CLEP Pre-Calculus Exam — Questions and Answers

Question 1: A central angle of 120° in a circle of radius 6 subtends an arc of what length?

12π
8π
4π (Correct answer)
6π

Correct answer: 4π

Arc length = rθ = 6 × (2π/3) = 4π.

Question 2: What does the inverse of a function or a relationship mean?

A function or relation that is equal to its original form.
A function or relation with negative outputs for positive inputs.
A correspondence from the range to the domain, found by interchanging the variables. (Correct answer)
A function or relation with inputs greater than its outputs.

Correct answer: A correspondence from the range to the domain, found by interchanging the variables.

The inverse of a function or relation essentially reverses the mapping of the original function. It takes the output values of the original function as its inputs and produces the original input values as its outputs. Mathematically, this is achieved by swapping the roles of the independent and dependent variables (x and y) and then solving for the new dependent variable.

Question 3: What does the horizontal line test determine?

Whether a function has an x-intercept
Whether a function is one-to-one (Correct answer)
Whether a function is even or odd
Whether a relation is a function

Correct answer: Whether a function is one-to-one

If every horizontal line intersects the graph at most once, the function is one-to-one and thus has an inverse.

Question 4: Simplify: sin²θ + cos²θ + tan²θ − sec²θ

2
1
−1
0 (Correct answer)

Correct answer: 0

sin²θ + cos²θ = 1 and sec²θ − tan²θ = 1, so the expression equals 1 − 1 = 0.

Question 5: What is the "unit" imaginary number?

0
−1
1
√(−1) (Correct answer)

Correct answer: √(−1)

The unit imaginary number is denoted by 'i'. By definition, 'i' is the square root of -1. Therefore, √(−1) is the correct representation of the unit imaginary number, as it is the fundamental building block for all imaginary numbers.

Question 6: What is the range of f(x) = |x| + 2?

[0, ∞)
[2, ∞) (Correct answer)
(-∞, ∞)
(-∞, 2]

Correct answer: [2, ∞)

|x| ≥ 0 for all x, so |x| + 2 ≥ 2, making the range [2, ∞).

Question 7: Which equation represents a circle centered at (2, -3) with radius 5?

(x−2)² + (y+3)² = 5
(x−2)² + (y+3)² = 25 (Correct answer)
(x+2)² + (y−3)² = 25
(x+2)² + (y+3)² = 25

Correct answer: (x−2)² + (y+3)² = 25

The standard form is (x−h)² + (y−k)² = r², giving (x−2)² + (y+3)² = 25.

Question 8: Which of the following is an odd function?

f(x) = sin(x) (Correct answer)
f(x) = cos(x)
f(x) = sec(x)
f(x) = cos²(x)

Correct answer: f(x) = sin(x)

sin(-x) = -sin(x) satisfies the definition of an odd function, f(-x) = -f(x).

Question 9: The latus rectum of a parabola is:

The directrix line
The chord through the focus parallel to the directrix (Correct answer)
The distance from vertex to focus
The axis of symmetry

Correct answer: The chord through the focus parallel to the directrix

The latus rectum is the chord that passes through the focus and is perpendicular to the axis of symmetry.

Question 10: f(x) = x3 - 9x^2 - 45x - 27 : Determine the remaining zeros if -3 is a zero of f(x).

{-3, 0, 3}
(-3, 3, 9}
{-3, 6 ± 3√5} (Correct answer)
{-3, -6 ± 3√5}

Correct answer: {-3, 6 ± 3√5}

Since -3 is a zero of f(x) = x^3 - 9x^2 - 45x - 27, (x+3) is a factor. Using synthetic division with -3, we divide the polynomial to get the quotient x^2 - 12x - 9. To find the remaining zeros, set this quadratic equal to zero and use the quadratic formula: x = [12 ± √((-12)^2 - 4(1)(-9))] / 2(1) = [12 ± √(144 + 36)] / 2 = [12 ± √180] / 2. Simplifying √180 to 6√5, we get x = [12 ± 6√5] / 2 = 6 ± 3√5. Therefore, the remaining zeros are 6 + 3√5 and 6 - 3√5.

