PRECALCULUS PRACTICE FINAL EXAM

This multiple-choice exam is based on the content of our Precalculus (MATH 1730) class, as well as the Calculus Readiness Test (Form 1E 1990) devised by the Mathematical Association of America. It is purely a diagnostic tool to help you judge your familiarity with precalculus topics - it is NOT an official testing instrument. To take this exam, for each question simply write down the letter corresponding to the answer of your choice. When you have completed the test, click on the SEE ANSWERS option at the bottom of the screen and score yourself. If you miss more than half of the problems, there is a good chance that you need to enroll in MATH 1730 before taking calculus. (However, we strongly recommend that you seek advice from us before making any decisions.)

(1)

If q is measured in radians and cos(q) = 3/5, then we know that cos(q+ 2p) =

(2)

For all angles q in degree or radian measure, we know that cos²(q) =

(a)

sin²(q) - 1.

(b)

2sin(q)cos(q).

(c)

1-sin(q).

(d)

1 - sin²(q).

(3)

A line that is perpendicular to y = 2x+5 will have slope

(a) m = -2

(b) m = -1/2

(c) m = 1/2

(d) m = 2

(4)

Let a > 0. If we know that log_a(3) = 1.8, then

(a)

we also know that a = 3^1.8.

(b)

we also know that 3 = a^1.8.

(c)

we also know that a = 3.

(d)

we also know that a = e, where e is euler's constant.

(5)

Which of the following equations is the same as 3/(x-1) = 8/(x+2)?

(a)

3x+2 = 8x-1

(b)

3(x+2)8(x-1) = 0

(c)

3x+6 = 8x-8

(d)

3/(x+2) - 8/(x-1) = 0

(6)

If the radian measure of q is 3, then we know that the terminal side of q lies in

(7)

If we write ln(xÖ[(x²-1)]) as a sum of natural logs, we obtain

(a)

ln(x) + ln(x) -ln(1)

(b)

ln(x) + (1/2)ln(x - 1)

(c)

ln(x) + (1/2)ln(x+1) + (1/2)ln(x-1)

(d)

the same expression, because it cannot be simplified.

(8)

For any positive real number a, we know the following about y = log_a(x):

(a)

its graph is always increasing.

(b)

its graph is always decreasing.

(c)

its graph never crosses the x-axis.

(d)

its graph never crosses the y-axis.

(9)

Suppose we know that a=5 cm, b = 3 cm, and A = 53^° in a certain triangle. According to the Law of Sines,

(a)

angle B must have approximate measure .48^°.

(b)

angle B must be obtuse.

(c)

there are two triangles which meet the criteria.

(d)

there is exactly one triangle which meets the criteria.

(10)

One form of the equation from the Law of Sines relating the angles A and C to the sides a and c is

(a)

sin(A) / c = sin(C) / a.

(b)

sin(A) / a = sin(C) / c.

(c)

sin(A) / sin(C) = c / a.

(d)

a² + c² = sin²(A) + sin²(C)

(11)

When the equation of a line is written in the form y = mx+b, the constant b represents

(a)

the slope of the line.

(b)

the y-coordinate of the point where the line crosses the y-axis.

(c)

the x-coordinate of the point where the line crosses the x-axis.

(d)

the change in y divided by the change in x.

(12)

The best first step in solving the equation 3^2x-1 = 5 would be

(a)

taking the 2x-1 root of both sides.

(b)

rewriting 3^2x-1 as 3^2x - 3

(c)

taking the cube root of both sides.

(d)

taking the natural log of both sides.

(13)

Which of the following equations is the same as 2x² - 3x - 1 = 0?

(a)

2(x² - (3/2)x + (9/16)) - 1 = 0

(b)

2(x² -(3/2)x + (9/16)) - 9/4 - 1 = 0

(c)

2(x + 3/2)² - 1 = 0

(d)

2(x-3/2)² - 1 = 0

(14)

A good first step in solving the equation 2x-1 = Ö(2x+1) would be to rewrite the equation as

(a)

2x = Ö[2x]

(b)

(2x-1) - Ö(2x+1) = 0

(c)

2x-1 = 2x+1

(d)

(2x-1)² = 2x+1

(15)

The method for solving log₂(x) + log₂(x+1) = 1 yields two possible solutions, namely x = 1 and x = -2. From this, we know

(a)

both x = 1 and x = -2 are solutions.

(b)

only x = 1 is a solution.

(c)

only x = -2 is a solution.

(d)

neither x = 1 nor x = -2 is a solution.

(16)

Which one of the following statements is true?

(a)

a^2/3 = -Ö(a³)

(b)

a^2/3 = (1/a²)³

(c)

a^2/3 = ³Ö(a²⁾

(d)

a^2/3 = 2a³

(17)

For all angles A,B both in degree or radian measure, we know that cos(A-B) =

(a)

cos(A)sin(B) + sin(A)cos(B).

(b)

cos(A)cos(B) + sin(A)sin(B).

(c)

2sin(A)cos(B).

(d)

1 - sin(A)sin(B).

(18)

For all angles q in degree measure, we know that sin(q) =

(a)

cos(90^° - q).

(b)

sin(90^° - q).

(c)

sec(90^° - q).

(d)

sin(q- 90^°).

(19)

If 0 < a < 1, then we know that the graph of y = a^x

(a)

always passes through the point (1,0).

