# Mathematics, Surveying and Transportation Engineering

Algebra, Trigonometry, Statistics, Geometry, Calculus, Differential Equations, Engineering Mechanics, Engineering Economy, Surveying, Transportation Engineering

## Two Gamblers Play Until One is Bankrupt: Chance That the Better Player Wins

**Problem**

Player *M* has Php1, and Player *N* has Php2. Each play gives one the players Php1 from the other. Player *M* is enough better than player *N* that he wins 2/3 of the plays. They play until one is bankrupt. What is the chance that Player *M* wins?

A. 3/4 | C. 4/7 |

B. 5/7 | D. 2/3 |

## Angle Between Two Zero-Based Vectors in XY-Plane

**Problem**

What is the angle between zero-based vectors ${\bf V_1} = (-\sqrt{3}, ~ 1)$ and ${\bf V_2} = (2\sqrt{3}, ~ 2)$ in an *x*-*y* coordinate system?

A. 0° | C. 150° |

B. 180° | D. 120° |

## Probability of Winning the Carnival Game of Tossing a Coin Into a Table

**Problem**

In a common carnival game, a player tosses a penny from a distance of about 5 feet onto the surface of a table ruled in 1-inch squares. If the penny (3/4 inch in diameter) falls entirely inside a square, the player receives 5 cents but does not get his penny back; otherwise he loses his penny. If the penny lands on the table, what is his chance to win?

A. 5/16 | C. 9/256 |

B. 1/16 | D. 3/128 |

## Random Steps of a Drunk Man: Probability of Escaping the Cliff

**Problem**

From where he stands, one step toward the cliff would send the drunken man over the edge. He takes random steps, either toward or away from the cliff. At any step his probability of taking a step away is 2/3, of a step toward the cliff 1/3. What is his chance of escaping the cliff?

A. 2/27 | C. 4/27 |

B. 107/243 | D. 1/2 |

## Length of Parabolic Sag Curve with Given Change in Grade Per Station

**Problem**

A grade of -5% is followed by a grade of 1%, the grades intersecting at the vertex (Sta. 10 + 060). The change of grade is restricted to 0.4% in 20 m. Compute the length of the vertical parabolic sag curve in meters.

A. 360 m | C. 300 m |

B. 320 m | D. 340 m |

## Compound Curves: Finding the Stationing of PCC with Given Stationing of PC

**Problems**

A compound curve has the following characteristics:

I_{1} = 24° |
D_{1} = 6° |

I_{2} = 36° |
D_{2} = 4° |

Stationing of P.C. = km 10 + 420 |

Compute the stationing of *P.C.C.*

A. km 10 + 560 | C. km 10 + 520 |

B. km 10 + 540 | D. km 10 + 500 |

## For Sn = 3^(2n - 1) + b; Find the Quotient a9 / a7

**Problem**

The sum of the first *n* terms of a series is 3^(2*n* - 1) + *b*. What is the quotient of the 9^{th} and the 7^{th} term?

A. 81 | C. 83 |

B. 82 | D. 84 |

## Y-component of Vector A if Vectors A and B are Orthogonal

**Problem**

Compute the value of *b* if **A** and **B** are orthogonal.

$${\bf A} = 2{\bf i} + b{\bf j} + {\bf k}$$

$${\bf B} = 4{\bf i} - 2{\bf j} - 2{\bf k}$$

A. 6 | C. 4 |

B. 5 | D. 3 |