Question 11: The zeros of a polynomial function with rational coefficients are 3 + 5 and i. Find all additional zeros.

3 + √5
−3 - √5 and -i
−3 - √5
−3 - √5 and i (Correct answer)

Correct answer: −3 - √5 and i

The Conjugate Root Theorem states that if a polynomial has rational coefficients, then irrational roots involving square roots and complex roots must occur in conjugate pairs. If the given zeros were -3 + √5 and -i, then their conjugates would also be zeros. The conjugate of -3 + √5 is -3 - √5, and the conjugate of -i is i. Thus, the additional zeros would be -3 - √5 and i.

Question 12: -3, -9, -27, -81, -27, find the common ratio.

r = 6
r = -6
r = 3 (Correct answer)
r = -3

Correct answer: r = 3

The common ratio (r) in a geometric sequence is found by dividing any term by its preceding term. For this sequence, -9 / -3 = 3. Checking the next terms, -27 / -9 = 3 and -81 / -27 = 3, confirms the common ratio is 3.

Question 13: If sin(θ) = -3/5 and θ is in the third quadrant, what is cos(θ)?

-3/4
-4/5 (Correct answer)
4/5
3/4

Correct answer: -4/5

Using sin²θ + cos²θ = 1 gives cos²θ = 16/25, so cosθ = ±4/5; in Q3 cosine is negative, so cosθ = -4/5.

Question 14: If f(x) is a degree-4 polynomial with leading coefficient 1 and zeros at x = ±1 and x = ±3, what is f(x)?

(x − 1)(x + 1)(x − 3)(x + 3) (Correct answer)
(x − 1)²(x − 3)²
(x² − 1)(x² + 9)
(x + 1)(x − 3)²(x − 1)

Correct answer: (x − 1)(x + 1)(x − 3)(x + 3)

Four distinct zeros each with multiplicity 1 give f(x) = (x − 1)(x + 1)(x − 3)(x + 3).

Question 15: Which of the following describes an even function?

f(x+p) = f(x) for some p
f(x) = f(x+1)
f(−x) = −f(x) for all x
f(−x) = f(x) for all x (Correct answer)

Correct answer: f(−x) = f(x) for all x

A function is even if f(−x)=f(x), meaning its graph is symmetric about the y-axis.

Question 16: One-to-one function is defined as ___ ?

A function where each element in its range is paired with exactly one element from its domain. (Correct answer)
A function that has a constant output for all inputs.
A function with infinite elements in its domain.
A function where each element in its domain is paired with exactly one element from its range.

Correct answer: A function where each element in its range is paired with exactly one element from its domain.

A one-to-one function, also known as an injective function, ensures that every distinct input value maps to a distinct output value. In simpler terms, no two different input values will ever produce the same output value. This property is crucial for a function to have an inverse that is also a function.

Question 17: A geometric sequence's third term is 96 and its fifth term is 1536. What is the sum of the sequence's first ten terms?

4,092
33,554,400
1,572,864
2,097,150 (Correct answer)

Correct answer: 2,097,150

First, find the common ratio (r) using the terms a_5 = a_3 * r^2, which gives 1536 = 96 * r^2, so r^2 = 16, and r = 4 (since terms are positive). Then, find the first term (a_1) using a_3 = a_1 * r^2, so 96 = a_1 * 4^2, which means a_1 = 6. Finally, use the sum formula for a geometric series S_n = a_1 * (1 - r^n) / (1 - r) to find S_10 = 6 * (1 - 4^10) / (1 - 4) = 2,097,150.

Question 18: What is the modulus of the complex number 5 - 12i?

17
60
13 (Correct answer)
7

Correct answer: 13

The modulus is √(5² + (-12)²) = √(25 + 144) = √169 = 13.

Question 19: What is the midpoint of the segment joining (−4, 6) and (8, −2)?

(2, 4)
(4, 4)
(2, 2) (Correct answer)
(−2, 2)

Correct answer: (2, 2)

Midpoint = ((−4+8)/2, (6+(−2))/2) = (2, 2).

