Practice Exercises For Mathematics Placement Test - Test 2 ...

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Practice Exercises For Mathematics Placement Test - Test 2. (Corresponds to Precalculus Competency - Preparedness for M151). The Test 2 Placement exam is ...

Practice Exercises For Mathematics Placement Test - Test 2 (Corresponds to Precalculus Competency - Preparedness for M151) The Test 2 Placement exam is a multiple choice exam covering topics typically found in a Precalculus course. Passing the exam means that you are prepared to take M151 - Calculus I. Below are practice exercises to review some of the material required for the exam. Answers to the practice exercises can be found on the last pages. Major Topics: A student taking the exam should be prepared to ∙ Factor and simplify polynomial, rational, radical and absolute value expressions. ∙ Recognize and graph equations of circles and parabolas (may need completing the square). ∙ Solve equations and inequalities involving absolute values, polynomial, and rational expressions. ∙ Work with function notation and function operations (including composition and the difference quotient). ∙ Work with quadratic and rational functions, as well as inverse functions. ∙ Graph polynomial, rational, exponential, and logarithmic functions. ∙ Use the Factor Theorem, the Remainder Theorem, and polynomial division (either long division or synthetic division) to find integer, rational, and irrational roots of polynomial equations. ∙ Solve exponential and logarithmic equations. ∙ Solve and graph systems of linear equations and inequalities in two or three variables. ∙ Know the unit-circle definition of trigonometric functions. ∙ Know right triangle trigonometry (opposite, adjacent, and hypotenuse). ∙ Evaluate trigonometric and inverse trigonometric expressions. ∙ Change radian measure to degree measure and vice versa. ∙ Graph trigonometric functions, including transformations. ∙ Know and use fundamental trigonometric identities to simplify expressions. ∙ Verify trigonometric identities. ∙ Solve trigonometric equations.

Revised May 16, 2010

1. Find all solutions to the following equations. (a) 𝑥2 + 3𝑥 − 7 = 0

(b)

𝑥2 − 5𝑥 − 6 =0 𝑥+2

(c)



4𝑥 − 3 + 2 = 7

(d) ∣𝑥 + 1∣ = 6

2. Determine the set of all solutions for each, graphing inequalities on a number line. (a) ∣𝑥 + 3∣ < 2

(b) ∣3𝑥 − 7∣ ≥ 1

(d) (2𝑥 − 5)(𝑥 + 3) > 0

(e)

(c) (𝑥 − 3)(𝑥 + 1) ≤ 0

(𝑥 + 2)(𝑥 − 1) ≤0 𝑥−3

(f)

1 2 > 𝑥+1 𝑥

3. Graph and state the coordinates of the vertex of each. (a) 𝑦 = 3(𝑥 − 5)2 + 2

(b) 𝑦 = 𝑥2 − 2𝑥 − 2

4. If 𝑓 (𝑥) = 2𝑥2 − 𝑥 and 𝑔(𝑥) = 𝑥 + 3, evaluate each of the following. (a) 𝑓 (5)

(b) (𝑓 + 𝑔)(𝑥)

(e) 𝑔(−3)

(f) (𝑓 − 𝑔)(3)

(i)

𝑓 (2 + ℎ) − 𝑓 (2) ℎ

(j)

(d) (𝑓 ∘ 𝑔)(𝑥)

(c) (𝑓 𝑔)(𝑥) ( ) 𝑓 (1) (g) 𝑔

(h) (𝑔 ∘ 𝑓 )(4)

𝑔(𝑥 + ℎ) − 𝑔(𝑥) ℎ

5. State the domain and range of each; then graph each. (a) 𝑓 (𝑥) =



7𝑥 + 4

(b) 𝑔(𝑥) = 𝑥3 + 1

6. Determine the inverse functions of 𝑓 (𝑥) =

(c) ℎ(𝑥) = ∣𝑥 − 2∣ + 1

1 . 𝑥−1

7. Graph each. (a) 𝑦 ≤ 3𝑥 + 2

(b) 𝑦 > 𝑥2

8. Find the set of solutions for each systems of equations. ⎧ { { ⎨ 2𝑥 − 𝑦 + 𝑧 = 1 2𝑥 − 𝑦 = −1 𝑦 = 𝑥2 + 1 𝑥 − 𝑦 + 2𝑧 = 3 (a) (b) (c) 𝑥+𝑦 =7 𝑦 = 3𝑥 + 5 ⎩ 𝑥−𝑦+𝑧 =1 9. Write an equation of the circle with center at the origin and radius of 3. 10. Determine the center and radius of the circle (𝑥 − 1)2 + (𝑦 + 2)2 = 4. 11. Determine the center and radius of the circle 𝑥2 + 𝑦 2 − 4𝑦 = 12. 12. Determine whether 𝑥 − 2 is a factor of 𝑥3 + 4𝑥2 − 2𝑥 + 4. [Use of the Factor Theorem would be a quick way to do this.] 13. Find the quotient and remainder if 3𝑥4 − 2𝑥2 − 3𝑥 + 7 is divided by 𝑥 + 1. [Use of synthetic division would be a quick way to do this.] 14. Based on the coefficient of the 𝑥4 term and the constant term, state the candidates for rational roots of 3𝑥4 + 2𝑥3 − 4𝑥 + 2 = 0.

