This Quiz will have 8 problems, each worth 2 points.  These problems will be on material that is considered prerequisite to this course.  You will have 30 minutes to complete the quiz. 

• You may not use a calculator as you work on this quiz. 
• If you do well on this quiz, you can earn up to 4 points extra credit on Test 1.  If you do poorly on this quiz, I will call you to discuss whether you have sufficient algebraic skills to continue in this course at this time.

Topics covered on this Quiz:

• You should be able to simplify an algebraic expression.
• Use the rules of exponents to evaluate or simplify an algebraic expression (A negative exponent denotes a reciprocal, and a fractional exponent denotes a root.)
• Calculate the sum or product of polynomials
• Solve a linear equation for a specified value of x or y
• Given some information about a line, you should be able to write the equation of the line.  In particular:
• Given two points, one point and the slope of the line, or the x- and y-intercepts, you should be able to write the equation of the line.
• Given the equation of a line, you should be able to give the equation of a line that is parallel or perpendicular to that line which goes through a specified point.  (If lines are parallel lines, they have the same slope.  If two lines are perpendicular, the product of their slopes will be -1.)
• You should be able to use the Pythagorean Theorem to find the length of the third side of a right triangle given the lengths of any two of its sides (that is, the lengths of two legs or the lengths of one leg and the hypotenuse).
• You should be able to calculate the area and perimeter (or circumference) of basic geometric figures (squares, rectangles, triangles, and circles).
• You should be familiar with basic right-triangle trigonometry for first quadrant angles.
• You should know the definitions of the sine, cosine, and tangent for an angle in a right triangle.
• Given a diagram of a right triangle with the lengths of sides marked, you should be able to give the values of the sine, cosine, and tangent of each of the angles.
• You should be able to convert between degree and radian measure for an angle. (`pi` radians corresponds to 180^o.) In particular, you should be able to convert 0^o, 30^o, 45^o, 60^o, 90^o, 180^o, 270^o and 360^o to radians. You should be able to convert `0,` `pi//6,` `pi//4,` `pi//3,` `pi//2,` `pi,` `3pi//2,` and `2pi` to degrees.
• You should be able to use the two standard triangles (30^o-60 ^o-90 ^o and 45 ^o-45 ^o-90 ^o) to find the value of each of the trigonometric functions for angles in the first quadrant (0^o, 30^o, 45^o, 60^o, and 90^o; and `0,` `pi//6,` `pi//4,` `pi//3,` `pi//2` radians).

Sample problems:

1.      Solve the formula `C = 2pi r` for `r.`

2.      Solve the formula `A = pir^2` for `r.`

3.      Write the expression `x^3 x^4` in the form `x^p.`

4.      Write the expression `x^3//x^4` in the form `x^p.`

5.      Write the expression `(x^3 x^4)^5` in the form `x^p.`

6.      Write the expression `sqrt(x^3)` in the form `x^p.`

7.      Write the expression `1//x^3` in the form `x^p.`

8.      Write the expression `x^4//x^7` in the form `x^p.`

9.      Find the product: `(t^2 + 3t + 3) (t^2 - 5t + 4).`

10.  Find the product: `t^2 (3t + 3) (t^2 - 5t).`

11.  Find the sum: `t^2 + (3t + 3) + (t^2 - 5t).`

12.  Find the equation of the line that goes through the points `(3, 0)` and `(0, -2).`

13.  Find the slope of the line that goes through the points `(3, -2)` and `(4, 1).`

14.  Write the equation of the line that goes through the points `(a, b)` and `(5, -2).`

15.  Find the equation of the line through `(3, 5)` and `(3, -4).`  What is the slope of this line? 

16.  What is the equation of the line through `(5, -6)` and `(10, -6)?`  What is the slope of this line?

17.  Find the `y-`intercept of the line that goes through the point `(2, 5)` and has slope `4.`

18.  What is the `x-`intercept of the line that goes through the point `(-2, 3)` and has slope `1?`

19.  Find the `y-`intercept of the line that goes through the point `(2, 5)` and has slope `0.`

20.  Write the equation of the line with `y-`intercept `-3` that is parallel to the line `x + 2y-5 = 0.`

21.  Find the equation of the line with `y-`intercept `7` that is perpendicular to `x + 2y = 0.`

22.  Write the equation of the line perpendicular to the line `x + 2y = 10` that passes through the origin.

23.  What is the length of the hypotenuse of a right triangle whose legs measure 12 cm and 15 cm?  (Leave your answer in radical form.)

