MATHEMATICS Leveled Exam 1: Easy

1. Find the discriminant of the quadratic equation 5𝑥𝑥2 + 7𝑥𝑥 = 10.

a. 249 b. 350 c. 49 d. 200 e. 560

2. Which of the following equations illustrates the associative law of multiplication?

a. 5 × (3 + 7) = 5 × 3 + 5 × 7 b. (40 + 20) + 5 = 40 + (20 + 5) c. 25 × 4 = 4 × 25 d. (5 × 3) × 7 = 5 × (3 × 7) e. 6 + 7 = 7 + 6

3. Solve the inequality: −16 ≤ 5𝑥𝑥 − 6 ≤ 9.

a. (0,4) b. (−2,3) c. (−1,4) d. [−1, 2] e. [−2, 3]

4. Evaluate 4cot2 30°

+ 1sin2 60°

− cos2 45°.

a. 1213

b. 1312

c. 16

d. 136

e. 13

5. Find the range of the function 𝑓𝑓(𝑥𝑥) = 3𝑥𝑥 − 4 where 𝑥𝑥𝑥𝑥{−1,0,1,3,4}.

a. {−7,−4,−1, 5, 8} b. {−8,−4, 1, 2, 5} c. {−8,−5,−2,0,1, 4, 7} d. { 4, 7} e. all real numbers

MATHEMATICS LEVELED EXAM 1 PAGE 2 DEMIDEC ©2016

6. Simplify (5𝑥𝑥5 + 7𝑥𝑥3 + 2𝑥𝑥2 + 3𝑥𝑥 + 9) + (4𝑥𝑥4 + 2𝑥𝑥3 + 9𝑥𝑥2 + 3𝑥𝑥).

a. 5𝑥𝑥5 + 𝑥𝑥4 + 11𝑥𝑥3 + 11𝑥𝑥2 + 6𝑥𝑥 + 6 b. 7𝑥𝑥5 + 3𝑥𝑥4 + 9𝑥𝑥3 + 9𝑥𝑥2 + 6𝑥𝑥 + 6 c. 𝑥𝑥5 + 3𝑥𝑥4 + 11𝑥𝑥3 + 9𝑥𝑥2 + 6𝑥𝑥 + 9 d. 5𝑥𝑥5 + 4𝑥𝑥4 + 11𝑥𝑥3 + 9𝑥𝑥2 + 9𝑥𝑥 + 9 e. 5𝑥𝑥5 + 4𝑥𝑥4 + 9𝑥𝑥3 + 11𝑥𝑥2 + 6𝑥𝑥 + 9

7. Find the domain of the function 𝑓𝑓(𝑥𝑥) = arcsin(2𝑥𝑥 − 5).

a. (2, 3) b. [2, 3] c. [2, 5] d. (3, 5) e. [3, 5]

8. If sin𝜃𝜃 = 𝑝𝑝2−𝑞𝑞2

𝑝𝑝2+𝑞𝑞2, cos 𝜃𝜃 =

a. 2𝑝𝑝𝑝𝑝2−𝑞𝑞2

b. 2𝑝𝑝𝑝𝑝2+𝑞𝑞2

c. 2𝑝𝑝𝑞𝑞𝑝𝑝2−𝑞𝑞2

d. 𝑝𝑝2+𝑞𝑞2

𝑝𝑝2−𝑞𝑞2

e. 2𝑝𝑝𝑞𝑞𝑝𝑝2+𝑞𝑞2

9. If sin𝜃𝜃 = 𝑝𝑝2−𝑞𝑞2

𝑝𝑝2+𝑞𝑞2, cot 𝜃𝜃 =

a. 2𝑝𝑝𝑝𝑝2−𝑞𝑞2

b. 2𝑝𝑝𝑞𝑞𝑝𝑝2+𝑞𝑞2

c. 2𝑝𝑝𝑞𝑞𝑝𝑝2−𝑞𝑞2

d. 𝑝𝑝2+𝑞𝑞2

𝑝𝑝2−𝑞𝑞2

e. 𝑝𝑝2−𝑞𝑞2

2𝑝𝑝𝑞𝑞

10. Solve 2𝑥𝑥20−𝑥𝑥

= 43 for 𝑥𝑥.

a. 8 b. 5 c. 1 d. 4 e. 3

MATHEMATICS LEVELED EXAM 1 PAGE 3 DEMIDEC ©2016

11. Solve log(5𝑥𝑥 − 4) − log(𝑥𝑥 + 1) = log 4 for 𝑥𝑥.

a. 1 b. 8 c. 3 d. 5 e. 2

12. Solve 64𝑥𝑥 = 1642𝑥𝑥

for 𝑥𝑥.

