Q: A triangle has sides of length $$a = 7$$, $$b = 9$$, and angle $$C = 60^\circ. $$Using the Law of Cosines, what is the length of side $$c$$?

$$c = \sqrt{49 + 81 - 2 \times 7 \times 9 \times \cos 60^\circ}$$

Question 1

Which of the following is the correct solution to the system of equations representing the intersection point of two roads, given by $$2x + 3y = 12$$ and $$x - y = 1$$?

Accepted Answer

$$x = 3,\ y = 2$$

Question 2

A triangle has sides of length $$a = 7$$, $$b = 9$$, and angle $$C = 60^\circ. $$Using the Law of Cosines, what is the length of side $$c$$?

Accepted Answer

$$c = \sqrt{49 + 81 - 2 	imes 7 	imes 9 	imes \cos 60^\circ}$$

Question 3

Which of the following expressions is equivalent to $$\dfrac{2x^2$ - 8}{4x}$$ for $$x 
eq 0$$?

Accepted Answer

$$\dfrac{x - 2}{2}$$

Question 4

A radioactive substance decays according to $$A = A_0$ e$$^{-kt}$$. If the initial amount is A_0$ = 100$$ grams and after 5 years only 60 grams remain, what is the value of $$k$$? (Round to 3 decimal places)

Accepted Answer

$$k = -\dfrac{1}{5} \ln \left( \dfrac{60}{100} \right)$$

Question 5

Which of the following is the correct factorization of $$x^2 - 5x + 6$?

Accepted Answer

$$(x - 2)(x - 3)$$

Question 6

If $$\sin 	heta = \dfrac{3}{5}$$ and $$	heta is in $$Quadrant II, what is $$\cos 	heta$$?

Accepted Answer

$$-\dfrac{4}{5}$$

Question 7

Which of the following is the correct solution to the quadratic equation $$2x^2 - 4x - 6 = 0$?

Accepted Answer

$$x = 3,\ x = -1$$

Question 8

A ladder leans against a wall making a $$70^\circ$$ angle with the ground. If the ladder is 10 ft long, how high up the wall does it reach? (Use $$h = 10 \sin 70^\circ)$$

Accepted Answer

$$10 \sin 70^\circ$$

Question 9

Simplify the expression $$\left( \dfrac{16x^4$}{y^2$} \$$right)^{1/2}$$.

Accepted Answer

$$\dfrac{4x^2$}{y}$$

Question 10

Which of the following is the correct value of $$\log_2$ 32$$?

Accepted Answer

$$5$$

Question 11

Given $$	an \alpha = 2$$ and $$\alpha is in $$Quadrant III, what is $$\sin \alpha$$?

Accepted Answer

$$-\dfrac{2}{\sqrt{5}}$$

Question 12

Which of the following is the correct solution for $$x in $$the equation $$3^{2x} = 81$?

Accepted Answer

$$x = 2$$

Question 13

A chemist mixes two solutions: one with 30% acid and one with 50% acid. If she wants 10 liters of a 40% acid solution, how many liters of the 30% solution should she use? (Let $$x be $$the amount of 30% solution.)

Accepted Answer

$$x = 5$$

Question 14

Which of the following is the correct simplification of $$\dfrac{1}{\sqrt{2} - 1}$$?

Accepted Answer

$$\sqrt{2} + 1$$

Question 15

If $$\log_{10} x = 3$$, what is the value of $$x$$?

Accepted Answer

$$1000$$

Question 16

Which of the following is the correct solution to the equation $$|2x - 5| = 7$$?

Accepted Answer

$$x = 6,\ x = -1$$

Question 17

A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?

Accepted Answer

$$13$$

Question 18

Which of the following is the correct value of $$\sin 2	heta if$$ $$\sin 	heta = \dfrac{3}{5}$$ and $$	heta is in $$Quadrant I?

Accepted Answer

$$\dfrac{24}{25}$$

Question 19

Which of the following is the correct solution to the equation $$x^2 + 4x + 8 = 0$?

Accepted Answer

$$x = -2 \pm 2i$$

Question 20

Which of the following is the correct value of $$\sqrt[3]{-27}$$?

Accepted Answer

$$-3$$

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