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AP Precalculus Full-length practice exam 1

40 questions

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Full-length practice exam 1

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Question 1

A parametric curve is given by $x (t) = 2 cos (t)$ , $y (t) = 2 sin (t)$ for $t \in [0, 2 π)$ . Which curve does it trace?

A.A circle of radius

1

centered at the origin.B.A circle of radius

2

centered at the origin.C.An ellipse with semi-axes

2

and

1

.D.A line of slope

1

through the origin.

Question 2

Find all $θ \in [0, 2 π)$ that satisfy $2 cos^{2} (θ) - cos (θ) - 1 = 0$ .

θ = 0, π /2, π

θ = 0, 2 π /3, 4 π /3

θ = π /3, 5 π /3

θ = π

2 π /3

4 π /3

Question 3

What is the unit vector in the direction of $w = ⟨ 6, - 8 ⟩$ ?

⟨ 6, - 8 ⟩

⟨ \frac{3}{5}, - \frac{4}{5} ⟩

⟨ \frac{6}{10}, \frac{- 8}{10} ⟩ \cdot 2

⟨ \frac{1}{6}, - \frac{1}{8} ⟩

Question 4

Eliminate the parameter from $x = t + 1$ , $y = t^{2}$ to express $y$ as a function of $x$ .

y = x^{2}

y = (x - 1)^{2}

y = (x + 1)^{2}

y = x^{2} - 1

Question 5

What is the period of $f (θ) = sin (θ)$ ?

π /2

π

2 π

4 π

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Question 6

Convert the rectangular point $(x, y) = (1, 3)$ to polar form $(r, θ)$ with $r > 0$ and $θ \in [0, 2 π)$ .

(2, π /6)

(2, π /3)

(4, π /3)

(2, 2 π /3)

Question 7

Compute the dot product of $u = ⟨ 2, 3 ⟩$ and $v = ⟨ 4, - 1 ⟩$ .

5

11

⟨ 8, - 3 ⟩

2

Question 8

If $sin (θ) = \frac{3}{5}$ and $θ$ is in Quadrant II, what is $cos (θ)$ ?

\frac{4}{5}

- \frac{4}{5}

\frac{3}{5}

- \frac{3}{5}

Question 9

Which best describes the behavior of $f (x) = \frac{x ^{2} - 1}{x - 1}$ at $x = 1$ ?

A.Vertical asymptote at

x = 1

.B.Hole (removable discontinuity) at

x = 1

.C.

x

-intercept at

x = 1

.D.Horizontal asymptote at

x = 1

Question 10

Solve the system using matrix methods: $2 x + y = 5$ and $x - y = 1$ .

x = 1

y = 2

x = 2

y = 1

x = 3

y = - 1

x = 0

y = 5

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Question 11

What is the inverse of $f (x) = 3^{x}$ ?

f^{- 1} (x) = \frac{1}{3 ^{x}}

f^{- 1} (x) = lo g_{3} (x)

f^{- 1} (x) = x^{1/3}

f^{- 1} (x) = ln (x)

Question 12

A linear function $g$ has constant rate of change $2$ , and a quadratic function $q (x) = x^{2}$ has rate of change that varies with $x$ . Over the interval $[1, 5]$ , what is the average rate of change of $q$ , and how does it compare to $g$ 's rate?

A.Average rate of

q

4

; faster than

g

's rate.B.Average rate of

q

6

; faster than

g

's rate.C.Average rate of

q

6

; slower than

g

's rate.D.Average rate of

q

24

; faster than

g

's rate.

Question 13

Compute $u + v$ where $u = ⟨ 2, - 1 ⟩$ and $v = ⟨ - 3, 4 ⟩$ .

⟨ - 1, 3 ⟩

⟨ 5, - 5 ⟩

⟨ - 6, - 4 ⟩

⟨ 1, 5 ⟩

Question 14

As $x \to \infty$ , what is the end behavior of $f (x) = - 2 x^{3} + 5 x^{2} - x + 4$ ?

f (x) \to \infty

f (x) \to - \infty

f (x) \to 4

f (x) \to 0

Question 15

An investment of $\$ 2000 $g r o w sco n t in u o u s l y a t anann u a l r a t eo f$ 6% $. A ccor d in g t o$ A(t) = 2000 \cdot e^{0.06 t} $, a pp r o x ima t e l y h o w l o n g u n t i l i t d o u b l es ? (U se$ \ln 2 \approx 0.693$.)

A.About

7.7

yearsB.About

11.6

yearsC.About

16.7

yearsD.About

33.3

years

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Question 16

Simplify $lo g_{5} (25) + lo g_{5} (125)$ using log properties.

3

5

lo g_{5} (150)

lo g_{5} (250)

Question 17

A polynomial $f (x)$ has degree $3$ , real coefficients, and zeros at $x = - 2$ , $x = 1$ , and $x = 4$ . If $f (0) = 16$ , what is $f (x)$ ?

f (x) = (x + 2) (x - 1) (x - 4)

f (x) = 2 (x + 2) (x - 1) (x - 4)

f (x) = (x - 2) (x + 1) (x + 4)

f (x) = 4 (x + 2) (x - 1) (x - 4)

Question 18

Identify the vertical asymptote(s) of $f (x) = \frac{x + 3}{x ^{2} - 4}$ .

x = - 3

onlyB.

x = 2

onlyC.

x = 2

and

x = - 2

x = \pm 3

Question 19

Find the magnitude of the vector $v = ⟨ 3, 4 ⟩$ .