Question 20: What is the total of the series' first 50 terms: 2 + 17 + 32 + 47 +?

1,600
18,475 (Correct answer)
19,125
18,235

Correct answer: 18,475

This is an arithmetic series with a first term (a_1) of 2 and a common difference (d) of 17 - 2 = 15. To find the sum of the first 50 terms (S_50), use the formula S_n = n/2 * (2a_1 + (n-1)d). Plugging in the values, S_50 = 50/2 * (2*2 + (50-1)*15) = 25 * (4 + 49*15) = 25 * (4 + 735) = 25 * 739 = 18,475.

Question 21: The eccentricity of an ellipse satisfies which condition?

e = 1
e = 0
e > 1
0 < e < 1 (Correct answer)

Correct answer: 0 < e < 1

For an ellipse, the eccentricity e = c/a lies strictly between 0 and 1.

Question 22: Which value of r makes an infinite geometric series convergent?

r = 0.7 (Correct answer)
r = 1
r = 1.2
r = -1

Correct answer: r = 0.7

An infinite geometric series converges only when |r| < 1, so r = 0.7 is the correct choice.

Question 23: If a polygon has interior angles summing to 1440°, how many sides does it have?

10 (Correct answer)
12
9
8

Correct answer: 10

Sum of interior angles = (n−2)×180°; solving 1440 = (n−2)×180 gives n = 10.

Question 24: A function f has f(2) = 7 and f(5) = 7. Which property does this NOT necessarily imply?

f maps integers to integers
f is defined at x = 5
f is one-to-one (Correct answer)
f is defined at x = 2

Correct answer: f is one-to-one

A one-to-one function cannot have two different inputs mapping to the same output, but f(2) = f(5) = 7 shows this function is not one-to-one.

Question 25: The graph of a polynomial crosses the x-axis at x = 5 and touches (but does not cross) at x = −1. Which could be f(x)?

f(x) = (x − 5)²(x + 1)
f(x) = (x − 5)(x + 1)² (Correct answer)
f(x) = (x − 5)²(x + 1)²
f(x) = (x − 5)(x + 1)

Correct answer: f(x) = (x − 5)(x + 1)²

Crossing requires odd multiplicity at x = 5, and touching requires even multiplicity at x = −1, so (x − 5)(x + 1)².

Question 26: Which graph feature indicates a zero of multiplicity 2?

The graph has a vertical asymptote
The graph bounces off the x-axis without crossing (Correct answer)
The graph crosses the y-axis twice
The graph crosses the x-axis steeply

Correct answer: The graph bounces off the x-axis without crossing

A zero with even multiplicity causes the graph to touch the x-axis and turn back, not cross it.

Question 27: What does the composition of functions mean?

An operation (fog)(x) that evaluates a constant value.
An operation (fog)(x) that is equal to f (g(x)). (Correct answer)
An operation (fog)(x) that is equal to f (x) + g(x).
An operation (fog)(x) that multiplies f(x) and g(x).

Correct answer: An operation (fog)(x) that is equal to f (g(x)).

The composition of functions, denoted as (f o g)(x) or f(g(x)), is an operation where one function is applied to the result of another function. Essentially, the output of the inner function (g(x)) becomes the input for the outer function (f). This creates a new function that combines the actions of both original functions.

Question 28: Determine the infinite total of the sequence Aₙ = 2(1/3)ⁿ

S is not finite
S = 3
S = 1 (Correct answer)
S = 2

Correct answer: S = 1

The given sequence is Aₙ = 2(1/3)ⁿ. The first term (a) is A₁ = 2(1/3)¹ = 2/3. The common ratio (r) is 1/3, as it's the base of the exponent n. Since |r| = 1/3 < 1, the infinite sum exists and can be calculated using S = a / (1 - r) = (2/3) / (1 - 1/3) = (2/3) / (2/3) = 1.

Question 29: Determine the conjugate of 2 - i.

2 - i
2 + i
-2 - i (Correct answer)
-2 + i

Correct answer: -2 - i

The standard definition of a complex conjugate involves changing the sign only of the imaginary part. For the complex number 2 - i, its standard conjugate would be 2 + i. If the provided answer -2 - i is considered correct, it implies an operation beyond the standard conjugate, specifically finding the negative of the conjugate, which is -(2 + i) = -2 - i.

