Transformations of Functions

Combinations of Functions Composite Functions

Inverse Functions

Mathematical Modeling and Variation

Quadratic Functions and Models

Polynomial Functions of Higher Degree

Polynomial and Synthetic Division

Complex Numbers

Zeros of Polynomial Functions

Rational Functions

Nonlinear Inequalities

Exponential Functions and Their Graphs

Logarithmic Functions and Their Graphs

Properties of Logarithms

Exponential and Logarithmic Equations

Exponential and Logarithmic Models

Radian and Degree Measure

Trigonometric Functions The Unit Circle

Right Triangle Trigonometry

Trigonometric Functions of Any Angle

Graphs of Sine and Cosine Functions

Graphs of Other Trigonometric Functions

Inverse Trigonometric Functions

Applications and Models

Using Fundamental Identities

Verifying Trigonometric Equations

Solving Trigonometric Equations

Sum and Difference Formulas

Multiple Angle and Product to Sum Formulas

Law of Sines

Law of Cosines

Vectors in the Plane

Vectors and Dot Products

Trigonometric Form of a Complex Number

Linear and Nonlinear Systems of Equations

Two Variable Linear Systems

Multivariable Linear Systems

Partial Fractions

Systems of Inequalities

Linear Programming

Matrices and Systems of Equations

Operations With Matrices

The Inverse of a Square Matrix

The Determinant of a Square Matrix

Applications of Matrices and Determinants

Sequences and Series

Arithmetic Sequences and Partial Sums

Geometric Sequences and Series

Mathematical Induction

The Binomial Theorem

Counting Principles

Probability

Lines

Introduction to Conics Parabolas

Ellipses

Select the curve represented by the parametric equations. x=5\theta y=6\theta

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. $x = t - 1$ $y = 5 t + 1$

Find a set of parametric equations for the rectangular equation. $t = 2 - x$ $y = \frac { 1 } { 3 x }$

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. $x = t - 2$ $y = \frac { t } { t - 2 }$

Select the curve represented by the parametric equations. $x = 1 + \cos \theta$ $y = 1 + 2 \sin \theta$

Find a set of parametric equations for the rectangular equation. $t = 4 - x$ $y = \frac { 1 } { x }$

Using following result find a set of parametric equation of conic. Hyperbola: $x = h + a \sec \theta , y = k + b \tan \theta$ Hyperbola: vertices: $( \pm 15,0 )$ ;foci: $( \pm 17,0 )$

Find a set of parametric equations for the rectangular equation. $t = 2 - x$ $y = 4 x - 2$

Select the parametric equations matching with the following graph.

Select the curve represented by the parametric equations.(indicate the orientation of the curve) $x = t - 1$ $y = 4 t + 1$

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. x=4\theta y=2\theta

Select the curve represented by the parametric equations. x=t y=

Select the curve represented by the parametric equations. Cycloid: $x = 5 ( \Theta - \sin \theta ) , y = 5 ( 1 - \cos \theta )$

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. x=52\Theta y=52\theta

Select the curve represented by the parametric equations. $x = \sqrt { t }$ $y = 1 - t$

Find a set of parametric equations for the rectangular equation. t=2-x y=1-6

Select the curve represented by the parametric equations. x=|t-2| y=t+6

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. $x = t + 8$ $y = \frac { t } { t + 8 }$