3

4

5

7

Question 20

Let $f (x) = \frac{2 x ^{2} + 3 x + 1}{x ^{2} - 1}$ . Identify all asymptotes and any holes.

A.Vertical asymptotes at

x = \pm 1

; horizontal asymptote

y = 2

.B.Hole at

x = - 1

; vertical asymptote

x = 1

; horizontal asymptote

y = 2

.C.Hole at

x = 1

; vertical asymptote

x = - 1

; horizontal asymptote

y = 2

.D.Vertical asymptotes at

x = \pm 1

; no horizontal asymptote.

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Question 21

Simplify $\frac{sin ( 2 θ )}{sin ( θ )}$ for $sin (θ) \neq = 0$ .

2

cos (θ)

2 cos (θ)

sin (θ)

Question 22

If $f (x) = e^{x}$ and $g (x) = ln (x)$ , what is $f (g (7))$ ?

0

1

7

e^{7}

Question 23

Evaluate the logarithm $lo g_{2} 16$ without using a calculator.

2

4

8

16

Question 24

Find the inverse of the matrix $A = [2111]$ .

[1 - 1 - 1 2]

[- 1 1 1 - 2]

[2 - 1 - 1 1]

[1121]

Question 25

The amplitude of $f (θ) = 3 sin (2 θ)$ is which of the following?

1

2

3

6

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Question 26

Find the quotient when $f (x) = x^{3} + 2 x^{2} - 5 x - 6$ is divided by $x - 2$ .

x^{2} + 4 x + 3

x^{2} - 4 x + 3

x^{2} + 2 x - 3

x^{2} + 4 x + 5

Question 27

A particle moves along the parametric curve $x (t) = t + 1$ , $y (t) = 2 t - 3$ . What is the position at $t = 2$ ?

(2, - 3)

(3, 1)

(2, 1)

(1, 2)

Question 28

What is the horizontal asymptote of $f (x) = \frac{3 x ^{2} + 7}{2 x ^{2} - x + 5}$ ?

y = 0

y = \frac{2}{3}

y = \frac{3}{2}

D.No horizontal asymptote.

Question 29

A projectile is launched with initial velocity $v_{0} = ⟨ 20, 30 ⟩$ m/s. Ignoring gravity, what is its position vector at $t = 4$ seconds, starting from the origin?

⟨ 24, 34 ⟩

mB.

⟨ 80, 120 ⟩

mC.

⟨ 20, 30 ⟩

mD.

⟨ 5, 7.5 ⟩

Question 30

Solve $ln (x + 2) - ln (x) = ln (3)$ for $x > 0$ .

x = - 1

x = 1

x = 2

x = 3

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Question 31

Solve $2^{x + 1} = 16$ for $x$ .

x = 2

x = 3

x = 4

x = 7

Question 32

What is the range of $g (θ) = - 2 sin (θ) + 1$ ?

[- 1, 1]

[- 1, 3]

[- 3, 1]

[1, 3]

Question 33

What is the degree of the polynomial $f (x) = 3 x^{4} - 2 x^{3} + 5 x - 7$ ?

3

4

5

7

Question 34

A radioactive substance decays according to $A (t) = A_{0} \cdot (1/2)^{t / H}$ where $H$ is the half-life. If $A_{0} = 80$ grams and $H = 5$ days, how many grams remain after $15$ days?

5

gB.

10

gC.

20

gD.

40

Question 35

What is the period of $f (θ) = cos (4 θ)$ ?

\frac{π}{2}

π

2 π

8 π

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Question 36

The polynomial $f (x) = (x - 2)^{2} (x + 1)$ has a zero at $x = 2$ . Which best describes the graph of $f$ at $x = 2$ ?

A.Crosses the

x

-axis at

x = 2

.B.Touches the

x

-axis and turns around at

x = 2

.C.Has a vertical asymptote at

x = 2

.D.Is undefined at

x = 2

Question 37

What are the real zeros of $f (x) = x^{2} - 9$ ?

x = 3

onlyB.

x = \pm 3

x = \pm 9

x = 0

Question 38

The parametric curve $x (t) = cos (t)$ , $y (t) = sin (2 t)$ has what behavior at $t = 0$ ?

A.Passes through

(1, 0)

moving in the

- y

direction.B.Passes through

(1, 0)

moving in the

+ y

direction.C.Passes through

(0, 0)

moving in the

+ x

direction.D.Passes through

(1, 0)

stationary.

Question 39

A bacterial culture grows according to $N (t) = 500 \cdot 2^{t /3}$ where $t$ is in hours. By what factor does the population grow every $3$ hours?

A.Adds

500

.B.Doubles.C.Triples.D.Multiplies by

2^{3} = 8

Question 40

What is the exact value of $cos (\frac{π}{3})$ ?

\frac{1}{2}

\frac{2}{2}

\frac{3}{2}

0

AP Precalculus Full-length practice exam 1 — Free with Answer Explanations | Test Practice Hub