a. 23

b. 12

c. 15

d. 25

e. 13

13. Find the y-intercept of 𝑓𝑓(𝑥𝑥) = 7𝑥𝑥2 − 8𝑥𝑥 + 5.

a. (0, 5) b. (0,−8) c. (0, 8) d. (0, 7) e. (0,−5)

14. Find the radian measure of 520°.

a. 16π5

b. 21π9

c. 26π9

d. 32π7

e. 35π6

15. Solve (𝑥𝑥+2)𝑥𝑥

− (𝑥𝑥−1)2𝑥𝑥

= 1 for 𝑥𝑥.

a. 6 b. 4 c. 1 d. 2 e. 5

MATHEMATICS LEVELED EXAM 1 PAGE 4 DEMIDEC ©2016

16. Find the horizontal asymptote of 𝑓𝑓(𝑥𝑥) = �12�2𝑥𝑥− 12.

a. 6 b. −6 c. 12 d. −12 e. −1

17. Find the solution of tan 3𝜃𝜃4

= 0.

a. 𝜃𝜃 = 2𝑛𝑛𝑛𝑛3

, 𝑛𝑛𝑥𝑥Ζ

b. 𝜃𝜃 = 4𝑛𝑛𝑛𝑛3

, 𝑛𝑛𝑥𝑥Ζ c. 𝜃𝜃 = 𝑛𝑛𝑛𝑛

3, 𝑛𝑛𝑥𝑥Ζ

d. 𝜃𝜃 = 3𝑛𝑛𝑛𝑛2

, 𝑛𝑛𝑥𝑥Ζ

e. 𝜃𝜃 = 3𝑛𝑛𝑛𝑛4

, 𝑛𝑛𝑥𝑥Ζ

18. Find the centroid of the triangle whose vertices are (1,6), (−1 − 2) and (3,−1).

a. (5,4) b. (2,5) c. (4,1) d. (3,2) e. (1,1)

19. Find the equation of the reflection of 𝑓𝑓(𝑥𝑥) = −9𝑥𝑥2 + 5𝑥𝑥 − 7 across the x-axis.

a. 9𝑥𝑥2 + 5𝑥𝑥 + 7 b. 9𝑥𝑥2 + 5𝑥𝑥 − 7 c. −9𝑥𝑥2 + 5𝑥𝑥 − 7 d. −9𝑥𝑥2 − 5𝑥𝑥 + 7 e. 9𝑥𝑥2 − 5𝑥𝑥 + 7

20. Solve: 𝑥𝑥2 − 144 < 0.

a. −4 ≤ 𝑥𝑥 < 4 b. −2 < 𝑥𝑥 < 4 c. −12 < 𝑥𝑥 < 12 d. −12 ≤ 𝑥𝑥 ≤ 12 e. −8 < 𝑥𝑥 ≤ 8

MATHEMATICS LEVELED EXAM 1 PAGE 5 DEMIDEC ©2016

21. Solve: cos2 66° − sin2 6°.

a. √5−18

b. √3−12

c. √5−34

d. 18

e. −12

22. Find the amplitude of the trigonometric function 𝑓𝑓(𝑥𝑥) = −2 sin �34𝑥𝑥�.

a. 2 b. 4 c. 3 d. −2 e. −1

23. Find the value of cos 210°.

a. √34

b. −12

c. −√32

d. −√35

e. 14

24. Find the distance AB where A (7, 8) and B (4, 4).

a. 10 units b. 8 units c. 4 units d. 6 units e. 5 units

25. Solve 𝑥𝑥4 − 61𝑥𝑥2 + 900 = 0 for x.

a. 𝑥𝑥 = ±3 ; 𝑥𝑥 = ±4 b. 𝑥𝑥 = ±5 ; 𝑥𝑥 = ±6 c. 𝑥𝑥 = ±2 ; 𝑥𝑥 = ±5 d. 𝑥𝑥 = ±4 ; 𝑥𝑥 = ±2 e. 𝑥𝑥 = ±7 ; 𝑥𝑥 = ±6

MATHEMATICS LEVELED EXAM 1 PAGE 6 DEMIDEC ©2016

26. 𝑓𝑓(𝑥𝑥) = 𝑘𝑘𝑥𝑥3 + 9𝑥𝑥2 + 4𝑥𝑥 − 10 has remainder 2 when divided by (𝑥𝑥 + 1). Find 𝑘𝑘.

a. −3 b. −7 c. 5 d. −1 e. 3

27. Find the value of cos(90°+𝜃𝜃) sec(−𝜃𝜃) tan(180°−𝜃𝜃)sec(360°−𝜃𝜃) sin(180°+𝜃𝜃) cot(90°−𝜃𝜃)

.