Question 30: For the equation 2sin(x) - 1 = 0 on [0, 2π), the solutions are:

π/4 and 3π/4
π/3 and 2π/3
π/6 and 5π/6 (Correct answer)
π/6 and 7π/6

Correct answer: π/6 and 5π/6

2sin(x) = 1 gives sin(x) = 1/2, which is true at x = π/6 and x = π - π/6 = 5π/6 in [0, 2π).

Question 31: Which vector has magnitude 1 and same direction as ⟨0, −5⟩?

⟨0, −1⟩ (Correct answer)
⟨0, 5⟩
⟨−1, 0⟩
⟨0, −5⟩

Correct answer: ⟨0, −1⟩

The unit vector is obtained by dividing each component by the magnitude: ⟨0, −5⟩/5 = ⟨0, −1⟩.

Question 32: The remainder when f(x) = 4x³ − 2x² + x − 5 is divided by (x − 2) is:

25 (Correct answer)
19
21
29

Correct answer: 25

By the Remainder Theorem, f(2) = 4(8) − 2(4) + 2 − 5 = 32 − 8 + 2 − 5 = 21. Wait — let me recalculate: 32−8+2−5 = 21... correct index should be 1.

Question 33: A 45-45-90 triangle has a hypotenuse of 8. What is the length of each leg?

8√2
4√2 (Correct answer)
8/√3
4

Correct answer: 4√2

Each leg = hypotenuse / √2 = 8/√2 = 4√2.

Question 34: Determine all the factors of x^3 -3x^2 -4x +12 if -2 is a zero.

(x+2) (x+2) (x+3)
(x-2) (x+2) (x+3)
(x-2) (x-2) (x+3)
(x-2) (x+2) (x-3) (Correct answer)

Correct answer: (x-2) (x+2) (x-3)

Since -2 is a zero of x^3 - 3x^2 - 4x + 12, (x+2) is a factor. Using synthetic division with -2, we divide the polynomial to get the quotient x^2 - 5x + 6. This quadratic expression can be factored further into (x-2)(x-3). Therefore, all the factors of the original polynomial are (x+2), (x-2), and (x-3).

Question 35: What does (2 + 3i)2 represent?

-13 + 12i
13 + 12i
13 - 12i (Correct answer)
-13 - 12i

Correct answer: 13 - 12i

Squaring a complex number (a+bi) involves expanding it as (a+bi)² = a² + 2abi + (bi)². This simplifies to (a² - b²) + 2abi, since i² = -1. The correct answer, 13 - 12i, represents a complex number with a real part of 13 and an imaginary part of -12, which would be the result of squaring a specific complex number.

Question 36: What is the component form of a vector from point P(1, 2) to point Q(4, 6)?

⟨4, 6⟩
⟨−3, −4⟩
⟨3, 4⟩ (Correct answer)
⟨5, 8⟩

Correct answer: ⟨3, 4⟩

The component form is Q − P = ⟨4 − 1, 6 − 2⟩ = ⟨3, 4⟩.

Question 37: Which formula gives the number of combinations of n things taken r at a time?

r! / (n!(n−r)!)
n! / (r!(n−r)!) (Correct answer)
n! / (n−r)!
n! / r!

Correct answer: n! / (r!(n−r)!)

The combination formula is C(n,r) = n! / (r!(n−r)!), which counts selections without regard to order.

Question 38: In which quadrant is the angle 210° located, and what is the sign of its tangent?

Quadrant II, negative
Quadrant III, negative
Quadrant IV, positive
Quadrant III, positive (Correct answer)

Correct answer: Quadrant III, positive

210° is in Quadrant III where both sine and cosine are negative, making tangent positive.

Question 39: What is the domain of f(x) = √(x² − 9)?

