Notes, summaries, assignments, exams, and problems for Mathematics

Answer Key

• 1: A
• 2: C
• 3: 50°
• 4: B
• 5: D
• 6: C
• 7: C
• 8: C
• 9: A
• 10: A

11. Congruency Proof

Given: OA = OD and OB = OC.

In ΔOBC, since OB = OC, then ∠OBC = ∠OCB. Consequently, ∠OCD = ∠OBA.

In ΔOAD, since OA = OD, then ∠OAD = ∠ODA.

Since ∠OCD = ∠OBA and ∠OAD = ∠ODA, we find that ∠AOB in ΔOAB is equal to ∠COD in ΔOCD.

Conclusion: By SAS congruency, ΔOAB ≅ ΔOCD. Therefore, AB = CD.

12. Conical Cloth Calculation

Given: r = 7m, h = 24m.

l = √(r² + h²) = √(49 + 576) = √625 = 25m.

Curved Surface Area (CSA) = πrl = (22/7) × 7 × 25 = 550m².

Length of cloth = 550m² / 5m = 110m.

13. Sphere Diameter

Surface area of sphere = 154 cm².

4πr² = 154

r² = 154 / (4π) = (154 / 4) × (7 / 22) = 49 / 4.

r = √(49 / 4) = 7/2 = 3.5 cm.

Diameter (D)... Continue reading "Mathematics Practice Solutions: Geometry and Mensuration" »

Matrices and Determinants

• Inverse Matrix: |A|⁻¹ = 1/|A|; A⁻¹ = (Adj A)ᵀ / |A|; A⁻¹ · A = I.
• Multiplication: Row by column (fila por columna).
• Determinants: |A₃,₃| = 4, det(2A) = 2³ · 4.
• Resolution (Cramer's Rule): x = |B c₂ c₃| / |A|, y = |c₁ B c₃| / |A|...

Geometry and Spatial Relationships

Areas and Volumes

• Area of a Parallelogram: |a × b|
• Area of a Triangle: |a × b| / 2
• Volume of a Parallelepiped: [u, v, w]
• Volume of a Tetrahedron: [u, v, w] / 6

Relative Positions

• Position of 3 Planes:
  • rg(A) = rg(A*) = 3: System Consistent Determined (1 point).
  • rg(A) = rg(A*) = 2: Line (recta).
  • rg(A) = rg(A*) = 1: Coincident.
  • rg(A) = 1, rg(A*) = 2: Parallel.
  • rg(A) = 2, rg(A*) = 3: Two cases: if two rows are proportional, two planes are parallel

Consider an European call option with strike equal to 10, T = 1, r = 0.05 and σ = 0.2. Using the following time-series:

1. For every time moment t = 0, 1/360, 2/360:

• Calculate the Black-Scholes price.
• Calculate the corresponding delta.
• Calculate the price of the corresponding replicating portfolio and its composition, if we consider a dynamic hedging.

2. Calculate a static hedging. What is the final value of the portfolio? Compare with the dynamic hedging results.

Solution

• At time 0, time to maturity is 1, and then
  • The Black-Scholes price is given by S0N(d1)−Ke−rT N(d2) = 10N(0.35)−10e−0.05N(0.15) = 1.0450
  • The Delta is given by N(d1) = N(0.35) = 0.6368
  • At time t = 0, the value of the replicating portfolio coincides with the Black-Scholes price

Mathematical Aptitude - Average

Question 1

The average age of 30 students is 9 years. If the age of their teacher is included, it becomes 10 years. The age of the teacher (in years) is

_______

(A) 31

(B) 35

(C) 40

(D) 43

Question 2

The average weight of 10 men is decreased by 3 kg when one of them whose weight is 80 kg is replaced by a new person. The weight of the new person is

_______

(A) 50

(B) 60

(C) 70

(D) 73

Question 3

The average of 17 numbers is 45. The average of the first 9 of these numbers is 51 and the last 9 of these numbers is 36. What is the ninth number?