a. 5 b. 2 c. −3 d. −1 e. 1

28. Solve √𝑥𝑥 + √9𝑥𝑥 = 12 for 𝑥𝑥.

a. 10 b. 16 c. 12 d. 9 e. 8

29. If 𝑧𝑧 = 8 − 6𝑖𝑖, find the value of |𝑧𝑧|.

a. 10 b. 7 c. 12 d. 5 e. 8

30. Find the domain of the real valued function 𝑓𝑓(𝑥𝑥) = 𝑥𝑥−1𝑥𝑥−2

.

a. all real numbers except 3 b. all real numbers except 1 c. all real numbers except 1 and 2 d. all real numbers e. all real numbers except 2

31. Solve sin𝜃𝜃−sin3𝜃𝜃sin2 𝜃𝜃−cos2 𝜃𝜃

.

a. cos 𝜃𝜃 b. 2 sin𝜃𝜃 c. 2 cos 𝜃𝜃 d. sin𝜃𝜃 e. 2 tan𝜃𝜃

MATHEMATICS LEVELED EXAM 1 PAGE 7 DEMIDEC ©2016

32. If (2𝑥𝑥 + 1) is a factor of 6𝑥𝑥3 + 5𝑥𝑥2 + 𝑘𝑘𝑥𝑥 − 2, find 𝑘𝑘.

a. −5 b. 3 c. 5 d. −3 e. −1

33. Fill in the missing term: (𝑥𝑥 + 4)(4𝑥𝑥3 + 2𝑥𝑥2 + 5) = 4𝑥𝑥4 + 18𝑥𝑥3 + _____ + 5𝑥𝑥 + 20.

a. 10𝑥𝑥2 b. 16𝑥𝑥2 c. 5𝑥𝑥2 d. 2𝑥𝑥2 e. 8𝑥𝑥2

34. If cos𝛼𝛼 = −12 and 𝜋𝜋 < 𝛼𝛼 < 3𝑛𝑛

2, find the value of 4 tan2 𝛼𝛼 − 3 csc2 𝛼𝛼.

a. 2 b. 8 c. 5 d. 7 e. 3

35. Express (1 − 𝑖𝑖)4 in the form of 𝑎𝑎 + 𝑏𝑏𝑖𝑖.

a. 4 + 1𝑖𝑖 b. −4 + 1𝑖𝑖 c. −4 + 0𝑖𝑖 d. 2 + 2𝑖𝑖 e. −2 + 2𝑖𝑖

1. If the points 𝑃(𝑎 + 1, −10), 𝑄(6, 𝑏 + 1), 𝑅(3,16) and 𝑆(2,2) are the vertices of a parallelogram 𝑃𝑄𝑅𝑆, find the values of 𝑎 and 𝑏.

a. 𝑎 = 2 𝑎𝑛𝑑 𝑏 = 4 b. 𝑎 = 4 𝑎𝑛𝑑 𝑏 = 3 c. 𝑎 = 1 𝑎𝑛𝑑 𝑏 = 2 d. 𝑎 = 4 𝑎𝑛𝑑 𝑏 = 1 e. 𝑎 = 2 𝑎𝑛𝑑 𝑏 = 5

2. Find the real number 𝑥 and 𝑦, if (𝑥 − 𝑖𝑦)(3 + 5𝑖) is the conjugate of −6 − 24𝑖.

a. 𝑥 = 2 𝑎𝑛𝑑 𝑦 = −4 b. 𝑥 = 2 𝑎𝑛𝑑 𝑦 = −2 c. 𝑥 = 1 𝑎𝑛𝑑 𝑦 = 3 d. 𝑥 = 3 𝑎𝑛𝑑 𝑦 = −3 e. 𝑥 = 6 𝑎𝑛𝑑 𝑦 = 3

3. Find the range of 𝑓(𝑥) = √16 − 𝑥2.

a. [0,4] b. (−4,4) c. [−4,4] d. [2,4] e. (2,4)

4. Solve: 𝑓(𝑥) − 𝑔(𝑥) − 2ℎ(𝑥), where 𝑓(𝑥) = 𝑥3 + 3𝑥2 + 5𝑥 − 4 , 𝑔(𝑥) = 3𝑥3 − 8𝑥2 − 5𝑥 +6 and ℎ(𝑥) = −𝑥3 + 5𝑥 − 5.

a. 11𝑥2 + 5 b. 15𝑥2 c. 11𝑥2 d. 12𝑥2 e. 15𝑥2 + 3𝑥 + 5

5. A pole casts a shadow √3 times longer the pole’s height. Find the angle of elevation to the sun.

a. 75° b. 15° c. 60° d. 45° e. 30°

6. Find the 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 and 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 of the line 𝑦 + 6 = −3(𝑥 − 8).

