Advanced Math Gateway Exam

You have 30 minutes. You may only use a pen/pencil, and the paper provided. There are ten problems, all of equal weight. You must score 80% or higher. There will be no partial credit, so check your work carefully.

1.

Put the equation for the parabola with focus at $${(1,2)}$$ and directrix $${y = -4}$$ into the standard form $${y = A + B(x-C)^2}$$.

2.

Give the coordinates of the vertices of the ellipse $${9x^2 - 36x + 4y^2-24y +36 = 0}$$.

3.

Give the equations of the asympotes of the hyperbola $${9x^2 - 36x - 4y^2+ 24y +36 = 0}$$.

4.

Write as the ratio of two integers in lowest terms: $${\sin(\arctan(4/3))}$$.

5.

Give the domain and range of arcsin.

6.

Solve for $${x}$$: $${\log_4(x) = 2}$$.

7.

For which real numbers is $${\ln((1-x)(x+3))}$$ defined?

8.

Suppose that $${\ln(a) = 4}$$ and $${\ln(b) = 3}$$. Write the following as the ratio of two integers in lowest terms:
$${\frac{\ln(ab^2)}{\ln(a^{-1}b^3)}}$$.

9.

Solve for $${x}$$. Express your answer in terms of logarithms:
$${e^{x-1} = 2}$$.

10.

Suppose that $${\ln(2) = 0.6931471806}$$,$${\ln(2) = 0.6931}$$,$${\ln(3) = 1.0986}$$,$${\ln(4) = 1.3863}$$,$${\ln(5) = 1.6094}$$,$${\ln(6) = 1.7918}$$,$${\ln(7) = 1.9459}$$,$${\ln(8) = 2.0794}$$,$${\ln(9) = 2.1972}$$,and $${\ln(10) = 2.3026}$$.Evaluate $${\log_7(2)}$$.