(A) 14

(B) 16

(C) 18

(D) 20

Question 4

The average weight of 9 mangoes increases by 20 gm if one of them weighing 120 gm is replaced by another. The weight of the new mango is

________

(A) 180 gm

(B)... Continue reading "Mathematical Aptitude Test: Average Practice Questions" »

Hypothesis Tests for Population Means

Test 1: Sample of 10 — Mean < 550

(a) In a random sample of 10 people who take a test, the average score is 532.4, and the standard deviation is 28.3. Do the data provide significant evidence, at the 5% level, that the population mean score is less than 550?

1. Step 1: Hypotheses.

   H₀: μ = 550 vs. H_a: μ < 550 (1 point) Where μ is the population mean score.

2. Step 2: Significance level.

   α = 0.05 (1 point)

3. Step 3: Test statistic.

   Test Statistic: t_obs = (X̄ − μ₀)/(s/√n) = (X̄ − 550)/(s/√n) (1 point)

4. Step 4: Rejection region.

   The test rejects H₀ at level α in favor of H_a if t_obs < −t_α,df=9, where t is based on 9 degrees of freedom. (1 point)

5. Step 5: Calculation and conclusion.

   Test Statistic:

Management of Interest Rate Risk

Major Risks in Bond Market

A) Interest Rate Risk (change in market prices of bonds due to varying interest rates)

• Increase in rates, decrease in market prices, increase in reinvestment rate risk (coupons reinvested at lower return)

B) Reinvestment Rate Risk (uncertainty of rate at which interim cash flows can be reinvested)

• High coupon rate, high reinvestment rate risk
• Greater for longer holding periods (high interim cash flows)

C) Default Risk (credit risk) (issuer unable/unwilling to pay interest and principal of bond)

• High credit rating, lower yield
• Short-term T-bills (almost risk-free, no default risk, and low return and reinvestment risk because of short duration)

D) Call Risk (risk bond issuer will redeem bond before... Continue reading "Interest Rate Risk Management: A Comprehensive Guide for Investors" »

Product Information and Constraints

Fran Farnsworth sells three health care products: Flex, Feelgood, and Firmup. Her costs, sales time, delivery costs, and commission for each product are as follows:

• Flex: $3 cost, 10 minutes to sell, $0.50 delivery cost, $2 commission
• Feelgood: $5 cost, 20 minutes to sell, no delivery cost, $4 commission
• Firmup: $4 cost, 12 minutes to sell, $1 delivery cost, $3 commission

Fran's weekly constraints are:

• Maximum $1500 worth of products
• Maximum $85 delivery expense
• Maximum 40 hours (2400 minutes) for sales

Linear Programming Model

Let x, y, and z represent the number of boxes of Flex, Feelgood, and Firmup sold per week, respectively. Let C be Fran's weekly commission. The LP model is:

Maximize:

C = 2x + 4y + 3z

Subject

What is a Shapefile?

A shapefile is an Esri vector data storage format for storing the location, shape, and attributes of geographic features. It is stored as a set of related files and contains one feature class. Google Earth, an Online Satellite Imagery (OSI) Technique, is a tool for generating shapefiles.

Geometric Analysis Tools in Projects

Example 1: Population Maps of To Kwa Wan using ArcGIS

We used ArcGIS to create population maps of To Kwa Wan.

1. Selected population and working population data from the C&SD department.
2. Opened TPUs with the population data in ArcGIS.
3. Calculated the area of the TPUs based on Hong Kong 1980 Grid, determining population densities.
4. Created maps with different population densities, using graduated colors to represent

Formal letter

We are writing in reference to your new products advised in March 23 edition of “Info Globe”. We are a software company based in Berlin which specializes in Games Development.

Our company is currently developing a new software to improve old games quality. We are interested in designing a new video game console which would works with old games with a better graphics quality.

 We would like to request a catalogue of your lasted collection that was advised in Info Globs. Please let us know if you would be interested in scheduling a meeting.

 I look forward to hearing from you.

Yours sincerely, M Walton