a. x-intercept (6, 0); y-intercept (0, 18) b. x-intercept (3, 0); y-intercept (0, 9) c. x-intercept (5, 0); y-intercept (0, 10) d. x-intercept (6, 0); y-intercept (0, 12) e. x-intercept (4, 0); y-intercept (0, 12)

7. Simplify arcsin (sin 𝑥+cos 𝑥

√2) , −

𝜋

4< 𝑥 <

𝜋

4.

a. 𝑥 +𝜋

3

b. 𝑥 −𝜋

4

c. 𝑥 +𝜋

2

d. 𝑥 +𝜋

4

e. 𝑥 +𝜋

6

8. Find the quotient when 𝑓(𝑥) = −𝑥3 + 3𝑥2 − 3𝑥 + 5 divides 𝑔(𝑥) = −𝑥2 + 𝑥 − 1 with remainder 3.

a. 𝑥 − 1 b. 𝑥 − 2 c. 𝑥 + 2 d. 𝑥 + 1 e. 𝑥 + 3

9. In any ∆𝐴𝐵𝐶, find the value of (𝑎 − 𝑏)2 cos2 𝐶

2+ (𝑎 + 𝑏)2 sin2 𝐶

2.

a. 𝑎2 + 𝑏2 b. 𝑐2 + 𝑏2 c. 𝑎2 d. 𝑏2 e. 𝑐2

10. Find the point on the 𝑥 − axis which is equidistant from the points 𝑃(−2,5) and 𝑄(2, −3).

a. (2,0) b. (−1,0) c. (−2,0) d. (1,0) e. (4,0)

11. The price of a notebook has been marked up by 20% and is being sold for $105. How much did the shopkeeper pay the manufacturer of the notebook?

a. $87.50 b. $92.50 c. $113.50 d. $850 e. $97.50

12. The sum of the squares of two positive integers is 650. If the square of the larger number is 25 times the square of the smaller number, find the numbers.

a. 5 𝑎𝑛𝑑 10 b. 8 𝑎𝑛𝑑 11 c. 7 𝑎𝑛𝑑 12 d. 5 𝑎𝑛𝑑 25 e. 8 𝑎𝑛𝑑 13

13. Simplify sin 5𝑥−2 sin 3𝑥+sin 𝑥

cos 5𝑥−cos 𝑥 .

a. sin 𝑥 b. tan 𝑥 c. cot 𝑥 d. sec 𝑥 e. cos 𝑥

14. Solve the inequality 𝑥2 + 13𝑥 + 12 > 0.

a. 𝑥 > −12 or 𝑥 ≤ −1 b. 𝑥 ≤ −10 or 𝑥 ≥ −2 c. 𝑥 ≤ −12 or 𝑥 ≥ −1 d. 𝑥 < −2 or 𝑥 > 1 e. 𝑥 < −12 or 𝑥 > −1

15. Evaluate: 2

3csc2 58° −

2

3cot 58° tan 32° −

5

3tan 13° tan 45° tan 77°.

a. 3 b. 1 c. −1 d. −2 e. 2

16. Find the axis of symmetry of the graph of 𝑓(𝑥) = 𝑥2 + 6𝑥 − 20.

a. 𝑥 = 3 b. 𝑥 = 7 c. 𝑦 = 2 d. 𝑦 = 1 e. 𝑥 = 4

17. Find the domain of 𝑓(𝑥) =|𝑥−4|

𝑥−4.

a. ℝ − {−4} b. ℝ − {8} c. ℝ − {2} d. ℝ − {4} e. ℝ

18. If 52𝑥−1 = 25𝑥−1 + 100, find the value of 31+𝑥.

a. 24 b. 27 c. 30 d. 21 e. 81

19. Evaluate sin(−420°) cos(390°) + cos(−660°) sin(330°).

a. 4 b. −2 c. 2 d. 1 e. −1

20. Find the life span of this person who spent 3

13 of her life in childhood and school,

4

39 of her life

in medical school, 1

2 of her life as a doctor, and died 13 years into retirement.