24.  What is the area of a circle whose circumference is `6pi` cm?

25.  What is the perimeter of a semicircle with diameter 5 cm? (Leave your answer in terms of `pi.)`

26.  What is the area of a semicircle of diameter 5 cm? (Leave your answer in terms of `pi.)`

27.  A pentagon is formed by adjoining an equilateral triangle to a square so that one side of the triangle fits exactly along one side of the square. What is the perimeter of this pentagon if the length of the side of the square is 5 cm?

28.  DABC has a right angle at A.  The legs of this triangle measure 6 inches and 8 inches.  What is the area of the circle which is constructed on the hypotenuse? 

29.  What is the altitude of an equilateral triangle with sides of length 3 units?

30.  What is the area of an equilateral triangle with sides of length 5 units?

31.  What is the perimeter of a regular octagon that has sides of length `sqrt(2)?`

32.  What is the area of a regular octagon that has sides of length `sqrt(2)?`

33.  Use one of the two standard triangles to find an angle whose tangent is 1.

34.  Use one of the two standard triangles to find the sine of a 60^o angle.

35.  Express 60^o in radian measure.

36.  Express 90^o in radian measure.

37.  Express 135^o in radian measure.

38.  If an angle is `pi//4` radians, what is its measure in degrees?

39.  If an angle is `pi//3` radians, what is its measure in degrees?

40.  If an angle is `2pi` radians, what is its measure in degrees?

41.  DABC has a right angle at A.  The lengths of sides AB and AC are 2 units and 3 units, respectively.  What is the cosine of angle B?

42.  If `tan (alpha) = 5,` find `sin (alpha).` 

43.  If `sin (alpha) = 0.5,` find `cos (alpha).` 

44.  If `cos (alpha) = 0.75,` find `tan (alpha).` 

Answers to sample problems:

If you get a different answer, check your work carefully; but it is possible that an error was made in typing these answers. 

1.      `r = C //(2pi)` (Note that the parentheses are necessary here.  Why?)

2.      `r = sqrt(A//pi)`

3.      `x^7`

4.      `x^(-1)` (that is, `1//x`) 

5.      `x^35`

6.      `x^(3//2)`

7.      `x^(-3)`

8.      `x^(-3)`

9.      `t^4 - 2t^3 - 8t^2 - 3t + 12`

10.  `3t^5 - 12t^4 - 15t^3`

11.  `2t^2- 2t + 3`

12.  `y=2/3(x-3)`

13.  The slope is `3.`

14.  `y-b=(b+2)/(a-5)(x-a)`

15.  The equation of the line is `x = 3.`  The slope of this line is undefined. (Why?)

16.  The equation of the line is `y = -6.`  The slope of this line is `0.`

17.  The y-intercept of this line is `-3.`

18.  The x-intercept of this line is `-5.`  

19.  The y-intercept of this line is `5.`

20.  `y=-1/2x-3`

21.  `y=2x+7`

22.  `y=2x`

23.  `sqrt(369)`;

24.  Area `= 9pi` cm²

25.  `5 + 5 pi//2` cm

26.  `25pi//8` cm²

27.  `25` cm 

28.  Area `= 25pi` square inches 

29.  Altitude `= sqrt(27/4) = 3sqrt(3)/2`

30.  Area `= 25 sqrt(3) // 4` square units

31.  `8sqrt(2)`

32.  `(2+sqrt(2))^2`

33.  The angle is `pi//4` or 45^o. [Both legs of the triangle have to be equal, so use the 45-45-90 triangle. `tan(45^o)=tan (pi//4) = 1.`]

34.  `sin(60^o)=sqrt(3)//2`  (Use the 30-60-90 triangle.)

35.  `pi//3`

36.  `pi//2`

37.  `3pi//4`

38.  45^o

39.  60^o

40.  360^o

41.  `cos(B)=2/sqrt(13)`

42.  `sin(alpha)=5/sqrt(26)`

43.  `cos(alpha)=sqrt(3)/2`

44.  `tan(alpha) = sqrt(7) / 3`