Q: Which of the following systems of equations is inconsistent?

$$\begin{cases} x + y = 2 \\ x + y = 5 \end{cases}$$

Question 1

Which of the following is the standard form equation of a circle with center at $$(2, -3)$$ and radius $$5$$?

Accepted Answer

$$ (x - 2)^2$ + (y + 3)^2$ = 25 $$

Question 2

Given the equation $$x^2$$ + $$y^2 - 4x + 6y = 12$, what are the center and radius of the circle?

Accepted Answer

Center: $$(2, -3)$$, Radius: $$5$$

Question 3

Which of the following equations represents a parabola that opens to the right?

Accepted Answer

$$ (y - 1)^2$ = 8(x + 2) $$

Question 4

A satellite dish is shaped like a paraboloid. If the vertex is at the origin and the focus is at $$(0, 2)$$, what is the equation of the parabola?

Accepted Answer

$$ x^2$ = 8y $$

Question 5

Which of the following is the standard form of the equation of an ellipse centered at the origin with a major axis along the $$x-$$axis, $$a = 5$$, and $$b = 3$$?

Accepted Answer

$$ \frac{x^2$}{25} + \frac{y^2$}{9} = 1 $$

Question 6

Given the equation $$\frac{x^2$}{16} - \frac{y^2$}{9} = 1$$, what are the coordinates of the vertices of the hyperbola?

Accepted Answer

$$(4, 0)$$ and $$(-4, 0)$$

Question 7

Which of the following best describes the graph of $$x^2$$ + $$y^2 - 6x + 8y + 9 = 0$?

Accepted Answer

A circle with center $$(3, -4)$$ and radius $$2$$

Question 8

Solve the following system of equations by elimination:

$$\begin{cases} 2x + 3y = 7 \ 4x - y = 5 \end{cases}$$

Accepted Answer

$$x = 2, y = 1$$

Question 9

Which of the following systems of equations is inconsistent?

Accepted Answer

$$\begin{cases} x + y = 2 \ x + y = 5 \end{cases}$$

Question 10

Given the matrix $$A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$$, what is $$\det(A)$$?

Accepted Answer

$$-2$$

Question 11

Which of the following is the correct partial fraction decomposition of $$\frac{5x+3}{(x+1)(x-2)}$$?

Accepted Answer

$$\frac{2}{x+1} + \frac{3}{x-2}$$

Question 12

A system of three equations in three variables has no solution. Which of the following best describes the geometric interpretation?

Accepted Answer

The three planes do not all intersect at a common point.

Question 13

If $$A$$ and $$B$$ are $$2 	imes 2$$ matrices, which of the following is always true?

Accepted Answer

$$\det(AB) = \det(A) \cdot \det(B)$$

Question 14

Which of the following is the correct equation for a hyperbola centered at the origin with vertices at $$(0, 4)$$ and $$(0, -4)$$ and foci at $$(0, 5)$$ and $$(0, -5)$$?

Accepted Answer

$$\frac{y^2$}{16} - \frac{x^2$}{9} = 1$$

Question 15

Which of the following is the sum of the partial fractions for $$\frac{3x+7}{(x-1)^2$}$$?

Accepted Answer

$$\frac{3}{x-1} + \frac{4}{(x-1)^2$}$$

Question 16

A medical imaging device uses the intersection of three planes to locate a tumor. If the system of equations representing the planes is dependent, what does this mean about the location?

Accepted Answer

There are infinitely many possible locations along a line.

Conic Sections, Systems of Equations, and Partial Fractions: Exam 3 Study Guide

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