Question 1

Which of the following equations represents an ellipse centered at the origin with a major axis length of 10 and a minor axis length of 6?

Accepted Answer

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

Question 2

Given the equation $$(x-2)^2$$ + $$(y+1)^2 = 9$, what is the center and radius of the circle?

Accepted Answer

Center: $$(2, -1)$$, Radius: $$3$$

Question 3

Which of the following equations represents a parabola that opens upward and has its vertex at $$(0, 0)$$?

Accepted Answer

$$y = x^2$

Question 4

What is the equation of a hyperbola centered at the origin, opening along the $$y-$$axis, with vertices at $$(0, 4)$$ and $$(0, -4)$$?

Accepted Answer

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

Question 5

Given the equation $$x^2$$ - $$3y^2 = 36$, what type of conic section does this represent?

Accepted Answer

Hyperbola

Question 6

Find the equation of a parabola with its focus on the positive $$x-$$axis, 2 units away from the directrix $$x = -2.$$

Accepted Answer

$$y^2 = 8x$

Question 7

What is the equation of an ellipse with foci at $$(\pm 5, 0)$$ and a major axis length of $$12$$?

Accepted Answer

$$\frac{x^2$}{36} + \frac{y^2$}{11} = 1$$

Question 8

Given the arithmetic sequence $$11, 8, 5, 2, \ldots$$, what is the common difference?

Accepted Answer

$$-3$$

Question 9

What is the formula for the $$nth $$term of the arithmetic sequence with first term $$a_1 = 101$ and common difference d = 3$?

Accepted Answer

$$a_n = 101 + 3(n-1)$$

Question 10

Determine the common ratio of the geometric sequence $$a_1 = 2$, a_2$ = 6$$, $$a_3 = 18$, a_4$ = 54.$$

Accepted Answer

$$3$$

Question 11

Find the sum of the first four terms of the sequence $$a_n = 2^n.$$

Accepted Answer

$$15$$

Question 12

What is the formula for the sum of the first $$n$$ terms of a geometric sequence with first term $$a_1$$ and common ratio $$r$$?

Accepted Answer

$$S_n = a_1$ \frac{1 - r^n}{1 - r}$$

Question 13

Expand $$(2x + 3)^4$ using Pascal's Triangle.

Accepted Answer

$$16x^4$$ + $$96x^3$$ + $$216x^2 + 216x + 81$

Question 14

Using the Binomial Theorem, what is the coefficient of $$x^5 in $$the expansion of $$(x - 3)^8$?

Accepted Answer

$$-15120$$

Question 15

Find the equation of a hyperbola with foci at $$(\pm 3, 0)$$ and passing through the point $$(4, 1).$$

Accepted Answer

$$\frac{x^2$}{5} - \frac{y^2$}{4} = 1$$

Question 16

What is the fourth term of the sequence defined by $$a_n = 2n^2$ - 1$$?

Accepted Answer

$$15$$

Question 17

If the sum of an infinite geometric series is $$S = \frac{a_1$}{1 - r}$$, which of the following is a valid condition for convergence?

Accepted Answer

$$|r| \u003c 1$$

Question 18

Expand $$(x - 3)^6$ using the Binomial Theorem.

Accepted Answer

$$x^6$$ - $$18x^5$$ + $$135x^4$$ - $$540x^3$$ + $$1215x^2 - 1458x + 729$

Question 19

Find the 100th term of the arithmetic sequence $$11, 8, 5, 2, \ldots.$$

Accepted Answer

$$-286$$

College Algebra Study Guide: Conic Sections, Sequences, and Series

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