15. Find all real solutions (giving exact values) of 2𝑥4 − 2𝑥3 − 11𝑥2 − 4𝑥 + 3 = 0. 16. Graph each. (b) 𝑦 = 2−𝑥

(a) 𝑦 = 2𝑥

(c) 𝑦 = log2 𝑥

17. Find all solutions for each. (c) 5𝑥 = 125

(a) log2 8 = 𝑥

(b) log7 𝑥 = 0

(d) 43𝑥−1 = 3𝑥−2

(e) log(𝑥 + 1) − log(𝑥) = log 4

18. Graph each. (a) 𝑓 (𝑥) =

3𝑥 − 2 𝑥+3

(b) 𝑔(𝑥) =

2𝑥 𝑥2 − 4

19. Find the coefficient of 𝑥3 𝑦. (a) (𝑥 + 𝑦)4

(b) (3𝑥 − 𝑦)4

20. Use the unit-circle definition of the trigonometric functions to obtain the values of each, if 𝑡 is as indicated in the figure to the right. (a) sin 𝑡

(b) cos 𝑡

𝑦

(c) tan 𝑡

𝑡 (𝑎, 𝑏) (d) cot 𝑡

(e) sec 𝑡

(f) csc 𝑡 1 𝑥

21. If sin 𝜃 = −

3 and 𝜃 is in quadrant III, determine each. 5

(a) cos 𝜃

(b) tan 𝜃

(c) cot 𝜃

(d) sec 𝜃

(e) csc 𝜃

22. Given 𝜃 as indicated in the figure at the right, determine each of the following. (a) sin 𝜃

(b) cos 𝜃

(c) tan 𝜃

(d) cot 𝜃

(e) sec 𝜃

(f) csc 𝜃

13 5 𝜃 12

23. State the values of the following. (𝜋) (a) sin (30∘ ) (b) cos 6 (𝜋) (𝜋) (f) tan (g) sin 4 3

(c) tan (30∘ ) (𝜋) (h) cos 3

(d) sin

(𝜋) 4

(i) tan (60∘ )

(e) cos (45∘ )

24. Write each in terms of a trigonometric function of an angle in the first quadrant. (a) sin (150∘ )

(b) cos (225∘ )

(c) tan (300∘ )

25. Change each radian measure to degree measure. 𝜋 3

𝜋 2

7𝜋 6 ( ) ( ) 3𝜋 3𝜋 26. Find the exact value of sin2 + cos2 . 7 7 (a)

(b)

(c)

3𝜋 4

(d)

(e) 𝜋

27. Graph each. (a) 𝑦 = sin 𝑥

(b) 𝑦 = cos 𝑥

(c) 𝑦 = tan 𝑥

3 cos 𝑥 2

(d) 𝑦 =

(e) 𝑦 = sin 3𝑥

( 𝜋) . 28. Find the period and amplitude of the graph of 𝑦 = 3 cos 2𝑥 − 4 29. Reduce each to a single function of the argument 𝜃. (b) sec 𝜃 − sin 𝜃 tan 𝜃

(a) cos 𝜃 csc 𝜃 √ 1 2 2 30. If sin 𝜃 = and cos 𝜃 = , find sin 2𝜃. 3 3 31. If tan 𝜙 =

4 , find cos 2𝜙. 3

32. Find the set of solutions in the interval 0 ≤ 𝑥 ≤ 2𝜋 for each equation. (a) sin 𝑥 =

1 2

(d) sin 2𝑥 + 1 = 0

(b) 2 cos 𝑥 + 1 = 2

(c) tan 𝑥 = 1

(e) 2 cos2 𝑥 − sin 𝑥 = 1

(f) tan2 𝑥 − 3 = 0

33. Let ∠𝐶 be the right angle of triangle △𝐴𝐵𝐶 . (a) If the length of the side opposite ∠𝐴 is 6 centimeters and the measure of ∠𝐵 is 60∘ , find the length of the hypotenuse. (b) If the measure of ∠𝐴 is 45∘ and the length of the side opposite ∠𝐵 is 25 feet, find the length of the side opposite ∠𝐴 . 34. Find the value of each. ( ) 1 (a) sin−1 − 2

(b) cos(tan−1



3)

You should know the following identities.