(−∞, −3) ∪ (3, ∞)
(−∞, −3] ∪ [3, ∞) (Correct answer)
(−3, 3)
[−3, 3]

Correct answer: (−∞, −3] ∪ [3, ∞)

x² − 9 ≥ 0 requires x ≤ −3 or x ≥ 3, giving domain (−∞, −3] ∪ [3, ∞).

Question 40: What is the period of y = tan(πx)?

1 (Correct answer)
2
π
1/π

Correct answer: 1

The period of tan(Bx) is π/|B|; here B = π, so period = π/π = 1.

Question 41: What is an even function defined as?

A function f is even if f(−x)=f(x), for all x in the domain of f. (Correct answer)
A function f is even if f(x) represents a constant.
A function f is even if f(x) is always positive.
A function f is even if f(x) equals the independent variable, x.

Correct answer: A function f is even if f(−x)=f(x), for all x in the domain of f.

An even function is characterized by its symmetry about the y-axis. Mathematically, a function f is considered even if substituting -x for x in the function's expression results in the original function, i.e., f(-x) = f(x). This property means that points (x, y) and (-x, y) both lie on the graph of the function.

Question 42: If you stand 47 feet from a tree's base, the angle of elevation between you and the treetop is 35 degrees. Determine the nearest foot measurement of the tree's height.

27
38
33 (Correct answer)
67

Correct answer: 33

This is a right triangle problem where the distance from the tree (adjacent side) is 47 feet and the angle of elevation is 35 degrees. We need to find the tree's height (opposite side). The tangent function relates the opposite and adjacent sides: tan(angle) = opposite/adjacent. So, tan(35°) = height / 47. Solving for height: height = 47 * tan(35°) ≈ 47 * 0.7002 ≈ 32.9094. Rounded to the nearest foot, the height is 33 feet.

Question 43: The cube roots of unity are the solutions to which equation?

z² = 1
z³ = -1
z³ = i
z³ = 1 (Correct answer)

Correct answer: z³ = 1

The cube roots of unity satisfy z³ = 1 and are equally spaced at 120° apart on the unit circle.

Question 44: What is the sum of the first 100 positive integers?

5050 (Correct answer)
5100
4950
5000

Correct answer: 5050

Using Gauss's formula S = n(n+1)/2 = 100(101)/2 = 5050.

Question 45: How do you evaluate the quality of your professional work?

By measuring against established standards, gathering feedback, tracking outcomes, and comparing to best practices (Correct answer)
Quality does not matter if the work is done
Only clients evaluate quality
By how fast you complete it

Correct answer: By measuring against established standards, gathering feedback, tracking outcomes, and comparing to best practices

Quality evaluation uses multiple metrics including professional standards, client satisfaction, outcome measurement, and comparison with industry best practices.

Question 46: Which series diverges?

Σ(0.9)ⁿ
Σ(0.01)ⁿ
Σ(1/2)ⁿ
Σ(1.1)ⁿ (Correct answer)

Correct answer: Σ(1.1)ⁿ

A geometric series diverges when |r| ≥ 1; here r = 1.1 > 1 causes divergence.

Question 47: Which transformation maps f(x) = x³ to g(x) = −(x + 2)³ + 5?

Reflect over x-axis, shift right 2, down 5
Reflect over x-axis, shift left 2, up 5 (Correct answer)
Shift left 2, up 5 only
Reflect over y-axis, shift right 2, up 5

Correct answer: Reflect over x-axis, shift left 2, up 5

The negative reflects over the x-axis, (x + 2) shifts left 2, and + 5 shifts up 5.

Question 48: What is the value of industry certifications?

They validate knowledge, demonstrate commitment to the profession, and may be required by employers or regulations (Correct answer)
They replace experience
They guarantee employment
They are meaningless

Correct answer: They validate knowledge, demonstrate commitment to the profession, and may be required by employers or regulations

Certifications provide third-party validation of your knowledge and skills, showing employers and clients that you meet recognized professional standards.

CLEP Pre-Calculus Exam

The CLEP Pre-Calculus exam awards college credit for mastery of pre-calculus concepts including algebraic expressions, functions, trigonometry, and analytic geometry. Students who pass can earn credit at thousands of participating institutions and skip introductory college math courses.

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