a. 82 𝑦𝑒𝑎𝑟𝑠 b. 80 𝑦𝑒𝑎𝑟𝑠 c. 78 𝑦𝑒𝑎𝑟𝑠 d. 85 𝑦𝑒𝑎𝑟𝑠 e. 75 𝑦𝑒𝑎𝑟𝑠

21. Evaluate sin 12° sin 48° sin 54°.

a. 1

3

b. 1

8

c. 1

6

d. 2

3

e. 5

8

22. Find the angle between the hour hand and the minute hand of a clock at half past three.

a. 5𝜋

6𝑟𝑎𝑑𝑖𝑎𝑛𝑠

b. 3𝜋

12𝑟𝑎𝑑𝑖𝑎𝑛𝑠

c. 𝜋

12𝑟𝑎𝑑𝑖𝑎𝑛𝑠

d. 5𝜋

12𝑟𝑎𝑑𝑖𝑎𝑛𝑠

e. 7𝜋

6𝑟𝑎𝑑𝑖𝑎𝑛𝑠

23. Solve √𝑥 + 6 + √10 − 𝑥 = √16.

a. 𝑥 = −6, 10 b. 𝑥 = −3, 8 c. 𝑥 = 6, 10 d. 𝑥 = 6 , −10 e. 𝑥 = 3, −8

24. Find the range of 𝑓(𝛾) = 4 sin 𝛾 − 3 cos 𝛾 + 7.

a. [3, 14] b. [4, 10] c. [4, 12] d. [2, 10] e. [2, 12]

25. If 𝑥 = log10 12 , 𝑦 = log4 2 × log10 9 and 𝑧 = log10 0.4, evaluate 𝑥 − 𝑦 − 𝑧.

a. 3 b. 1 c. 5 d. 2 e. 4

26. Simplify 1+sin 2𝜃+cos 2𝜃

1+sin 2𝜃−cos 2𝜃 .

a. sin 𝜃 b. tan 𝜃 c. cos 𝜃 d. cot 𝜃 e. sec 𝜃

27. If 10 sin4 𝛼 + 15 cos4 𝛼 = 6, evaluate 27 csc6 𝛼 + 8 sec6 𝛼.

a. 300 b. 125 c. 250 d. 100 e. 275

28. Let 𝑓(𝑥) = 𝑥. If 𝑓 was further stretched vertically by a factor of 3, then reflected over the 𝑥 −axis and then translated vertically by 5 units, find the new equation of the graph.

a. −6𝑥 + 5 b. −3𝑥 − 5 c. 3𝑥 − 5 d. 5 + 3𝑥 e. −3𝑥 + 5

29. Solve 𝑥5 − 29𝑥3 + 100𝑥 = 0 for x.

a. 𝑥 = 0, ±1, ±15 b. 𝑥 = 0, ±2, ±5 c. 𝑥 = 0, ±4, ±3 d. 𝑥 = 0, ±2, ±3 e. 𝑥 = 0, ±3, ±5

30. Evaluate cos2 33°−cos2 57°

sin221°

2− sin269°

2

.

a. √3

b. √2 c. 1

d. −√2

e. 3√2

31. Find the equation of the graph of 𝑓(𝑥) = 𝑥3 − 2𝑥2 + 6𝑥 − 1 reflected across the y-axis.

a. 𝑥3 − 2𝑥2 − 6𝑥 − 1 b. −𝑥3 + 2𝑥2 − 6𝑥 + 1 c. −𝑥3 − 2𝑥2 − 6𝑥 + 1 d. −𝑥3 − 2𝑥2 − 6𝑥 − 1 e. 𝑥3 + 2𝑥2 + 6𝑥 + 1

32. Solve for 𝑥: 𝑒2𝑥 + 𝑒𝑥 − 12 = 0.

a. ln 3 b. ln 5 c. −ln 5 d. 2ln 3 e. ln 5

33. Find a and b if (𝑥 − 2) and (𝑥 − 3) are both factors of 𝑓(𝑥) = 𝑥3 + 𝑎𝑥2 + 2𝑏𝑥 − 7.

a. 𝑎 = −14

3 and 𝑏 =

74

3

b. 𝑎 = −73

4 and 𝑏 =

71

6

c. 𝑎 = −37

6 and 𝑏 =

71

12

d. 𝑎 = −71

3 and 𝑏 =

74

12

e. 𝑎 = −74

30 and 𝑏 =

71

6

34. Which of the following expressions has the GREATEST period?

a. 2 cos 𝑥 b. tan 7𝑥 c. sin 4𝑥

d. 1

3cos 6𝑥

e. cos1

3𝑥

35. Solve tan 𝛽 + tan 2𝛽 + tan 3𝛽 = tan 𝛽 tan 2𝛽 tan 3𝛽 for 𝛽.

a. 𝜋

3, 𝑛𝜖Ζ

b. 𝑛𝜋

3, 𝑛𝜖Ζ

c. 2𝑛𝜋

3, 𝑛𝜖Ζ

d. 𝑛𝜋

2, 𝑛𝜖Ζ

e. 5𝑛𝜋

3, 𝑛𝜖Ζ

MATHEMATICS Leveled Exam 3: Hard

1. A two digit number is four times the sum and three times the product of its digits. Find the number.

a. 20 b. 24 c. 18 d. 15 e. 22

2. If 𝐴𝐴(0,1) is equidistant from 𝐵𝐵(5,−3) and 𝐶𝐶(𝑥𝑥, 6). The sine of the angle to C from the origin is positive. Find x.