Logarithmic Identities

Exponent Identities

log(𝐴𝐵) = log 𝐴 + log 𝐵

𝐴𝑚 𝐴𝑛 = 𝐴𝑚+𝑛

( log

𝐴 𝐵

) = log 𝐴 − log 𝐵

log(𝐴𝑛 ) = 𝑛 log 𝐴

𝐴𝑚 = 𝐴𝑚−𝑛 𝐴𝑛 (𝐴𝑚 )𝑛 = 𝐴𝑚𝑛 (𝐴𝐵)𝑛 = 𝐴𝑛 𝐵 𝑛 (

𝐴 𝐵

)𝑛 =

𝐴𝑛 𝐵𝑛

Trigonometric Identities sin2 𝑥 + cos2 𝑥 = 1

sin(2𝑥) = 2 sin 𝑥 cos 𝑥

1 + tan2 𝑥 = sec2 𝑥

cos(2𝑥) = cos2 𝑥 − sin2 𝑥

1 + cot2 𝑥 = csc2 𝑥

sin(−𝑥) = − sin 𝑥

tan 𝑥 =

sin 𝑥 cos 𝑥

cos(−𝑥) = cos 𝑥

cot 𝑥 =

cos 𝑥 1 = sin 𝑥 tan 𝑥

tan(−𝑥) = − tan 𝑥

sec 𝑥 =

1 cos 𝑥

csc 𝑥 =

1 sin 𝑥

Answers For Test 2 √ −3 ± 37 1. (a) 𝑥 = ≈ 1.54, −4.54 2

(b) 𝑥 = 6, −1

2. (a) −5 < 𝑥 < −1 −1

0

0

(c) −1 ≤ 𝑥 ≤ 3

0

−3

3

−2

0

1

−2 −1

3

(b) 2𝑥2 + 3 (g)

(c) 2𝑥3 + 5𝑥2 − 3𝑥

0

1 4

(h) 31

(d) 2(𝑥 + 3)2 − (𝑥 + 3) = 2𝑥2 + 11𝑥 + 15

(i) 7 + 2ℎ

(j) 1

5. (a) Domain: [−4/3, ∞), Range: [0, ∞) (b) Domain: all real numbers (ℝ), Range: all real numbers (ℝ) (c) Domain: all real numbers (ℝ), Range: [1, ∞) 1 𝑥

5 2

(b) Vertex (1, −3)

3. (a) Vertex (5, 2)

6. 𝑓 −1 (𝑥) = 1 +

0

(f) 𝑥 < −2 or −1 < 𝑥 < 0

(e) 𝑥 ≤ −2 or 1 ≤ 𝑥 < 3

(f) 9

8 3

2

(d) 𝑥 < −3 or 𝑥 > 5/2

−1

(e) 0

(d) 𝑥 = 5, −7

(b) 𝑥 ≤ 2 or 𝑥 ≥ 8/3

−5

4. (a) 45

(c) 𝑥 = 7

7. (a)

(b)

8. (a) 𝑥 = 2, 𝑦 = 5

(b) 𝑥 = 0, 𝑦 = 1, 𝑧 = 2

(c) 𝑥 = 4, 𝑦 = 17 and 𝑥 = −1, 𝑦 = 2

9. 𝑥2 + 𝑦 2 = 9 10. Center (1, −2), radius 2 11. Center (0, 2), radius 4 12. Not a factor. 13. Quotient: 3𝑥3 − 3𝑥2 + 𝑥 − 4, Remainder: 11 2 1 14. ±2, ± ± 1, ± 3 3 15. 𝑥 = −1, 3,

√ √ −2 ± 12 −1 ± 3 = 4 2

16. (a)

17. (a) 𝑥 = 3

(b)

(b) 𝑥 = 1

(c) 𝑥 = 3

(c)

(d) 𝑥 =

ln 94 ln 4 − 2 ln 3 = 3 ln 4 − ln 3 ln 64 3

(e) 𝑥 =

1 3

18. (a)

(b)

3 −2

−3

(b) −108

19. (a) 4 20. (a) 𝑏 21. (a) − 22. (a)

2

(b) 𝑎 4 5

5 13

1 23. (a) 2

(b)

𝑏 𝑎

3 4

12 13 √ 3 (b) 2 (b)

24. (a) sin 30∘ 25. (a) 60∘

(c)

(d) 4 3

(c) (c)

5 12 √

3 (c) 3

(b) − cos 45∘ (b) 90∘

(c) 210∘

𝑎 𝑏

(e)

(d) − (d)

1 𝑎

5 4

12 5 √

1 𝑏

(f) (e) − (e)

2 (d) 2

5 3

13 12

(f)

5 13



2 (e) 2

√ (f) 1

(g)

3 2

(h)

1 2

(i)



3

(c) − tan 60∘ (d) 135∘

(e) 180∘

26. 1 27. (a) −𝜋

(b) −𝜋

(c)

1 −1

𝜋

√ 4 2 30. 9

(b) cos 𝜃

−𝜋 − 𝜋2

𝜋

28. Amplitude = 3, Period = 𝜋 29. (a) cot 𝜃

1.5 𝜋 −1.5

1 −1

(d)

𝜋 2

(e) − 𝜋3

1 −1

𝜋 3

31. −

7 25

32. (a)

𝜋 5𝜋 , 6 6

33. (a) 12 cm 34. (a) −

𝜋 6

(b)

𝜋 5𝜋 , 3 3

(b) 25 ft (b)

1 2

(c)

𝜋 5𝜋 , 4 4

(d)

3𝜋 7𝜋 , 4 4

(e)

𝜋 5𝜋 3𝜋 , , 6 6 2

(f)

𝜋 2𝜋 4𝜋 5𝜋 , , , 3 3 3 3

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