Unit 5: Trigonometry, Seating and Coded Inequalities - Practice Quiz

1 A pole casts a shadow m long when the angle of elevation of the Sun is . What is the height of the pole?

2 From a point on the ground, the angle formed between the horizontal line and the line of sight to the top of a tower is called the:

3 A person stands m from a tower. If the angle of elevation of its top is , what is the tower's height, ignoring the person's height?

4 Which trigonometric ratio directly relates the height of an object to its horizontal distance from an observer?

5 A car moves toward a tower at a constant speed. What generally happens to the angle of elevation of the tower's top?

6 A boat travels directly toward a lighthouse at m/s for seconds. By how much does its distance from the lighthouse decrease?

7 An observer in a stationary tower watches a train move away along a straight horizontal track. What happens to the angle of depression of the train?

8 Five people sit in a row facing north. A sits immediately to the left of B. Who sits immediately to the right of A?

9 P, Q, and R sit in a row facing north. Q sits between P and R, and P sits to the left of Q. What is their order from left to right?

10 Seven students sit in a straight row. How many students sit between the students in the first and fifth positions?

11 Four people A, B, C, and D sit in a row facing north. A is at the left end, B is immediately right of A, and D is at the right end. Who is third from the left?

12 Six people sit around a circular table facing the center. If B sits immediately clockwise from A, who sits immediately counterclockwise from B?

13 Eight people are seated equally around a circular table. How many people sit between a person and the person directly opposite?

14 When people sit around a circular table facing the center, a person's right-hand direction is generally:

15 Which statement is true if ?

16 If and , which relation must be true?

17 What happens to the inequality sign when both sides of are multiplied by ?

18 Let be the positive root of and be the positive root of . Which relation is correct?

19 Let satisfy and satisfy . Which relation is correct?

20 The larger root of is compared with the positive root of . Which relation is correct?

21 From a point on level ground, the angle of elevation of the top of a tower is . After moving m toward the tower, the angle becomes . What is the height of the tower?

22 Two points on the same side of a tower are m apart. The angles of elevation of the top from the farther and nearer points are and , respectively. What is the height of the tower?

23 A vertical pole casts a shadow m long when the angle of elevation of the Sun is . What is the height of the pole?

24 From the roof of a m building, the angle of depression of the base of a tower is , while the angle of elevation of its top is . What is the height of the tower?

25 A car moves directly toward a tower of height m. The angle of elevation of the top changes from to in seconds. What is the speed of the car?

26 From the top of a m lighthouse, the angle of depression of a ship changes from to as it sails directly away for minutes. What is the ship's speed?

27 An aircraft flies horizontally at an altitude of m toward an observer. Its angle of elevation changes from to in seconds. What is its horizontal speed?

28 Eight people A, B, C, D, E, F, G, and H sit in a row facing north. C is at the left end, H is second to the right of C, A is third to the left of F, B is immediately to the right of A, D is immediately to the left of F, and E is at the right end. Who sits second to the left of D?

29 Seven people P, Q, R, S, T, U, and V sit in a row facing north. T is at the left end. P is third to the left of Q, R is immediately to the right of P, S is second to the right of Q, and U is immediately to the left of Q. What is V's position relative to U?

30 Six people K, L, M, N, O, and P sit in a row facing north. L is at the right end, K is second to the left of L, M is immediately to the left of K, N is second to the right of M, and O is at the left end. Who sits immediately to the right of P?

31 Nine students sit in a row facing north. R is fourth from the left, S is third to the right of R, and T is second from the right. How many students sit between S and T?

32 Eight people A, B, C, D, E, F, G, and H sit around a circular table facing the center. A sits opposite E, B sits second clockwise from A, C sits immediately anticlockwise from E, and D sits immediately clockwise from A. Who sits immediately anticlockwise from E?

33 Six people P, Q, R, S, T, and U sit around a circular table facing the center. P sits immediately clockwise from Q, R sits opposite Q, S sits second clockwise from R, and T sits immediately clockwise from R. Who sits opposite S?

34 Eight people M, N, O, P, Q, R, S, and T sit around a circular table facing the center. M is third clockwise from N, O is opposite M, P is immediately anticlockwise from O, Q is second anticlockwise from P, R is immediately clockwise from N, and S is immediately anticlockwise from M. Who sits opposite R?

35 Given , which relationship is definitely true?

36 If , which conclusion must be true?

37 Which interval contains all real values of satisfying both and ?

38 In the equations below, and denote any roots of their respective equations. Compare and .

39 In the equations below, and denote any roots of their respective equations. Compare and .

40 In the equations below, and denote any roots of their respective equations. Compare and .

41 Two observation points and lie on opposite sides of the foot of a vertical tower and on the same straight line. The angles of elevation of the top from and are and , respectively. If m, what is the height of the tower?

42 An observer is m horizontally from the foot of a vertical mast. The angle of depression of the foot is , while the angle of elevation of the top is . What is the height of the mast?

43 The angle of elevation of the top of a pole from a point on level ground is . After moving m directly toward the pole, the angle becomes . What is the height of the pole?

44 Points and lie due east and due north, respectively, of the foot of a vertical tower. The angles of elevation of the top from and are and . If m, what is the tower's height?

45 A car approaches the foot of a tower at constant speed along a straight level road. The angle of elevation of the top changes from to in seconds. How many more seconds will the car take to reach the foot of the tower?

46 A bird flies horizontally at a constant altitude of m toward a point directly above an observer. Its angle of elevation changes from to in seconds. How long after the second observation will it be directly above the observer?

47 A ship sails directly away from the foot of a m lighthouse at constant speed. The angle of depression from the top changes from to in minutes. What is the ship's speed?

48 Nine people through sit in a row facing north. sits fourth to the left of . is immediately right of , and is immediately left of . sits exactly midway between and . is immediately right of , and is immediately right of . and occupy the two ends. Who sits third to the right of ?

49 Six people through sit in a row facing north. is somewhere left of , is somewhere right of , is immediately left of , and is adjacent to . Exactly three people sit between and . Which arrangement from left to right is possible?

50 Eight people through sit in a row facing north. is third to the right of . is immediately left of , is immediately left of , is immediately right of , and is immediately right of . and occupy the two ends. Who sits second to the right of ?

51 Seven distinct books are arranged in a row. is fourth to the right of , is immediately right of , and is immediately left of . Book is at one end, and is not adjacent to . How many valid arrangements are possible?

52 Eight people through sit around a circular table facing the center. sits opposite . is immediately clockwise from , and is second clockwise from . sits opposite . sits between and and is adjacent to both. is immediately clockwise from . Where is relative to ?

53 Seven distinct people through are seated around a circular table, with rotations treated as identical. If and must sit together while and must not sit together, how many arrangements are possible?

54 Eight people sit around a circular table facing the center. sits opposite . is adjacent to , is second clockwise from , and sits opposite . Which statement must be true?

55 Six distinct people through sit around a circular table facing the center. and must not be adjacent, while must sit adjacent to both and . How many circular arrangements are possible if rotations are considered identical?

56 Define the coded relations as follows: means , means , and means . Given , , and , consider: I. ; II. ; III. . Which conclusion necessarily follows?

57 Let and be distinct positive real numbers satisfying . Consider: I. ; II. . Which statement is correct?

58 Find the complete solution set of the inequality

59 The variable is any root of , and is any root of . Which relation between and is always valid?

60 The variable is any real root of , and is any real root of . Which relation must hold?