a. 5 b. -4 c. -5 d. 4 e. 2√41

3. In a ∆𝑃𝑃𝑃𝑃𝑃𝑃, find the value of �𝑞𝑞2−𝑟𝑟2

𝑝𝑝2� sin 2𝑃𝑃 + �𝑟𝑟

2−𝑝𝑝2

𝑞𝑞2� sin 2𝑃𝑃 + �𝑝𝑝

2−𝑞𝑞2

𝑟𝑟2� sin 2𝑃𝑃.

a. 0 b. 2 c. 7 d. 4 e. 1

4. Evaluate: �𝑥𝑥𝑝𝑝+𝑞𝑞�2�𝑥𝑥𝑞𝑞+𝑟𝑟�

2�𝑥𝑥𝑟𝑟+𝑝𝑝�2

(𝑥𝑥𝑝𝑝𝑥𝑥𝑞𝑞𝑥𝑥𝑟𝑟)4 .

a. 0 b. 2 c. 3 d. 4 e. 1

5. Find the value of tan 6° tan 42° tan 66° tan 78°.

a. 4 b. 1 c. 4 d. 6 e. 5

MATHEMATICS LEVELED EXAM 3 PAGE 16 DEMIDEC ©2016

6. The line joining 𝑃𝑃(3,3) and 𝑆𝑆(6,−9) is trisected at the points 𝐴𝐴 and 𝐵𝐵 where 𝐴𝐴 divides the line in the ratio 1:2. If point 𝐴𝐴 lies on the line 2𝑥𝑥 − 𝑦𝑦 + ℎ = 0, find the value of ℎ.

a. −2 b. −4 c. −9 d. −7 e. −3

7. Which of the following expressions is equivalent to tan𝛼𝛼1−cot𝛼𝛼

+ cot𝛼𝛼1−tan𝛼𝛼

?

a. 1 + tan𝛼𝛼 + cot𝛼𝛼 b. 1 + sin𝛼𝛼 + cos𝛼𝛼 c. 1 + cot𝛼𝛼 + sec𝛼𝛼 d. tan𝛼𝛼 + cos𝛼𝛼 e. 1 + cot𝛼𝛼 + csc𝛼𝛼

8. Solve for 𝛽𝛽: tan𝛽𝛽 + tan 2𝛽𝛽 + √3 tan𝛽𝛽 tan 2𝛽𝛽 = √3.

a. 2𝑛𝑛𝑛𝑛3

+ 2𝑛𝑛9

, 𝑛𝑛𝑛𝑛Ζ b. 𝑛𝑛𝑛𝑛

3,𝑛𝑛𝑛𝑛Ζ

c. 𝑛𝑛𝑛𝑛3

+ 𝑛𝑛9

, 𝑛𝑛𝑛𝑛Ζ

d. 2𝑛𝑛𝑛𝑛3

+ 𝑛𝑛4

, 𝑛𝑛𝑛𝑛Ζ e. 𝑛𝑛𝑛𝑛

3+ 𝑛𝑛

2, 𝑛𝑛𝑛𝑛Ζ

9. If 𝑓𝑓(𝑥𝑥) = (𝑎𝑎 − 𝑥𝑥𝑚𝑚)1/𝑚𝑚,𝑎𝑎 > 0 𝑎𝑎𝑛𝑛𝑎𝑎 𝑚𝑚 ∈ Ν, then find the value of 𝑓𝑓(𝑓𝑓(𝑥𝑥)).

a. 𝑥𝑥2 b. 2𝑥𝑥 c. 1 d. 𝑥𝑥𝑚𝑚 e. 𝑥𝑥

10. If sin𝑝𝑝 + sin 𝑞𝑞 = 𝑚𝑚 𝑎𝑎𝑛𝑛𝑎𝑎 cos 𝑝𝑝 + cos 𝑞𝑞 = 𝑛𝑛, find the value of cos(𝑝𝑝 − 𝑞𝑞).

a. 2𝑚𝑚2+𝑛𝑛2+22

b. 𝑚𝑚2−𝑛𝑛2−22

c. 𝑚𝑚2+𝑛𝑛2−22

d. 3𝑚𝑚2+2𝑛𝑛2−22

e. 𝑚𝑚2+2𝑛𝑛2+22

MATHEMATICS LEVELED EXAM 3 PAGE 17 DEMIDEC ©2016

11. What must be added to the polynomial 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥4 − 4𝑥𝑥3 − 𝑥𝑥2 + 3𝑥𝑥 − 1 so that the resulting polynomial is exactly divisible by 𝑔𝑔(𝑥𝑥) = 𝑥𝑥2 + 2𝑥𝑥 − 3?