(b)

always passes through the point (-1,0).

(c)

has a horizontal asymptote along the x-axis.

(d)

has a vertical asymptote along the y-axis.

(20)

A point (a,b) is known to lie on the graph of a line. If we reach another point on the line by moving three units to the right and two units down from (a,b), then the slope of this line is

(a)

m = -3/2

(b)

m = 2/3

(c)

m = -3/2

(d)

m = -2/3

(21)

If we know that q is such that sin(q) = -3/5 and tan(q) = -3/4, then we know

(a)

cos(q) = 4/5.

(b)

sec(q) = -5/4.

(c)

csc(q) = 4/3.

(d)

cos(q) = 5/4.

(22)

The midline of y = 5 + 3cos[4(x-1)] is

(23)

The terminal side of q = 23p/3 lies in

(24)

If we know that the solutions of the equation u² - 5u + 6 = 0 are u = 2 and u = 3, then what are the solutions to the equation (3x-2)² - 5(3x-2) + 6 = 0?

(a)

x = 4/3 and x = 5/3

(b)

x = ±4/3 and x = ±5/3

(c)

x=2 and x=3

(d)

only x=2

(25)

Let a > 0. As long as m and n are both positive, we know

(a)

log_a(mn) = log_a(m+n)

(b)

log_a(mn) = log_a(m) + log_a(n)

(c)

log_a(mn) = nlog_a(m)

(d)

log_a(mn) = mlog_a(n)

(26)

Let a > 0. If we know that (2,5) lies on the graph of y = a^x, then we know that

(27)

If f(x) = 5x+4, then the inverse of f will

(a)

subtract 4 from its input, then divide by 5.

(b)

divide its input by 5, then subtract 4.

(c)

divide its input by 4, then subtract 5.

(d)

subtract 5 from its input, then divide by 4.

(28)

If a population of lemmings is growing at a relative annual rate of 2.2%, how many lemmings will there be in five years, assuming the initial population is 500? Round to the nearest lemming.

(29)

A function g is the inverse of a function f provided

(a)

the graph of g is the reflection of the graph of f about the y-axis

(b)

the graph of g never crosses the x-axis

(c)

f( g(x) ) = x and g( f(x) ) = x whenever these expressions are defined

(d)

both f and g have the same domain

(30)

If f(x) = x² -1, then (f ° f)(x) is given by the formula

(a)

y=(x²-1)(x²-1)(x)

(b)

y=2x²-2

(c)

y = x⁴+2x²

(d)

y = x⁴-2x²

(31)

If we were to graph the function y = 3x²-1 on the interval -1 < x £ 2, then we would

(a)

place an open circle at the point (-1,2) and an open circle at the point (2,11)

(b)

place a closed circle at the point (-1,2) and a closed circle at the point (2,11)

(c)

place a closed circle at the point (-1,2) and an open circle at the point (2,11)

(d)

place an open circle at the point (-1,2) and a closed circle at the point (2,11)

(32)

If Arctan(3/5) = q, then we know

(a)

cot(q) = -3/5.

(b)

tan(q) = 3/5.

(c)

sin(q) = 3/5.

(d)

tan(q) = -3/5.

(33)

If y = 2 + 3sin(4(x-1)), then we know

(a)

the midline of the sinusoid is y = 3.

(b)

the amplitude of the sinusoid is 2.

(c)

the period of the sinusoid is p/2.

(d)

the horizontal translation of the sinusoid is one unit left.

(34)

If the average rate of change for a function f on the interval [2,5] is -3, then we know that

(a)

the function is increasing on the interval [2,5].

(b)

the function is decreasing on the interval [2,5].

(c)

the function f has a turning point in the interval [2,5].

(d)

the slope of the line connecting the points (2,f(2)) and (5,f(5)) is -3.

(35)

If f(x) = Ö(x+1), then (f * (f ° f))(8)=

(a) 6

(b) Ö3

(c) 27

(d) 2

(36)

Suppose an ant is sitting on the perimeter of the unit circle at the point (0,-1). If the ant travels a distance of 2p/3 in the clockwise direction, then the coordinates of the point where the ant stops will be

(37)

What can be said about the function y = (x²-1)/(x²-x-2)?

(a)

The function has two vertical asymptotes, one at x=-1, the other at x=2.

(b)

The function has exactly one vertical asymptote at x=2 and a horizontal asymptote at y=1.

(c)

The function has exactly one vertical asymptote at x=-1 and no horizontal asymptotes.

(d)

The function has no vertical asymptotes and a horizontal asymptote at y=1.

(38)

The horizontal translation of the function f(x) = -2 + 4cos(4x-5) is

(a)

Exactly two units to the right compared to the basic cosine function.

(b)

Exactly five units to the left compared to the basic cosine function.

(c)

Exactly 1.25 units to the right compared to the basic cosine function.

(d)

Exactly p/2 units to the left, compared to the basic cosine function.

(39)

Suppose you deposit $1,000 into an account which pays 4% annual interest, compounded quarterly. Approximately how long will it take for the amount of money in the account to double?

(a)

About 25 years

(b)

About 17.4 years

(c)

About 17.3 years

(d)

About 25.2 years

(40)

In a triangle, suppose we know that side b = 3 feet, side c = 2 feet, and that angle A = 140^°. According to the Law of Cosines, the length of side a is approximately