a. −63𝑥𝑥 − 61 b. 63𝑥𝑥 − 62 c. −60𝑥𝑥 − 62 d. −63𝑥𝑥 + 62 e. −60𝑥𝑥 + 62

12. Find the value of 3(sin𝛼𝛼 − cos𝛼𝛼)4 + 6(sin𝛼𝛼 + cos𝛼𝛼)2 + 4(sin6 𝛼𝛼 + cos6 𝛼𝛼)4 + sin2 𝛼𝛼 +cos2 𝛼𝛼 − 13.

a. 3 b. 0 c. 4 d. 1 e. 5

13. If 𝑓𝑓(𝑥𝑥) = 𝑥𝑥−1𝑥𝑥+1

. 𝑥𝑥 ≠ −1, 𝑓𝑓�𝑓𝑓(𝑥𝑥)�=

a. −4𝑥𝑥

b. 1𝑥𝑥

c. −32𝑥𝑥

d. −1𝑥𝑥

e. −13𝑥𝑥

14. Simplify �4𝑎𝑎3

+ 8𝑏𝑏� + (−2𝑎𝑎 + 6𝑏𝑏) − �10𝑎𝑎6− 40𝑏𝑏� + �14𝑎𝑎

6− 24𝑏𝑏�.

a. 30𝑏𝑏 b. 40𝑏𝑏 c. −7𝑎𝑎 + 33𝑏𝑏 d. 33𝑏𝑏 e. 5𝑎𝑎 + 30𝑏𝑏

15. Which of the following expressions is equivalent to sin 2𝑦𝑦 + 2 sin 4𝑦𝑦 + sin 6𝑦𝑦?

a. sin2 4𝑦𝑦 cos2 𝑦𝑦 b. 4 sin2 4𝑦𝑦 cos2 𝑦𝑦 c. 4 sin 4𝑦𝑦 cos2 𝑦𝑦 d. 2 sin 4𝑦𝑦 cos2 𝑦𝑦 e. sin 4𝑦𝑦 cos2 𝑦𝑦

MATHEMATICS LEVELED EXAM 3 PAGE 18 DEMIDEC ©2016

16. Find the value of cos 𝑛𝑛8

.

a. 2√2 − 1

b. �√2−12√2

c. �√2−1√2

d. 2√2

e. �√2+12√2

17. Evaluate 3𝑥𝑥5 + 5𝑥𝑥4 − 75𝑥𝑥 − 125 = 0.

a. 𝑥𝑥 = ±5 ; 𝑥𝑥 = −53

b. 𝑥𝑥 = +√5 ;𝑥𝑥 = −√5𝑖𝑖 ; 𝑥𝑥 = 53

c. 𝑥𝑥 = ±√3 ;𝑥𝑥 = ±√3𝑖𝑖 ; 𝑥𝑥 = 43

d. 𝑥𝑥 = ±√5 ;𝑥𝑥 = ±√5𝑖𝑖 ; 𝑥𝑥 = −53

e. 𝑥𝑥 = ±√3 ;𝑥𝑥 = ±√3𝑖𝑖 ; 𝑥𝑥 = −43

18. Solve the inequality: 𝑥𝑥2 + 15𝑥𝑥 < −14.

a. −15 < 𝑥𝑥 < −1 b. −14 ≤ 𝑥𝑥 ≤ 1 c. −13 < 𝑥𝑥 < 1 d. −13 < 𝑥𝑥 < 1 e. −14 < 𝑥𝑥 < −1

19. Convert 𝑥𝑥 = 0.235353535 … to fractional form.

a. 23399

b. 2399

c. 233660

d. 33690

e. 233990

20. In an election for the mayor of a city, there were two candidates. A total of 10,000 votes were polled. 253 votes were declared invalid. The successful candidate got 5 votes for every 4 votes his opponent had. By what margin did the successful candidate win?

a. 1050 votes b. 1080 votes c. 1083 votes d. 1081 votes e. 1085 votes

MATHEMATICS LEVELED EXAM 3 PAGE 19 DEMIDEC ©2016

21. Find the period of the trigonometric function 𝑓𝑓(𝑥𝑥) = cos 4𝑥𝑥 + cot 3𝑥𝑥.

a. 2𝜋𝜋 b. 𝜋𝜋 c. 3𝜋𝜋 d. 7𝜋𝜋 e. 5𝜋𝜋

22. Solve for 𝑥𝑥: 𝑥𝑥𝑥𝑥−12

+ 5𝑥𝑥−2

= 𝑥𝑥2

𝑥𝑥2−14𝑥𝑥+24.

a. 25 b. 20 c. 30 d. 28 e. 31

23. Which of the following equations represents the graph given below?

a. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2 − 4 b. 𝑓𝑓(𝑥𝑥) = √𝑥𝑥 + 4 c. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2 + 4 d. 𝑓𝑓(𝑥𝑥) = √𝑥𝑥 + 4 e. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥3 + 4

24. A kite is flying at a height of 95 meters from ground level, attached to a string inclined at 60° to the horizontal. Find the length of the string to the nearest meter.

a. 110 meters b. 115 meters c. 102 meters d. 105 meters e. 112 meters

(0,4)

Y

X

MATHEMATICS LEVELED EXAM 3 PAGE 20 DEMIDEC ©2016

25. If three consecutive natural numbers are such that the sum of one-third of the first, one-fourth of the second, and one-fifth of the third is at most 22, find the LARGEST of the numbers.

a. 20 b. 24 c. 26 d. 29 e. 31

26. Find the range of the function 𝑓𝑓(𝑥𝑥) = �(𝑥𝑥 − 1).

a. (1,∞) b. (−1,∞) c. [0,∞) d. [0,1) e. [−1, 1]

27. If 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥3 + 𝑚𝑚𝑥𝑥2 + 𝑛𝑛𝑥𝑥 − 2 has a factor (𝑥𝑥 + 2) and leaves a remainder 7 when divided by 2𝑥𝑥 − 3, find the values of 𝑚𝑚 and 𝑛𝑛.

a. 𝑚𝑚 = 3 and 𝑛𝑛 = −1 b. 𝑚𝑚 = 1 and 𝑛𝑛 = −3 c. 𝑚𝑚 = −3 and 𝑛𝑛 = −1 d. 𝑚𝑚 = 3 and 𝑛𝑛 = 3 e. 𝑚𝑚 = 3 and 𝑛𝑛 = −3

28. Find the domain and range of the function 𝑓𝑓(𝑥𝑥) = 4 − |𝑥𝑥 − 10|.

a. Domain = ℝ and Range = (−∞, 2) b. Domain = ℝ and Range = (−∞, 4] c. Domain = ℝ and Range = (−∞, 1) d. Domain = ℝ and Range = (−∞, 10) e. Domain = ℝ and Range = (−∞, 8]

29. Solve for 𝑥𝑥: 421− 2(𝑥𝑥+8)

12= 6(𝑥𝑥−3)

15− 6

10.

a. 𝑥𝑥 = 7131

b. 𝑥𝑥 = 1211

c. 𝑥𝑥 = 11113

d. 𝑥𝑥 = 138119

e. 𝑥𝑥 = 17121

MATHEMATICS LEVELED EXAM 3 PAGE 21 DEMIDEC ©2016

30. Evaluate cot(−15𝑛𝑛4

).

a. 3 b. 4 c. 1 d. 2 e. 5

31. Find 𝑥𝑥 if 𝑥𝑥 cot(90° + 𝑎𝑎) + tan(90° + 𝑎𝑎) sin𝑎𝑎 + csc(90° + 𝑎𝑎) + tan 45° = 1.

a. cos 𝑎𝑎 b. sin𝑎𝑎 c. tan𝑎𝑎 d. cot 𝑎𝑎 e. csc 𝑎𝑎

32. Evaluate 1log2 126

+ 1log7 126

+ 1log9 126

.

a. 3 b. 7 c. 9 d. 1 e. 5

33. Express the angular measurement of the angle of a regular octagon in radians.

a. 4𝑛𝑛3

radians

b. 3𝑛𝑛5

radians

c. 2𝑛𝑛5

radians

d. 2𝑛𝑛3

radians

e. 3𝑛𝑛4

radians

34. Solve for : √3𝑥𝑥2 + 4𝑥𝑥 + 32 = 4 − 𝑥𝑥.

a. −8 𝑜𝑜𝑜𝑜 − 4 b. 4 𝑜𝑜𝑜𝑜 2 c. −4 𝑜𝑜𝑜𝑜 − 2 d. −8 𝑜𝑜𝑜𝑜 2 e. −4 𝑜𝑜𝑜𝑜 8

35. If 𝑖𝑖𝑧𝑧3 + 𝑧𝑧2 − 𝑧𝑧 − 𝑖𝑖2 + 𝑖𝑖 = 1, then find the value of |𝑧𝑧|.

a. 1 b. 3 c. 2 d. 0 e. 4